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ISSN:2164-6457 (print)

ISSN:2164-6473 (online)

Volume 7 (2018)

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Journal of Applied Nonlinear Dynamics

JAND Volume 1, Issue 1

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Lie Algebraic Approach to Nonlinear Integrable Couplings of Evolution Type

pp. 1-28 | DOI:10.5890/JAND.2011.12.001

Yufeng Zhang, Wen-Xiu Ma

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Abstract

Based on two higher-dimensional extensions of Lie algebras, three kinds of specific Lie algebras are introduced. Upon constructing proper loop algebras, six isospectral matrix spectral problems are presented and they yield nonlinear integrable couplings of the Ablowitz- Kaup-Newell-Segur hierarchy, the Broer-Kaup hierarchy and the Kaup-Newell hierarchy. Especially, the reduced cases of the resulting integrable couplings give nonlinear integrable couplings of the nonlinear Schr¨odinger equation and the classical Boussinesq equation. Two linear functionals are introduced on two loop algebras of dimension 6 and Hamiltonian structures of the obtained nonlinear integrable couplings are worked out by employing the associated variational identity. The proposed approach can also be used to generate nonlinear integrable couplings for other integrable hierarchies.

Adaptive Emission Control of Freeway Traffic Using Quasi-Stationary Solutions of an Approximate Hydrodynamic Model

pp. 29-50 | DOI:10.5890/JAND.2011.12.002

Jozsef K. Tar, Laszlo Nadai, Imre J. Rudas, Terez A. Varkonyi

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Abstract

Precise emission models are so complex that their details practically are not available. The “physics” of hydrodynamic traffic models provides only ambiguousmathematical formulae. Therefore adaptive controllers iteratively improving the forecasts of a rough initial model are required that work without tuning any particular model parameters. For this purpose a simple numerical technique is reported. The stable stationary solutions of a particular model are determined. Then a simple iterative adaptive emission control grasping only the main factors is developed. It is shown that small modification of the parameters of the dynamic model have drastic influence on the emission so the use of adaptive techniques is essential.

Two Kinds of Multiple Wave Solutions for the Potential YTSF Equation and a Potential YTSF-Type Equation

pp. 51-58 | DOI:10.5890/JAND.2012.01.001

Abdul-MajidWazwaz

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Abstract

In this work, we study the (3+1)-dimensional YTSF equation and a YTSF-type equation. We derive two kinds of multiple wave solutions for each equation. The simplified form of the direct method will be used to conduct the analysis.

On Controlling Milling Instability and Chatter at High Speed

pp. 59-72 | DOI:10.5890/JAND.2012.02.001

Meng-Kun Liu, C. Steve Suh

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Abstract

A highly interrupted machining process, milling at high speed can be dynamically unstable and chattering with aberrational tool vibrations. While its associated response is still bounded in the time domain, however, milling could become unstably broadband and chaotic in the frequency domain, inadvertently causing poor tolerance, substandard surface finish and tool damage. Instantaneous frequency along with marginal spectrum is employed to investigate the route-to-chaos process of a nonlinear, time-delayed milling model. It is shown that marginal spectra are the tool of choice over Fourier spectra in identifying milling stability boundary. A novel discrete-wavelet-based adaptive controller is explored to stabilize the nonlinear response of the milling tool in the time and frequency domains simultaneously. As a powerful feature, an adaptive controller along with an adaptive filter effective for on-line system identification is implemented in the wavelet domain. By exerting proper mitigation schemes to both the time and frequency responses, the controller is demonstrated to effectively deny milling chatter and restore milling stability as a limit cycle of extremely low tool vibrations.

Analytical Routes of Period-1 Motions to Chaos in a Periodically Forced Duffing Oscillator with a Twin-well Potential

pp. 73-108 | DOI:10.5890/JAND.2012.02.002

Albert C. J. Luo and Jianzhe Huang

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Abstract

In this paper, analytical routes of period-1 motions to chaos in the Duffing oscillator with a twin-potential well are investigated through the generalized harmonic balance method. The analytical solutions of period-m motions are presented by the Fourier series, and the corresponding Hopf bifurcation of periodic motions leads to new periodic motions with period-doubling. Three analytical routes of asymmetric period-1 motions to chaos are presented comprehensively. To verify approximate, analytical periodic solutions, numerical simulations are carried out. In the analytical routes, the unstable periodic motions are presented, and such analytical routes with unstable periodic motions can help one find unstable chaos. Such unstable chaos cannot be obtained simply via the time going to infinity.

Entropy Analysis of Fractional Derivatives and Their Approximation

pp. 109-112 | DOI:10.5890/JAND.2012.03.001

J.A. Tenreiro Machado

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Abstract

Fractional derivatives (FDs) need to be calculated using series or rational fractions. The effect of such approximations are usually analysed by means of the frequency or the time responses. This paper takes advantage of a probabilistic interpretation of FDs and adopts the Shannon entropy for assessing the truncation effect produced by the approximations.

JAND Volume 1, Issue 2

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On the Usefulness of Riemann-Liouville and Caputo Derivatives in Describing Fractional Shift-invariant Linear Systems

pp. 113-124 | DOI:10.5890/JAND.2012.05.001

Manuel D. Ortigueira, Fernando J. Coito

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Abstract

The description of shift-invariant systems in terms of Riemann- Liouville and Caputo derivatives is studied according to their “initial conditions”. The situation of a past excitation of a linear system is considered and shown that the referred initial conditions may be either null or unavailable. This may lead to question the use of such derivatives.

Soliton Solutions of the Long-Short Wave Equation with Power Law Nonlinearity

pp. 125-140 | DOI:10.5890/JAND.2012.05.002

Manel Labidi, Houria Triki, E.V. Krishnan, Anjan Biswas

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Abstract

This paper studies the generalized long-short wave equation with power law nonlinearity. There are several approaches that are used to solve this coupled system nonlinear evolution equations. The series solution approach yields the topological 1-soliton solution or shock wave solution. The ansatz method and the semiinverse variational principle leads to the non-topological 1-soliton of the equation. Additionally, the variational iteration method is used to study the equation. Finally, numerical simulations are also given to this equation.

Numeric-Analytic Solutions of Dynamical Systems Using a New Iterative Method

pp. 141-158 | DOI:10.5890/JAND.2012.05.003

Sachin Bhalekar, Varsha Daftardar-Gejji

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Abstract

In this article, we couple the New Iterative Method proposed by Daftardar-Gejji and Jafari [J. Math. Anal. Appl. 2006;316:753– 763] with classical discretization technique to integrate some linear and nonlinear systems. The numerical examples are presented to explain the method. Examples include some nonlinear dynamical systems such as Financial system, Duffing oscillator, Van der Pol oscillator and a non-autonomous system.

Asymptotic Analysis of the Hopf-Hopf Bifurcation in a Time-delay System

pp. 159-171 | DOI:10.5890/JAND.2012.05.004

Christoffer R. Heckman, Richard H. Rand

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Abstract

A delay differential equation (DDE) which exhibits a double Hopf or Hopf-Hopf bifurcation [1] is studied using both Lindstedt’s method and center manifold reduction. Results are checked by comparison with a numerical continuation program (DDEBIFTOOL). This work has application to the dynamics of two interacting microbubbles.

Is it Possible to Replace the Probability Distribution Function Describing a Random Process by the Prony’s Spectrum? (I)

pp. 173-194 | DOI:10.5890/JAND.2012.05.005

Raoul R. Nigmatullin

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Abstract

The new law that governs by the generation of frequencies for the strongly-correlated systems (having a memory) has been found. The generalization of the present idea is based on detailed analysis of the previous results obtained in paper [1] that were devoted to new solutions of the Prony’s problem. It was turned out that many complex systems with memory generate new set of frequencies based on frequencies that have been generated in the nearest past. For justification of this relationship we collected different data that confirm this statement. We created also a special mathematical program, which selects (based on some criteria) a desired hypothesis that is chosen from other six similar ones. For all available data considered there is an optimal hypothesis that describes the distribution of frequencies that follows from the recurrence relationship including in itself the neighboring frequencies. The found hypothesis provides the optimal fit of the random smoothed sequence with high accuracy (the relative error less that 10%) including also the fit of the remnant function.The physical interpretation of this law is given also. This “unexpected” discovery found for a wide class of the strongly-correlated systems with memory allows to replace the probability distribution function associated with some process by its Prony’s spectrum. From mathematical point of view it will help to obtain new solutions of the old Prony’s problem and replace also the Fourier spectrum containing usually the excess of artifact frequencies by the informative-significant band of frequencies obtained from new general law that, in turn, was found for the strongly-correlated systems.

Controllability, Observability, Duality for Fractional Differential-Algebraic Systems with Delay

pp. 195-205 | DOI:10.5890/JAND.2012.05.006

Zbigniew Zaczkiewicz

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Abstract

The paper deals with problems of relative controllability and Rn1 - observability for linear stationary fractional differential-algebraic system with delay (FDAD). FDAD system consists of fractional differential in the Caputo sense and difference equations. We present control systems and observation systems. We introduce the determining equation systems and their properties. By solution representations into series of their determining equation solutions we obtain effective parametric rank criteria for relative controllability and Rn1 -observability. A dual controllability result is also formulated.

JAND Volume 1, Issue 3

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Fluctuation Metrology Based on the Prony's Spectroscopy (II)

pp. 207-226 | DOI:10.5890/JAND.2012.06.001

Raoul R. Nigmatullin

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Abstract

The basic purpose of this paper is related to creation of the basis of new metrology based on quantitative analysis of the Prony’s spectra. The Prony’s spectroscopy gives a possibility to transform a wide class of multi-periodic and random signals (associated with a clearly expressed trend) to their amplitudefrequency response (AFR). For the strongly-correlated complex systems with memory the corresponding AFR is generated by a few initial frequencies that allow realizing the further compression of the initial random signal and reading it in terms of the limited number of quantitative parameters. This important peculiarity of the Prony’s spectroscopy allows in creating of a “universal” language for reading of different fluctuations and comparing them with each other. One interesting example of reading of the infinite sequences that are formed from the transcendental numbers (the Euler’s constant E = 2.71828... and number PI = 3.14159...) is considered. The new elements that are added to calculation scheme/algorithm increase the stability and accuracy of the Prony’s spectra considered. In general, it opens new possibilities in creation of the basis of the fluctuation metrology that enables to read a wide class of random signals and compare them with other signals in terms of the quantitative parameters entering into the corresponding AFRs that, in turn, are governed by the corresponding distribution of frequencies.

Linear Sampling Reconstructions Using a Singular Perturbation Technique

pp. 227-237 | DOI:10.5890/JAND.2012.07.001

Keehwan Kim, Koung Hee Leem, George Pelekanos

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Abstract

In this paper, we propose a new and efficient regularization technique (Singular Perturbation Technique-SPT) for the reconstruction of the shape of a scatterer from far field measurements. The approach used is based on the linear sampling method and uses singular perturbations of the identity along with Neumann series approximations to recover the shape of the unknown object. In comparison with the well established Tikhonov-Morozov regularization techniques our new algorithm appears to be more computationally efficient as it doesn’t require computation of the regularization parameter for each point in the grid. Moreover reconstructions obtained using (SPT) compare well with those obtained using other existing regularization methods.

Path Tracking Design by Fractional Prefilter Using a Combined QFT/H∞ Design for TDOF Uncertain Feedback Systems

pp. 239-261 | DOI:10.5890/JAND.2012.07.003

N. Yousfi, P. Melchior, C. Rekik, N. Derbe , A. Oustaloup

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Abstract

Robotics Abstract The objective of this work concerns the motion control and robust path tracking. A method based on fractional prefilter has been developed and extended to multivariable systems. It is based on the MIMO-QFT robust synthesis methodology. This method specifically deals with the appropriate transformation of the MIMO system to the equivalent SISO system. This paper presents a way to incorporate Quantitative Feedback Theory (QFT) principles to H∞ control design procedure to solve the Two-Degree-Of-Freedom (TDOF) with Highly Uncertain Plants. The SISO-QFT synthesis method can be used for each SISO loop. Then the controllers are synthesized using the H design. After that, synthesis of prefilters is given with optimization of its parameters using integral error minimization. Finally a numerical example

On the Stability of a Rotating Blade with Geometric Nonlinearity

263-286 | DOI:10.5890/JAND.2012.06.002

Fengxia Wang, Albert C.J. Luo

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Abstract

The stability of a rotating, nonlinear blade under a time-varying torque is investigated. The nonlinear blade is presented based on the geometric nonlinearity.Compared to the nonlinear model, the linear model of a rotating blade with effective load is investigated. The two models of rotating blades are reduced to the ordinary differential equations through the Galerkin method, and gyroscopic systems with parametric excitations are obtained. From such gyroscope systems, the stability of the two models is studied by the generalized harmonic balance method, and the approximate, analytical solutions for the two models are obtained. The stability regions in parameter maps are presented for a better understanding of the stability of such parametric, time-varying systems. The analytical and numerical solutions are illustrated. This study provides an efficient and accurate way to determine the stability of gyroscope systems with time-varying excitations.

On the Fuzzy Sliding Mode Control of Nonlinear Motions in a Laminated Beam

pp. 287-307 | DOI:10.5890/JAND.2012.07.004

L. Dai, L. Sun

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Abstract

A modified fuzzy sliding mode control (MFSMC) strategy is devel-oped in this research for actively controlling and stabilizing the non-linear vibration of a laminated composite cantilever beam. The cantilever beam model, which is wildly seen in engineering applications, is established based on Hamilton’s principle through the application of Reddy’s third-order theory together with von Karman-type equations. Geometric nonlinearity of the beam is considered.The governing equations for the beam are derived corresponding to the higher order discretization of Galerkin method. By the model developed with n degrees of freedom, a vibration control strategy is developed on the basis of the modification of the fuzzy sliding mode control (FSMC). The vibration control strategy of the present research provides the availability for controlling the vibrations of such beam in n degrees of freedom. The approach is also proven to be effective in stabilizing the vibration of the nonlinear beam in a desired manner.

JAND Volume 1, Issue 4

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Existence of Solutions for Fractional Delay Integrodifferential Equations

pp. 309--319 | DOI:10.5890/JAND.2012.10.001

K. Balachandran , S. Kiruthika, M. Rivero, J.J. Trujillo

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Abstract

In this article, we study the existence of solutions for a class of fractional integrodifferential equations with time varying delay by using the resolvent operators and fixed point technique. The main results present in this paper improve some results on this issue that have been studied recently. An example is provided to illustrate our main theoretical results.

Describing an Axisymmetric Notch in a Pipe Using Singularities

pp. 321--339 | DOI:10.5890/JAND.2012.09.003

D.K.Stoyko, N.Popplewell,A.H.Shah

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Abstract

A hybrid Semi-Analytical Finite Element (SAFE) and standard finite element procedure is adopted to ultrasonically but locally detect and characterize an axisymmetric open notch in an infinitely long steel pipe. Interactions between incident, dispersive guided waves and the notch are shown numerically to change a radial displacement’s temporal history and introduce additional singularities in the previously unblemished pipe’s Frequency Response Function (FRF). Differences between the frequencies of the singularities of the unblemished and notched pipes appear to reflect the notch’s dimensions. They also indicate the location of a nearby notch.

Smart Passive Vibration Isolation: Requirements and Unsolved Problems

pp. 341--386 | DOI:10.5890/JAND.2012.09.002

H. Marzbani, Reza H. Jazar, A. Khazaei

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Abstract

Vibration isolators are needed to control the relative displacement, acceleration, or most importantly the transmitted force to the base or to the isolated device. As a general rule, a good vibration isolator should be as soft as possible to reduce the transmitted force, and it should be hard to limit the relative displacement. Soft suspension provides large relative displacement in resonance zone. To limit the relative displacement while having a soft suspension, a limiting displacement design is needed. There are two practical designs, Hydraulic Engine Mount (HEM) and Piecewise Linear Suspension (PLS) which are being used in industry since 80s. Hydraulic engine mounts were invented to passively produce a low damping at low amplitude and a high damping at high amplitude. Similarly, piecewise linear suspension were introduced to provide a soft suspension at low amplitude and a hard suspension at high amplitude. Having dual behavior puts both HEM and PLS in the domain of nonlinear system which in turn brings many new phenomena never appeared in linear analysis. This article will review the necessity of having a dual behavior suspension and the design of the two practical designs. However, both of these designs have some challenging and unsolved problems. Introducing the problems opens avenue for supervisors, researchers, and students to direct their research practice and hopefully solve them.

Stability Analysis of Flow Pattern in Flow around Body by POD

387--399 | DOI:10.5890/JAND.2012.09.001

J.Z, Zhang, K.L. Li, W. Kang

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Abstract

A method is presented for the numerical analysis of instabilities and bifurcations of the complex flows around body. First, the formations, developments and evolvements of the complicated flow pattern in flow around a cylinder are numerically simulated using finite element method combined with explicit characteristic based split scheme (CBS) method, and the flow patterns are classified tentatively by streamline topology, in comparison with the existing results. The results show streamline topology method is powerful to investigate the instability and nature of the streamline patterns, especially, the vortex shedding. Further, the model reduction of fluid dynamics based on proper orthogonal decomposition (POD) and the stability analysis are given. Following POD, a low-dimensional dynamic system is derived to study numerically the stability and bifurcation of pattern. As an example, the stability analysis of flow around a circular cylinder is carried out, and is verified by the existing results. As a conclusion, the results show that the numerical method presented is powerful for the stability and bifurcation analysis of the complicated flows around body, and some nonlinear phenomena can be captured by the method.

A Few Notes on Lax Integrability, Integrable Couplings and Computing Formula of the Constant γ

pp. 401--406 | DOI:10.5890/JAND.2012.06.003

F.K. Guo, B.L. Feng ,T.T. Guo

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Abstract

We point out in the paper that the current available definition on Lax integrability is necessary to be modified; the existed method for generating integrable couplings by the approach “original equations + symmetric equations” is wrong. Besides, a simple and efficient formula for calculating the constant γ appearing in the trace identity and the quadratic-form identity is proposed, which is universal for finding Hamiltonian structures of integrable dynamics.

Traveling Waves, Impulses and Diffusion Chaos in Excitable Media

pp. 407-412 | DOI:10.5890/JAND.2012.07.002

T.V. Karamysheva, N.A. Magnitskii

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Abstract

In the present work it is shown, that the FitzHugh-Nagumo type system of partial differential equations with fixed parameters can have an infinite number of different stable wave solutions, traveling along the space axis with arbitrary speeds, and also traveling impulses and an infinite number of different states of spatiotemporal (diffusion) chaos.Those solutions are generated by cascades of bifurcations of cycles and singular attractors according to the FSM theory (Feigenbaum-Sharkovskii-Magnitskii) in the threedimensional system of Ordinary Differential Equations (ODEs), to which the FitzHugh-Nagumo type system of equations with self-similar change of variables can be reduced.

JAND Volume 2, Issue 1

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Analytical Maximum Response of a Rigid Spindle-Angular Contact Ball Bearings Assembly Subject to Rotating Unbalance and Base Harmonic Excitations

pp. 1--32 | DOI: 10.5890/JAND.2012.09.004

F.M.A. El-Saeidy

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Abstract

Analytical vibration of a rigid machine spindle-angular contact ball bearings assembly subject to base harmonic excitations plus mass imbalance is investigated. Equations of motion for a complicated ball bearing assembly system are derived using Lagrange’s equations. For time-invariant bearing stiffness and damping matrices, analytical solutions are obtained for the maximum and minimum radii of disk orbit/bearing orbit (i.e., orbit maximum and minimum amplitudes). Analytical and numerical results are in excellent agreement. With increasing bearing number of balls and/or axial preload, response amplitude and shifts resonance peaks decrease in high frequency regions. For time-varying bearing stiffness and damping matrices, numerical results are obtained from Runge-Kutta method, and such numerical results are discussed with regard to time domain and fast Fourier transform (FFT).With increasing base excitation frequency, nonlinear spindle-bearings assembly becomes much stiffer and decreases the corresponding response amplitude. In addition, chaotic motions of axial vibration are observed.

On Dynamics and Control of an Adaptable Wheeled Climbing Robot

pp. 33--57 | DOI: 10.5890/JAND.2012.08.001

A. H. Bazargan, M. Mehrandezh , L. Dai

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Abstract

This paper introduces a re-configurable wheeled climbing robot, which is capable of doing a multitude of tasks that no other single robot could do in the past. It can climb staircases, move inside empty ducts and pipes, climb up the ropes and poles of varying cross sections, and even jump over obstacles with proper motion coordination. It can also move inside narrow passageways by reconfiguring itself. In this paper, a comprehensive dynamic model of the robot is derived for the first time. A real-time simulator to test different control strategies by a human operator using conventional human-machine interfaces has been developed. This simulator can be employed to quickly size the electromechanical actuators and synthesize different control strategies.The data obtained was further used to design a human-analogous autonomous controller. A novel human-analogous control strategy based on an Adaptive Network-based Fuzzy Inference System (ANFIS) was implemented to control the position of the robot climbing a straight pole against gravity, based on data obtained from the real-time Human-In-The-Loop (HITL) simulator. Relevant input/output data were stored, filtered, and used offline to tune the parameters of an ANFIS-based controller. The ANFIS controller whose parameters were optimized was then implemented on the real system autonomously. Based on the information obtained via the HITL simulator system, the controller can extrapolate needed data for untrained cases. This control strategy provides some advantages over conventional controllers; for example, this avoids actuator saturation, minimizes the energy expenditure, and circumvents the gain-scheduling process.

Vectorial Splitting Rational Lp Inequalities for Integral Operators

pp. 59--81 | DOI: 10.5890/JAND.2012.09.005

George A. Anastassiou

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Abstract

Here we present Lp, p>1, vectorial integral inequalitites for products of multivariate convex and increasing functions applied to vectors of ratios of functions. As applications we derive a wide range of vectorial fractional inequalities of Hardy type. They involve the left and right Erdelyi-Kober fractional integrals and left and right mixed Riemann-Liouville fractional multiple integrals. Also we give vectorial inequalities for Riemann-Liouville, Caputo, Canavati radial fractional derivatives. Some inequalities are of exponential type.

Hidden Oscillations in Drilling Systems: Torsional Vibrations

pp. 83--94 | DOI: 10.5890/JAND.2012.09.006

G.A. Leonov, M.A. Kiseleva, N.V. Kuznetsov, P. Neittaanm¨aki

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Abstract

Study of drilling systems plays important role in drilling industry. During the operation mode these systems experience different kinds of vibration, which may cause malfunctioning (i.e. dissipation of kinetic energy, noise, excessive wear, machine parts premature failure etc). In this article the most common type of vibrations is considered - torsional vibrations. The two mass model of a drilling system and modified version of it, supplemented by equations of induction motor, are studied. Both systems experience so-called hidden oscillations. It is extremely difficult to analyze such hidden oscillations since they cannot be found with the help of standard analytical procedures of trajectory modeling in the equilibrium state domain. Hidden oscillations correspond to torsional vibrations in real systems, thus they may cause breakdowns.

Multiple Kink Solutions for the (2+1)-dimensional Sharma--Tasso--Olver and the Sharma--Tasso--Olver--Burgers Equations

pp. 95-102 | DOI: 10.5890/JAND.2012.09.007

Abdul-Majid Wazwaz

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Abstract

In this work, we investigate the (2+1)-dimensional third-order and fourth-order Sharma–Tasso–Olver (STO) equations. Moreover, we also study the (2+1)-dimensional generalized Sharma– Tasso–Olver-Burgers (STO-B) equation. Multiple kink solutions are formally derived for each equation. The Hereman-Nuseir method, a simplified form of Hirota’s direct method is applied to carry out this analysis.

JAND Volume 2, Issue 2

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Stability Boundaries of Period-1 Rotation for a Pendulum Under Combined Vertical and Horizontal Excitation

pp. 103--126 | DOI: 10.5890/JAND.2013.04.001

B. Horton, S. Lenci, E. Pavlovskaia, F. Romeo, G. Rega, and M. Wiercigroch

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Abstract

The aim of this work is to study the dynamics of pendulum driven through its pivot moving in both horizontal and vertical directions. It expands the results obtained for the parametric pendulum by Lenci et al. to two other cases, i.e. the elliptically excited pendulum and the pendulum, with an inclined rectilinear base motion (the tilted pendulum). Here we derive approximate analytical expressions representing the position of the saddle-node bifurcation associated with period-1 rotations in the excitation amplitude/frequency plane in the presence of damping by using the perturbation method proposed by Lenci et al. This includes development of a procedure for deducing expressions for the period doubling, creating a pair of stable period-2 rotational attractors. The obtained approximations are plotted on the excitation parameters plane and compared with numerical results. Simple Pad´e approximations for the analytical expressions relating to the position of the saddle-node bifurcation are also obtained.

Simple Geometric Techniques to Delineate the Location, Extent, and Approximate Shapes of Attractors in Chaotic Systems

pp. 127--139 | DOI: 10.5890/JAND.2013.04.002

S. Roy Choudhury

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Abstract

In this paper we examine the chaotic regimes of a variety of recently discovered hyperchaotic systems using a completely novel geometrical interpretation of the Competitive Modes analysis as simple criteria to map out the spatial location and extent, as well as the rough general shape, of the system attractor for any parameter sets corresponding to chaos. The accuracy of this mapping adds further evidence to the growing body of recent work on the correctness and usefulness of these Competitive Modes conjectures. Indeed, this may be taken as an ’a posteriori’ validation of the Competitive Modes conjectures.

Self-Similar Property of Random Signals: Solution of Inverse Problem

pp. 141--150 | DOI: 10.5890/JAND.2013.04.003

Raoul R. Nigmatullin and J.A. Tenreiro Machado

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Abstract

Many random signals with clearly expressed trends can have selfsimilar properties. In order to see this self-similar property new presentation of signals is suggested. A novel algorithm for inverse solution of the scaling equation is developed. This original algorithm allows finding the scaling parameters, the corresponding power-law exponent and the unknown log-periodic function from the fitting procedure. The effectiveness of algorithm is tested in financial data revealing season fluctuations of annual, monthly and weekly prices. The general recommendations are given that allow the verification of this algorithm in general data series.

Some Remarks on a Multi Point Boundary Value Problem for a Fractional Order Differential Inclusion

pp. 151--160 | DOI: 10.5890/JAND.2013.04.004

Aurelian Cernea

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Abstract

We study a multi point boundary value problem for a fractional order differential inclusion involving a nonconvex set-valued map. We establish a Filippov type existence theorem and we prove the arcwise connectedness of the solution set of the problem considered.

Rolling of a Rigid Body Without Slipping and Spinning: Kinematics and Dynamics

pp. 161-173 | DOI: 10.5890/JAND.2013.04.005

A.V. Borisov, I.S. Mamaev, and D.V. Treschev

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Abstract

In this paper we investigate various kinematic properties of rolling of one rigid body on another both for the classical model of rolling without slipping (the velocities of bodies at the point of contact coincide) and for the model of rubber-rolling (with the additional condition that the spinning of the bodies relative to each other be excluded). Furthermore, in the case where both bodies are bounded by spherical surfaces and one of them is fixed, the equations of motion for a moving ball are represented in the form of the Chaplygin system. When the center of mass of the moving ball coincides with its geometric center, the equations of motion are represented in conformally Hamiltonian form, and in the case where the radii of the moving and fixed spheres coincides, they are written in Hamiltonian form.

Influence of Embedded Material on Natural Frequencies of Double Segment Rotating Disk

pp. 175-192 | DOI: 10.5890/JAND.2013.04.006

Ehsan Sarfaraz and Hamid R. Hamidzadeh

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Abstract

An analytical method is presented to determine the effect of adding different materials at one of the edges of an annular rotating disk on its in-plane natural frequencies and critical speeds. The proposed analysis is based on the linear in-plane free vibration of a compound disk with material discontinuity, by adopting the two-dimensional plane stress theory. The frequency equation was achieved by satisfying the compatibilities of the displacements and stresses at the interfaces of the different segments. The materials used in each segments of the disk are assumed to be homogenous,elastic, and isotropic. Furthermore, the annular disk is considered to be clamped at the inner side and free at the outer edge with a radius ratio of 0.3, and rotates with a constant angular speed. The variation of non-dimensional natural frequencies in fixed coordinates for different modes and different segment radiuses at the inner or outer side with respect to speed of rotation are computed. Presented results indicated that by adding additional segment, undesirable natural frequencies of the rotating disk can be modified to be within the acceptable range.

Alternate Models of Replicator Dynamics

pp. 193-206 | DOI: 10.5890/JAND.2013.04.007

Elizabeth N. Wesson and Richard H. Rand

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Abstract

Models of evolutionary dynamics are often approached via the replicator equation, which in its standard form is given by ˙ xi = xi ( fi (x)−φ ) , i = 1, . . . ,n, where xi is the frequency of strategy i, fi is its fitness, and φ = Σn i=1 xi fi is the average fitness. A game-theoretic aspect is introduced to the model via the payoff matrix A by taking fi(x) = (A · x)i. This model is based on the exponential model of population growth, ˙ xi = xi fi, with φ introduced in order both to hold the total population constant and to model competition between strategies. We analyze the dynamics of analogous models for the replicator equation of the form ˙ xi = g(xi)( fi −φ ), for selected growth functions g.

JAND Volume 2, Issue 3

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CRONE Control : Principles, Extensions and Applications

pp. 207--223 | DOI: 10.5890/JAND.2013.08.001

A. Oustaloup, P. Lanusse, J. Sabatier, and P. Melchior

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Abstract

This paper is a review paper on CRONE control, a frequency-domain robust control design methodology based on fractional differentiation. Principles of CRONE control are presented. The recent extension of this design methodology to multi-input, multi-output systems and to time varying systems are described. Major applications done during industrial collaborations are briefly described.

Fuzzy Fractional Neural Network Approximation by Fuzzy Quasi-interpolation Operators

pp. 235--259 | DOI: 10.5890/JAND.2013.08.002

George A. Anastassiou

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Abstract

Here we consider the univariate fuzzy fractional quantitative approximation of fuzzy real valued functions on a compact interval by quasi-interpolation sigmoidal and hyperbolic tangent fuzzy neural network operators. These approximations are derived by establishing fuzzy Jackson type inequalities involving the fuzzy moduli of continuity of the right and left Caputo fuzzy frac-tional derivatives of the engaged function. The approximations are fuzzy pointwise and fuzzy uniform. The related feed- forward fuzzy neural networks are with one hidden layer. Our fuzzy frac- tional approximation results into higher order converges better than the fuzzy ordinary ones.

Modal Method for Solving the Nonlinear Sloshing of Two Superposed Fluids in a Rectangular Tank

pp. 261--283 | DOI: 10.5890/JAND.2013.08.003

Bachir Meziani and Ouerdia Ourrad

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Abstract

The nonlinear sloshing of two superposed fluids in a rectangular tank is studied. The equations governing the nonlinear motion of the interface and the free surface have been resolved at first and second order. The shape of these surfaces depends on the perturbation parameter ε and density ratio ρ of the fluids. The analysis of nonlinear effects shows new aspects that are not pre-viously observed. Linear theory is still valid for very small sloshing amplitude. When the amplitude increases, nonlinear effects can not be neglected. Superposed fluids with neighbors densities are more sensitive to nonlinear effects. they occurs in this case, by sub harmonic instabilities in space and beats phenomena in time. The first non-linear effects are observed at the interface. when ε increases, the influence of terms related to the first order eigenfrequency ω11 decreases and the influence of terms related to the eigenfrequency 2ω11 increases faster than those relating to the second order eigenfrequency ω21. At the free surface, the influence of terms related to the second order eigenfrequency ω21 is more pronounced when they appear as the influence of those relating to the eigenfrequency 2ω11.

Numerical Study on Bray-Liebhafsky Oscillatory Reaction: Bifurcations

pp. 285--301 | DOI: 10.5890/JAND.2013.08.004

Branislav Stankovi ́c, Zˇeljko C ̆upi ́c, Nataˇsa Peji ́c and Ljiljana Kolar-Ani ́c

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Abstract

The time series obtained by numerical simulations of the model of the Bray-Liebhafsky oscillatory reaction is analyzed under the continuously fed well stirred tank reactor (CSTR) conditions, with the aim to find bifurcation points in which system of Bray-Liebhafsky oscillatory reaction transforms from stable to unsta- ble state and vice versa. Types of bifurcation points, supercrit- ical and subcritical Andronov-Hopf bifurcation and saddle-loop bifurcation are determined from characteristic scaling laws.

Bistability and Bursting Oscillations in Electromechanical Butterfly Valves

pp. 303-314 | DOI: 10.5890/JAND.2013.08.005

C.A. Kitio Kwuimy and C. Nataraj

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Abstract

This papers considers an electromechanical butterfly valve and discusses the conditions for appearance of bistability and burst- ing oscillations. The critical nonlinear stiffness coefficient and inlet velocities leading to such dynamics are obtained as a func- tion of the system parameters, and the effects of external per- turbation on the bursting response of the system are illustrated. It is observed that the driving parameters and the direct current voltage strongly affect the sharpness, the number of strikes, the amplitude of the strikes and the time interval between the strikes. Combinations of large-amplitude oscillations and small- amplitude oscillations are obtained for the electric circuit, while combination of fast-slow dynamics is obtained for the mechani- cal part. The results of the paper provide potential criteria for evaluating and optimizing system performance.

JAND Volume 2, Issue 4

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The Effect of Slow Flow Dynamics on the Oscillations of a Singular Damped System with an Essentially Nonlinear Attachment

pp. 315-328 | DOI: 10.5890/JAND.2013.11.001

J.O. Maaita, E. Meletlidou, A.F. Vakakis, and V. Rothos

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Abstract

We study a three degree of freedom autonomous system with damping, composed of two linear coupled oscillators with an essentially nonlinear lightweight attachment. In particular, we are interested in strongly nonlinear interactions between the linear oscillators and the essentially nonlinear attachment. First, we reduce our system to a non-autonomous second order nonlinear damped oscillator. Then, we introduce a slow-fast partition of the dynamics and average out the main frequency components in order to obtain a reduced system that is studied through the Slow Invariant Manifold (SIM) approach. Depending on the pa- rameters of the system we find different interesting nonlinear dynamical phenomena. With the help of the SIM approach we can study how the parameters of the original problem influence the asymptotic behavior of the orbits of the system. This is accomplished with the application of Tikhonov’s theorem. We classify the different cases of the dynamics according to the values of the parameters and the theoretically predicted asymptotic behavior of the orbits. Interesting phenomena are reported such as orbit capture, relaxation oscillations and complex structure of the basins of attraction.

Fractional Order Level Control of a System with Communicating Vessels

pp. 329-342 | DOI: 10.5890/JAND.2013.11.002

Cosmin Copot, Clara M. Ionescu, and Robin De Keyser

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This paper illustrates the advantages and disadvantages of using integer- and fractional-order control algorithms for an example of a system with S-shape dynamics. A laboratory setup of communicating vessels, where the level is regulated by means of a pump, is described. Both PI and PID control are designed, im- plemented and tested on the real setup. The results show that fractional order control may outperform classical PID control, under specific conditions.

Model Reduction of Nonlinear Continuous Shallow Arch and Dynamic Buckling Simulations on Approximate Inertial Manifolds with Time Delay

pp. 343-354 | DOI: 10.5890/JAND.2013.11.003

Jiazhong Zhang, Liying Chen, and Sheng Ren

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Abstract

In comparison with Traditional Galerkin’ s Method (TGM), an Approx- imate Inertial Manifolds with Time Delay (AIMTDs) is presented for the model reduction of shallow arch with large deformation, a kind of nonlinear continuous dynamic system governed by partial differ- ential equation with second order in time, and the numerical simula- tion is also presented for the dynamic buckling analysis of shallow arch under impact. By this method, the nonlinear governing equations, which is a kind of infinite dimensional dynamic system, are studied in the phase space, and the solutions of the governing equations are projected onto the complete space spanned by the eigenfunctions of the linear operator of the governing equations. With the introduction of AIMTDs, it decomposes the infinite-dimensional phase space into two complementary subspaces, namely, a finite-dimensional one and its infinite-dimensional complement, and the relation between these two subspaces is the one with a time delay, that is, the evolution of the high modes is not only relevant to the instantaneous low modes, but also to the past high modes. Hence, the system can be projected onto the subspace with finite-dimension, in combination with the relation between the two subspaces, implying the model is reduced. Then, the nonlinear Galerkin’s procedure is adapted to approach the solution on AIMTDs in which the final equilibrium positions are included. Finally, the method is applied to the numerical simulation of dynamic buck- ling of the shallow arch under impact, and some comparisons between Traditional Galerkin’s procedure and Approximate Inertial Manifolds with Time Delay are given. It can be concluded that the method pre- sented give a guarantee for the truncation of buckling modes as mode expansion is used, and are generally feasible for the model reduction of the nonlinear continuous dynamic systems with second order in time, and it is an efficient numerical method requiring less computing time and high accuracy. More, the AIMTDs can be easily approached by finite element method.

Control of a Hydro-electromechanical System Using Fractional-order Controllers: A Comparative Study

pp. 355-371 | DOI: 10.5890/JAND.2013.11.004

Roy Abi Zeid Daou, Xavier Moreau, and Clovis Francis

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Abstract

This article presents a comparative study of the regulators of a hydro-electromechanical system, composed of two pumps, a uniform tank and a level sensor. Both controllers are fractional orders where the first one is the generalized PID controller and the second one is the CRONE (French acronym: Commande Robuste d’Ordre Non Entier) controller. In a first time, the transfer function of the plant is presented after identification (using the graybox method) and simplification processes. Then, the realization of both regulators is shown where the first controller (generalized PID) is obtained after imposing the regulator model whereas the second one is deduced using the open-loop constraints. At the end, a comparison between the behavior of both controllers is made in the frequency domain around some functional points of the plant as its behavior is nonlinear.

Chatter Dynamics on Impulse Surfaces in Impulsive Differential Systems

pp. 373–396 | DOI: 10.5890/JAND.2013.11.005

Shasha Zheng, Xilin Fu

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Abstract

In this article, we discuss the pulse phenomena for a class of impulsive differential systems from the angle of discontinuity, consider the systems as a global discontinuous one consisting of two uniquely-continuous subsystems and investigate its chatter dynamics. We introduce the method of flow theory and focus on the dynamical behaviors on the impulse surface. By deliberating analytical criteria for the transversality of a flow to the boundary surface, some sufficient conditions that guarantee the absence of pulse phenomena are obtained. We generalize several known results and some examples are given to apply the theory.

Vibrational Resonance in a Duffing System with a Generalized Delayed Feedback

pp. 397–408 | DOI: 10.5890/JAND.2013.11.006

J.H. Yang, Miguel A.F. Sanjuán, C.J. Wang, and H. Zhu

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Abstract

We investigate the vibrational resonance in the Duffing system with different kinds of delayed feedback. Our approach is to consider the delayed feedback as a generalized delayed feedback in a fractional-order differential version. For three special cases, the generalized delayed feedback corresponds to displacement delayed feedback, velocity delayed feedback, and acceleration delayed feedback respectively. At first, based on the vibrational mechanism, the approximate solution of the system is obtained. Then, we give conditions for all resonance patterns. The the oretical predictions are verified by numerical simulations. Furthermore, the theoretical results are in good agreement with the numerical simulations. Since the delayed feedback is in a generalized form, our results can be regarded as universal for the vibrational resonance in nonlinear systems with different kinds of delayed feedback.

Fractional Differential Equations System for Commercial Fishing under Predator-Prey Interaction

pp. 409–417 | DOI: 10.5890/JAND.2013.11.007

G.H. Erjaee, M.H. Ostadzad, K. Okuguchi, and E. Rahimi

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Abstract

In this article we introduce and analyze a mathematical model of commercial fishing as a dynamical system presented by fractional differential equations (FDE). In the model two different species of fish are interacting as predator and prey. We discuss the long-run properties of this system, including stability of equilibrium points, non-existence of limit cycle and periodic solution for the movement of fish stocks for both species. In deriving our mathematical model, instead of customary use of classical positive integer order of differential equations, we use fractional order derivatives. As we will discuss, non-local properties of FDE and its advantages in modeling problems with non-smooth domains will yield more accurate results in comparison with ordinary differential equation counterpart.

JAND Volume 3, Issue 1

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Comparison Between Davidson-Cole and Frequency-Band Limited Fractional Differentiator I/O Type Transfer Function with Speed and Acceleration Inputs in Path Tracking Design

pp. 1–16 | DOI: 10.5890/JAND.2014.03.001

N. Yousfi, P. Melchior, C. Rekik, N. Derbel, and A. Oustaloup

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Abstract

A new approach to path tracking design based on fractional prefilter was developed in this paper. In path tracking design, the dynamic of actuators must be taken into account in order to reduce over- shoots appearing for small displacements. Taking into consideration the maximum velocity, acceleration, jerk, and the bandwidth of the closed-loop on which the input is applied, it permits the generation of an optimal movement reference-input giving a minimum path comple- tion time. An approach to path tracking based on fractional prefilter has been developed. This approach based on a Davidson-Cole (DC) and Frequency Band Limited Fractional Differentiator (FBLFD) pre- filters, with position input. This work describes an extension of this method. It consists of a path tracking using fractional differentiation and comparison between different types of prefilters by direct opti- mization of an Input/Output (I/O) transfer function with speed and acceleration inputs. Fractional differentiation has been used through a Davidson-Cole and frequency band-limited fractional differentiator (FBLFD) prefilters. A simulation on a motor model validates the developed methodology.

Dynamics of Bimodality in Vehicular Traffic Flows

pp. 17–26 | DOI: 10.5890/JAND.2014.03.002

Arjun Mullick and Arnab K. Ray

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Abstract

A model equation has been proposed to describe bimodal features in vehicular traffic flows. The dynamics of the bimodal distribution reveals the existence of a fixed point that is connected to itself by a homoclinic trajectory. The mathematical conditions associated with bimodality have been established. The critical factors necessary for both a breaking of symmetry and a transition from bimodal to uni- modal behaviour, in the manner of a bifurcation, have been analysed.

ODE Admitting Two-dimensional Algebras of Dynamic Symmetries

pp. 27–36 | DOI: 10.5890/JAND.2014.03.003

M.I. Timoshin

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Abstract

A generalization of S. Lie’s classification of second order ODEs on two-dimensional algebras of point symmetries is constructed. First integrals for found types second order ODEs are reduced. The pos- sibility of the determination of two-dimensional algebras of dynamic symmetries over number field is considered. Interconnection of dy- namic and contact symmetries is demonstrated. On a concrete ex- ample it is shown the procedure of the decomposition of a contact transformation into superposition of point transformation and Leg- endre transformation.

The Dynamics of the Slow Flow of a Singular Damped Nonlinear System and It Parametric Study

pp. 37–49 | DOI: 10.5890/JAND.2014.03.004

J.O. Maaita, E. Meletlidou, A.F. Vakakis, and V. Rothos

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Abstract

We study the dynamical behavior of the slow flow of a three degree of freedom dissipative system of linear coupled oscillators with an essentially nonlinear attachment and compare the behavior of the initial system to the Slow Invariant Manifold (SIM). The dynamics of the slow flow can be simple, making regular oscillations in the region of the stable branches of the SIM, having relaxation oscillations or chaotic behavior.The initial system oscillates in the region of the SIM, verifying that the SIM plays an essential role for the dynamics of the initial system.

Synchronization and Stability of Surface Acoustic Wave (SAW) Coupled Phase Oscillators and Sensing Applications

pp. 51–72 | DOI: 10.5890/JAND.2014.03.005

Shashank S. Jha and R.D.S. Yadava

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Abstract

We present an analysis of phase dynamics of two coupled limit cy- cle oscillators where coupling is provided by a simple surface acoustic wave (SAW) delay line and the coupled oscillators are either SAW de- layed or direct self-feedback type. Synchronization and stability anal- yses are carried out with primary motivation to explore whether cou- pling SAW device in synchronization mode could make better sensing platform compared to usual SAW feedback oscillator. Also, the para- metric dependencies of the phase dynamics are analyzed to determine whether SAW-coupled oscillators could become stable chaotic code generators for secure communication. Both limit cycle and chaotic dynamics are seen to occur in different regions of parametric space. It is found that by proper tuning of system parameters sensitivity can be enhanced by several orders of magnitude resulting in possibility for making advanced SAW sensor system.

Transmission Model for the Co-infection of HIV/AIDS and Tuberculosis

pp. 73–84 | DOI: 10.5890/JAND.2014.03.006

Carla MA Pinto and Ana Carvalho

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Abstract

A mathematical model for the dynamics of co-infection of HIV/AIDS and tuberculosis is developed. The model includes treatment for both HIV and tuberculosis and vertical transmission for HIV/AIDS. The disease-free equilibrium of the model is computed and its local stabil- ity is proved. The reproduction numbers of the full model and of its two submodels, the HIV only model and the TB only model, are also calculated. Numerical simulations show the disease-free equilibrium. Future work will focus on computing the stability of the endemic equilibria.

Low-Frequency Free Vibration of Rods with Finite Strain

pp. 85–93 | DOI: 10.5890/JAND.2014.03.007

A.M. Baghestani, S.J. Fariborz, and S.M. Mousavi

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Abstract

The governing differential equation for the free vibration of a rod undergoing finite strain is obtained by means of Hamilton’s principle. The equation contains quadratic as well as cubic nonlinear terms. For the low-frequency analysis of rods, the two harmonics solution is considered for the equation. The Galerkin method is employed to convert the partial differential equation to a system of two nonlinear ordinary differential equations. These equations are solved utilizing generalized differential quadrature(GDQ) and continuation methods to obtain the backbone curves and also mode shapes of vibration for rods with two different kinds of boundary conditions.

Soliton Solutions for the Modified KdV6, Modified (2+1)-dimensional Boussinesq, and (3+1)-dimensional KdV Equations

pp. 95–104 | DOI: 10.5890/JAND.2014.03.008

Abdul-Majid Wazwaz

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Abstract

We study soliton solutions for a modified KdV6 equation, modified Boussinesq equation, and KdV equation in (1+1), (2+1) and (3+1) dimensions respectively. Three distinct new dependent variable trans- formations are combined with the simplified form of Hirota’s direct method is used to achieve these soliton solutions. One soliton solution is formally established for each equation together with its associated dispersion relation and dispersion variable.

JAND Volume 3, Issue 2

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Hunting in Groups—Dynamics of Chase-and-Escape

pp. 105–113 | DOI: 10.5890/JAND.2014.06.001

Kishore Dutta

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Abstract

Recently the dynamics of group chase-and-escape has attracted many physicists. We highlight the biological behaviors and the underlying mathematical rules and physical laws involved in the dynamics of chase-and-escape during the process of hunting in groups. We discuss some recently proposed stochastic models for such a system which revealed various universal statistical features and show how slight variation of the behavioral rules or the interaction parameters can lead to very different statistical behaviors. A number of possible challenging questions are addressed that might led to develop a more life-like model for this novel enthralling dynamical problem.

Synchronization in Richards’ Chaotic Systems

pp. 115–130 | DOI: 10.5890/JAND.2014.06.002

J.L. Rocha, S. Aleixo, and A. Caneco

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Abstract

In this work we establish new one-dimensional discrete dynamical systems: a family of unimodal maps that is proportional to the right hand side of Richards’ growth equation. We investigate in detail the bifurcation structure of Richards’ functions, on the twodimensional parameter space (β,r), whereβis the shape parameter, related with the growth-retardation phenomena, andris the intrinsic growth rate. Sufficient conditions are provided for the occurrence of extinction, stability, period doubling, chaos and non admissibility of Richards’ dynamics. We consider networks having in each node a Richards’ function. We prove some results about the synchronization level when fixing the network topology and changing the local dynamics expressed by the Lyapunov exponents, which depends on the (β,r) parameters. Moreover, we fix the local dynamics and prove results about the synchronization when the network topology change for some kind of networks. Finally, using numerical simulations, we compute the Lyapunov exponents to measure the system complexity, we obtain synchronization intervals of these networks, and we discuss the evolution of the synchronization level in terms of the parameters Richard’s function.

Survey on Micro-Raman Spectroscopy Data Analysis

pp. 131–137 | DOI: 10.5890/JAND.2014.06.003

Elsa Ferreira Gomes and Cristina Castro Ribeiro

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Abstract

The interest in the use of Raman spectroscopy in cancer diagnosis is due to its capability to unveil the evolution of the biochemical composition of tissue and cells throughout disease progression. Differences between Raman spectra of normal and malignant tissues and cells, both in vivo and in vitro, are being used as a method for the early detection of cancer. In this survey paper we review recently published works related to the analysis of data resulting from Raman spectroscopy. We structure our analysis according with the three steps of the process: data preprocessing, data reduction and data classification.

Tau Method for Linear Quadratic Regulator Problems

pp. 139–146 | DOI: 10.5890/JAND.2014.06.004

A. Gavina1, J. Matos, and P.B. Vasconcelos

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Abstract

We are interested in investigating the applicability of the Operational Tau method to solve optimal control problems. The present work focuses on quadratic regulator problems defined on infinite time. For the numerical approximation of the solution, domain transformation technique is studied as well as a special treatment of orthogonal polynomial basis. Numerical resultsobtained via a MATLAB implementation of the proposed approach are shown, illustrating its effectiveness.

Complexity, Chaos, and the Duffing-Oscillator Model: An Analysis of Inventory Fluctuations in Markets

pp. 147–158 | DOI: 10.5890/JAND.2014.06.005

Varsha S. Kulkarni

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Abstract

Apparently random financial fluctuations often exhibit complexity, chaos. Predictability of limited length time series is hard to infer. Knowledge about the process driving the dynamics facilitates such analysis. This paper shows that quarterly inventory changes of wheat in the global market, during 1974-2012, follow a nonlinear deterministic process. Weakly chaotic behavior alternates with non-chaotic behavior. Cubic dependence of price changes on inventory changes leads to establishment of Duffing Oscillator model as suitable for examining the inventory changes. Endowing parameters with suitable meanings, one may infer temporary speculation changes reflect inventory volatility that drives the transitions between chaotic and non-chaotic behaviors.

Solution of New Generalized Diffusion-Wave Equation Defined in a Bounded Domain

pp. 159–171 | DOI: 10.5890/JAND.2014.06.006

Yufeng Xu, Om P. Agrawal, and Nimisha Pathak

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Abstract

This paper concerns with solutionof a Generalized Diffusion-Wave Equation (GDWE) defined in a bounded space domain. In the model proposed, the GDWE is defined using operator B^α_P introduced recently. In contrast to fractional derivatives which employ fractional power kernels in their definitions, operator B^α_P allows the kernels to be arbitrary. Therefore, it offers more generality to a diffusion-wave equation than a fractional derivativedoes. Intheschemeproposed, the method of separation of variables is used to separate the space and time domains. The space equation in conjunction with boundary conditions are used to identify the eigenfunctions. The time dependent equation is solved analytically for a specific kernel. For a general kernel, a closed form solution for time equation may not be available. For this reason, we present a numerical scheme to solve this equation. The analytical solution for an exponential kernel is used to verify the numerical scheme. Two examples are presented to show applications of these models. It is hoped that operator B^α_P will allow us to model diffusion-wave behaviors of a system for which fractional derivatives may not be suitable, and the analytical and numerical schemes will allow us to solve these equations.

Feedback Linearization Synchronization of Unified Chaotic Systems

pp. 173–186 | DOI: 10.5890/JAND.2014.06.007

Chang-Zhong Chen, TaoFan, Bang-Rong Wang, Hassan Saberi Nik, and Ping He

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Abstract

The goal of this paper is to present a method that can be used to synchronize unified chaotic systems. The method is mainly based on the technology of feedback Linearization of nonlinear control systems. Numerical simulations are then provided to show the effectiveness and feasibility of the proposed chaos control and synchronization schemes.

On the Production Of Energy From Sea Waves By a Rotating Pendulum: A Preliminary Experimental Study

pp. 187–201 | DOI: 10.5890/JAND.2014.06.008

S. Lenci

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Abstract

The paper is aimed at experimentally showing the possibility of energy production by water waves using a new mean of production of green energy which has been proposed recently. Rotations of a floating pendulum, constrained to move vertically, are studied experimentally in the wave flume of the Laboratory of Hydraulics of the Polytechnic University of Marche. After checking the robustness of the rotation for different values of the wave frequency and amplitude, a generator applied to the pendulum pivot is switched on, and it is shown that it is able to produce electric energy. This represents the experimental proof of the feasibility of the underlying idea. The extracted energy is still small, and further developments are required to optimize the system, likely involving control for rotation activation and sustain in irregular waves, but the goal of showing that it is possible to produce energy from sea waves by a pendulum is reached, and the way for new development is opened.

JAND Volume 3, Issue 3

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Fractional Modeling of Driver’s Dynamics. Part 1: Passive Feedback and Steering Wheel/Hand Link

pp. 203–214 | DOI: 10.5890/JAND.2014.09.001

Xavier Moreau, Firas Khemane, Rachid Malti, and Jean-Luc Mermoz

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Abstract

This paper is the first part of a study related to modeling drivers dynamics in the overall driving loop in the context of disturbance rejection and trajectory tracking. The final objective is to dispose of a library of drivers models usable in a simulation context for Global Chassis Control (GCC) of automotive vehicles. Small angle variations are applied on the steering wheel and hence the use of a linear model is fully justified. The driver’s dynamics is first of all modeled by using a rational transfer function and then a fractional one. More specifically, the first part of the study presents the experimental set-up, named instrumented chair, and describes the experimental protocol. Then different models parts of the experimental set-up are established. Next, passive feedback and steering wheel/hand link is modeled. It is shown that a fractional model, as compared to a rational one, allows to enhance by a factor of 2.7 the criterion in modeling dissipative phenomena. The second part of the study treats both driver steering feel and visual feedback modeling, using a set-membership approach.

Fractional Modeling of Driver’s Dynamics. Part2: Set Membership Approach for Steering Feel and Visual Feedback

pp. 215–226 | DOI: 10.5890/JAND.2014.09.002

Firas Khemane, Rachid Malti, and Xavier Moreau

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Abstract

This paper is the second part of a study related to modeling drivers dynamics in the overall driving loop in the context of disturbance rejection and trajectory tracking. After a brief description of the experimental set-up, a fractional model for steering feel and visual feedback cases are proposed, and their parametersestimated. As expected in human reaction, data recorded from different experiments present a considerable dispersion, due to varying human reactions from one experiment to another. Such time-variant systems can be modeled using set membership methods which allow identifying a set of feasible models for healthy persons.

Simultaneous Time-Frequency Control of Friction-Induced Instability

pp. 227–244 | DOI: 10.5890/JAND.2014.09.003

Meng-Kun Liu and C. Steve Suh

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Abstract

An elastic cantilever beam pressed against a rigid rotating disk is explored for studying the mitigation of self-excited friction-induced vibrations that are inherently unstable due to alternating friction conditions and decreasing dynamic friction characteristics. Because no linearization or approximation scheme is followed, the genuine characteristics of the system including stick-slip and inherent discontinuities are fully disclosed without any distortion. It is shown that the system dynamics is stable only within certain ranges of the relative velocity. With increasing relative velocity, the response loses its stability with diverging amplitude and broadening spectrum. A novel time-frequency controller is subsequently applied to negate the chaotic vibration at high relative velocity by adjusting the applied normal force. The controller design requires no closed-form solution or transfer function, hence allowing the underlying features of the discontinuous system to be fully established and properly controlled. The inception of chaotic response at high relative velocity is effectively denied to result in the restoration of the system back to a relatively stable state of limit-cycle.

Synchronization of Coupled Map Lattice Using Delayed Variable Feedback

pp. 245–253 | DOI: 10.5890/JAND.2014.09.004

Siddharth Arora and M.S. Santhanam

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Abstract

We apply the method of variable feedback to obtain complete synchronization in a coupled map lattice. The conditions under which such a synchronization is possible are obtained analytically. We show that synchronization is robust against noise and parameter mismatches. This method leads to synchronized state quite rapidly and we discuss its applications for near-real-time multi-channel communications.

Nonlinear Oscillations of an Articulated Pipe System Subjected to Oil Flow and External Excitations

pp. 255–269 | DOI: 10.5890/JAND.2014.09.005

L. Dai, Y. Zhang, X. Wang, and T. Huang

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Abstract

Articulated pipes conveying fluids are widely used in industries especially petroleum industries. The nonlinear oscillations of an articulated pipe jointed with an oil conveying system is studies and characterized in this research. The responses of the articulated pipe without external excitation are analyzed. The oscillations of the pipe corresponding to the fluid flow and joint stiffness are investigated in detail. The nonlinear oscillations of the pipe subjected to external excitation are also studied in this research. The periodic, quasi-periodic and chaotic oscillations of the pipe are found in the study. The nonlinear behaviors of the pipe’s responses are diagnosed and characterized with employment of the Periodicity Ratio (P-R) method. A regularirregular region diagram is developed corresponding to wide ranges of system parametric values, reflecting those used in engineering practices. The results of the research provide practically sound guidance for industrial application and research in this field.

Dynamics of a System of Two Coupled Oscillators Driven by a Third Oscillator

pp. 271–282 | DOI: 10.5890/JAND.2014.06.006

Lauren Lazarus and Richard H. Rand

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Abstract

Analytical and numerical methods are applied to a pair of coupled nonidentical phase-only oscillators, where each is driven by the same independent third oscillator. The presence of numerous bifurcation curves defines parameter regions with 2, 4, or 6 solutions corresponding to phase locking. In all cases, only one solution is stable. Elsewhere, phase locking to the driver does not occur, but the average frequencies of the drifting oscillators are in the ratio of m:n.These behaviors are shown analytically to exist in the case of no coupling, and are identified using numerical integration when coupling is included.

Spectral Density Prediction for Response of Nonlinear Gear Pairs under Random Excitation

pp. 283–294 | DOI: 10.5890/JAND.2014.09.007

Jianming Yang and Ping Yang

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Abstract

This paper investigates the dynamic response of a gear pair under random excitation from the view point of spectral density. Two methods are used to calculate the response spectral density function (SDF). The first method uses the statistical linearization (SL) technique to find an equivalent linear system to the original nonlinear one, then the response SDF is calculated with the equivalent linear system. While the second method regards the natural frequency of the system as a function of the response amplitude which is a random variable and its probability density function (PDF) is computed through stochastic averaging (SA). Then the response SDF is computed as a probabilistic averaging over the whole range of amplitude. Simulation result shows that both methods can predictthe resonant frequency very well and consistently in the case of weaknonlinearity. But with increasing of nonlinearity, the SDF predicted from the SL technique tends to be narrower than that from method II.

JAND Volume 3, Issue 4

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Transient Responses in Nonlinear Dynamics of Structures

pp.295–297 | DOI: 10.5890/JAND.2014.12.013

Jan Awrejcewicz

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Abstract

Innumerable mechanical structures including vibrating profiles, sandwich plates, platforms, thin uniperiodic shells, laminated windshells, isotropic and orthotropic platesor even multibody systems exhibit complex dynamic behavior. Analysis of the corresponding phenomena involves interesting mathematical and experimental methods that are sometimes used in parallel. Mostly, a dynamical transient response of any of the enumerated complex structures leads to the unpredictable states that need to be described and predicted. For this reason, the Special Issue extends selected papers presented during the international conference on “Dynamical Systems-Theory and Applications” hold in December 2–5, 2013 in L´od´z, Poland. Main areas of modern experimental and numerical analysis taken into consideration by authors of these papers could be mentioned: bifurcations and chaos in dynamical systems, stability of dynamical systems, original numerical methods of vibration analysis, non-smooth systems, engineering systems and differential equations, control in dynamical systems, asymptotic methods in nonlinear dynamics,vibrations of lumped and continuous systems,dynamics in life sciences and bioengineering. A brief description of contents of this Special Issue follows.

Artificial Neural Networks in Oil Production Problems

pp. 229–306 | DOI: 10.5890/JAND.2014.12.001

Yuliya Lind et al.

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Abstract

The general approach to engineering of systems in oil and gas industry from the aspect of their automation and use of information technologies including design and experiment result analysis on the base of mathematical models requires involving newest technologies of artificial intelligence to obtain the most effective results. In this paper application of artificial intelligence methods such as neural networks for optimization of drilling process and automation of log curves digitization has been proposed. A hybrid neural network on the base of radial basis network learning by k-means algorithm has demonstrated the highest efficiency for solution of these problems regarding classification and pattern recognition.

About the Structure of the Vortex Flow Around Cylinder With Viscous Fluid

pp. 307–315 | DOI: 10.5890/JAND.2014.12.002

Rustyam G . Akhmetov and Ruslan R . Kutluev

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Abstract

The problem of stationary viscous in compressible fluid flow around the cylinder has been analyze d by means of the asymptotic methods. The fluid flow equations are considered in the variables "stream function-a vortex". Asymptotic vortex in the boundary layer near the boundary of the cylinder for average and large Reynolds numbers has been investigated. The equation of the interior boundary layer for stream function has been made by means of using the method of matched asymptotic expansions. The properties solution of the given equation are investigated by means of numerical methods under the additional condition of slipping on the boundary of the cylinder.

Vibrating Profile in the Aerodynamic Tunnel – Identification of the Start of Flutter

pp. 317–323 | DOI: 10.5890/JAND.2014.12.003

Jan Kozanek, Vaclav Vlcek, and Igor Zolotarev

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Abstract

The experimental results for the vibrating NACA0015 profile elastically supported in a translation and rotation for a set of increasing Mach numbers of the airflow are presented. The profile was placed in the aerodynamic tunnel of the Institute of Thermomechanics ASCR. The support properties of the profile were modified by three additional masses to control the eigenfrequencies corresponding to transversal vibrations and verified by the identification of the complex eigenvalues (eigenfrequencies, damping) for zero flow velocity in laboratory. The transversal free vibrations as the time functions are graphically depicted for stable vibrations and in the cases of the starting flutter. The start of the flutter was determined from free vibrations of kinematically excited profile. The complex eigenvalues were identified as the functions of Mach number including corresponding limits of the system aeroelastic stability.

Analytical Design of Multivariable Control Systems by Dynamical Decomposition Method

pp. 325–332 | DOI: 10.5890/JAND.2014.12.004

Anatoly R. Gaiduk, Kseniya V. Besklubova, and Elena A. Plaksienko

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Abstract

Design of multivariable control systems for complex technical plants with several interrelated channels and controlled variables is difficult problem. Usually here it is necessary to provide either independent (autonomous, unrelated) or coherent control of output variables. This design problem can be solved by using the analytical method implying exact, dynamical decomposition of the multivariable plant to the set of an independent single input–single output channels. The dynamic decomposition method is based on decomposing property of the adjunct matrix and provides realized of control action as function of output variables, reference input and measured disturbances (control on output and inputs) of the system. The efficiency of this analytical method is shown by the numerical example of control system design for the turbojet engine of the gas-pumping station with three interrelated channels.

The Rigid Finite Element Method Applied to a Riser Handling Analysis

pp. 333–345 | DOI: 10.5890/JAND.2014.12.005

Marek Szczotka

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Abstract

Riser handling is a special operation performed by a dedicated, specialized lifting equipment operated from a production platform or vessel. A mathematical model of the system is developed, which allow us to simulate the loads acting on the lifting equipment as well as riser elements. The rigid finite element method (RFEM) is applied to discretize the system, consisting of long and flexible components like riser and various rope systems. Large deformations are common conditions in such systems and operations, which have to be accounted in the analysis. For the problem described in the paper, an FPSO (Floating Production, Storage and Offloading Unit) vessel consists of a dedicated riser pull-in winch. An another pipeline installation vessel (PLV) is used for the construction work related to riser handling. The load transfer between FPSO pull-in winch an PLV lifting winch is critical for the project specification and the design input. On the basis of calculated loads, the design of the riser pull-in winch can be concluded. A dedicated software developed enables for an user-friendly definition and execution of the riser handling analysis. Based on developed models and results obtained, the design conditions can be formulated. With the aid of developed mathematical model and the computer program, an example simulation of the riser installation is performed.

Vertical Oscillation of a Simplified Model a Mechanical System

pp. 347–358 | DOI: 10.5890/JAND.2014.09.006

Josef Soukup and Martin Svoboda

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Abstract

This paper deals with vertical mechanical oscillation of symmetric and asymmetric systems with different kinematic excitation. The solution was carried out using numerical and experimental methods. The subject of analysis were the influence of geometry, production asymmetry and unbalanced excitation on the vibration of solid bodies in the system of flexibly bonded solids taking into account the boundary conditions and the type of loading (application to the oscillation of vehicles). Numerical solutions were performed using the ADAMS program. Experiments were performed on a laboratory model. The aim was to verify the applicability and suitability of different methods and procedures for the investigation of vertical oscillation of the mechanical system of bodies.

Nonlinear Irregular Vibration in the Systems with an Elastomeric Friction Element

pp. 359–368 | DOI: 10.5890/JAND.2014.12.007

Jarosław Dyk, Adam Jungowski, and Jerzy Osiński

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Abstract

The main focus of interest in the report is the specific irregular character of the motion in the system with an elastomeric element being a part of the frictional damper. The solution to this problem is based on the assumption that the material of the elastomeric element is non-compressible and its properties are described by Mooney-Rivlin model of the first order. Additionally, the values of the coefficients are chosen on the basis of the experiments described in references. The form of the equation of motion, which has been written by use of the non-compressiblity condition, testifies that vibrations of the system have the strong non-linear character. The numerical calculations by use of Gear's method, have given the evidence that after more than ten thousands of periods the irregular motion is present. For the case of the calculation of dissipative energy by acting dry friction between the elastomer and an aluminum element, as well as by the viscous damping, the equation of motion has been supplemented with suitable elements. It has been affirmed that the irregular motion is present only in the case of the system that exhibits the low level of damping. The detailed experiment research work on the elastomeric friction damper, in the frequency range up to 10Hz and different levels of temperature, confirms the strongly non-linear character of the motion.

Simulations of Post-impact Skin/core Debond Growth in Sandwich Plates Under Impulsive Loading

pp. 369–379 | DOI: 10.5890/JAND.2014.12.008

Vyacheslav N. Burlayenko and Tomasz Sadowski

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Abstract

In this study, the transient dynamics of a sandwich plate with post impact zone is examined. The dynamic response of the sandwich plate is simulated by using the finite element code ABAQUS. The damage mechanics approach implemented into ABAQUS via cohesive elements is applied to model the debonding propagation under impulsive loading. To demonstrate the effectiveness of the FE model developed, a foam-cored sandwich plate with an initial penny-shaped impacted region is modelled. The influence of the skin-to-core debond on the global nonlinear dynamics of the sandwich plate is studied in detail.

Tolerance Models of Dynamic Problems for Microheterogeneous Cylindrical Shells

pp. 381–391 | DOI: 10.5890/JAND.2014.12.009

Barbara Tomczyk

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Abstract

Thin linearly elastic Kirchhoff-Love-type circular cylindrical shells with a periodically micro - inhomogeneous structure in circumferential direction (uniperiodic shells) are investigated. At the same time, these shells have constant structure in axial direction. The aim of this contribution is to formulate some new mathematical non-asymptotic averaged models for the analysis of selected dynamic problems for the shells under consideration. These, so-called, tolerance models are derived by means of a certain extended version of the known tolerance (non-asymptotic) modelling of micro-heterogeneous media presented in [Tomczyk, B. and Wo?zniak, C. (2012), Tolerance models in elastodynamics of certain reinforced thin-walled structures. In: Ko lakowski, Z. & Kowal-Michalska, K. (eds.), Statics, Dynamics and Stability of Structures, vol.2, Technical University of Lodz Press, Lodz, 123-153]. Contrary to the starting exact shell equations with highly oscillating, non-continuous and periodic coefficients, the tolerance model equations proposed here have constant coefficients depending also on a cell size. Hence, these models make it possible to investigate the effect of a length scale on the global shell dynamics. Moreover, a certain homogenized (asymptotic) model, being independent of a microstructure size, is also derived applying the extended tolerance averaging technique proposed in the above mentioned monograph.

Motion Equations Isotropic and Orthotropic Plate Impacted by Elastic Rod

pp. 393–401 | DOI: 10.5890/JAND.2014.12.010

J. Soukup, J. Skočilas, B. Skočilasová, and L. Ryhlíková

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Abstract

The article deals with solution of the motion equations of plate. The rectangular plate is loaded by elastic rod which influences rectangular plate with force. The direction of the force is perpendicular to the face upper surface of the rectangular plate. The solution is derived for isotropic and orthotropic material. The solution of the motion equations has no analytical solution. The new approximation of the solution based on the transformation of integral equations to system of algebraic equations is presented.

Finite Element Modeling of Headform Impactor Crash with Laminated Windshield

pp. 403–412 | DOI: 10.5890/JAND.2014.12.011

Piotr Kosiński and Jerzy Osiński

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Abstract

The main object of the present study is to investigate the mechanical behavior of laminated car windshield in the case of a pedestrian head impact. Windshield Finite Element model consist one layer o f P VB and two layer s of glass. The layer of P VB was modeled in a few cases: as elastic, elasto-plastic, "low-density foam" and hyper elastic material model. The layers of glass we remodeled as "brittle cracking" material model included in Abaqus simulation environment. Both of layers was modeled as solid. Examined also two type of PVB mesh elements: " 3 D Stress " and " Cohesive ". In addition, consideration was two types of connection between glass and PVB layers: first merge and second tie connection. Analysis was performed with different types of mesh. Radial mesh proved to be better than rectangular mesh to reflect the phenomenon of cracking glass. Furthermore, the peak value of the head form linear acceleration has been tested for different thickness and material models of PVB layer.

Aeroelastic Stability Analysis of High Aspect Ratio Aircraft Wings

pp.413–422 | DOI: 10.5890/JAND.2014.12.012

J.R. Banerjee, X. Liu, and H.I. Kassem

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Abstract

Free vibration and flutter analyses of two types of high aspect ratio aircraft wings are presented. The wing is idealised as an assembly of bending-torsion coupled beams using the dynamic stiffness method leading to a nonlinear eigenvalue problem. This problem is solved using the Wattrick-Williams algorithm yielding natural frequencies and mode shapes. The flutter analysis is carried out using the normal mode method in conjunction with generalised coordinates and two - dimensional unsteady aero dynamic theory of Theodorsen. This is essentially a complex eigenvalue problem in terms of both air-speed and frequency. The flutter determinant is solved by an iterative procedure covering a wide range of air-speeds and frequencies. The computed natural frequencies, mode shapes, flutter speeds and flutter frequencies are compared and contrasted for the two type of aircraft wings and some conclusions are drawn.

JAND Volume 4, Issue 1

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Disappearance of Resonance Tongues

pp.1–9 | DOI: 10.5890/JAND.2015.03.001

Rocio E. Ruelas and Richard H. Rand

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Abstract

We investigate a phenomenon observed in systems of the form dx/dt = a1 (t)x + a2(t)y, dy/dt = a3(t)x + a4(t)y, where ai(t) = Pi + εQicos2t, where Pi, Qi and ε are given constants, and where it is assumed that when ε=0 this system exhibits a pair of linearly independent solutions of period 2π. Since the driver cos2t has period π, we have the ingredients for a 2:1 subharmonic resonance which typically results in a tongue of instability involving unbounded solutions when ε >0. We present conditions on the coefficients Pi, Qi such that the expected instability does not occur, i.e., the tongue of instability has disappeared.

Numerical Solution of Fuzzy Delay Functional Differential Equations by Euler Method

pp. 11–19 | DOI: 10.5890/JAND.2015.03.002

P. Prakash and S. Senthilvelavan

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Abstract

In this paper, we study the numerical solution of fuzzy delay functional differential equations by using Euler method. Examples are presented to illustrate the computational aspects of the above method.

A Low-pass-equivalent,State-space Model for the Nonlinear Coupling Dynamics in Mechatronic Transducers

pp. 21–42 | DOI: 10.5890/JAND.2015.03.003

Nikolaos I.Xiros and Ioannis T.Georgiou

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Abstract

The nonlinear analysis of a typical electrical oscillator coupled nonlinearly to a mechanical one, as encountered in mechatronics applications for sensing, actuation and energy harvesting, is approached by using a state-space decomposition inspired by Volterra theory representation. The equation of motion of the mechanical subsystem includes an electromagnetic force directly proportional to the electric current squared. The nonlinear coupled dynamics is investigated systematically by partitioning the coupled system state vector in such a way as to fully exploit the mechanical low-pass and the electrical band-pass intrinsic features of free dynamics. In particular, by employing the Hilbert Transform, a low-pass equivalent system is derived and verified by using standard perturbation analysis. Then, a typical case is investigated thoroughly by means of numerical simulation of the original coupled low and band-pass, real-state-variable system and the low-pass-equivalent, complex-state-variable derived one. The nonlinear model equations considered here pave the way for a systematic investigation of nonlinear feedback control options designed to operate mechatronic transducers in energy harvesting, sensing or actuation modes.

A Neural Network for Solving Nonlinear Convex Programming with Linear Equality and Bounded Constraints

pp. 43–52 | DOI: 10.5890/JAND.2015.03.004

Sitian Qin,Yiming Liu,and Chang fengShao

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Abstract

In this paper, to solve the nonlinear convex programming problems with linear equality and bounded constraints, a new neural network model is constructed. It is proved that if the initial point lies in the linear equality region, the state of the proposed neural network is convergent to an exact optimal solution of the optimization problem. Compared with the existed neural networks, the proposed in this paper has a low model complexity and avoid estimating the penalty parameters in advance. In the end, several numerical simulations illustrate the effectiveness of the proposed neural network

Accuracy Assessment of Fractional Order Derivatives and Integrals Numerical Computations

pp. 53–65 | DOI: 10.5890/JAND.2015.03.005

Dariusz W.Brzezinski and P.Ostalczyk

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Abstract

This paper presents results of a numerical experiment, during which different numerical criteria of exact values setting in the accuracy assessment of fractional order derivatives / integrals numerical calculations are tested. Although traditional accuracy criteria in form of relative error expressed in % are applied, the values assumed as exact, necessary for comparison, are now: value of a function, classical 1st derivative, integral of the 1st order and Mittag-Leffler function’s values. For that purpose, fractional order differentiation and integration operators concatenation rules are applied. The methods allow to assess the accuracy of numerical calculations of fractional derivatives and integrals for each required function and not only for ones, for which mathematical formulas are available. The proposed measures are employed to determine proper operation and assess the accuracy of fractional order derivatives and integrals numerical algorithms.The algorithms utilize Riemann-Liouville and Grunwald-Letnikov frac-¨ tional order derivatives and integrals formulas.

Non-Orthogonal Amplitude-Frequency Analysis of the Smoothed Signals(NAFASS): Dynamics and the Fine Structure of the Sunspots

pp. 67–80 | DOI: 10.5890/JAND.2015.03.006

Nigmatullin R.R and Toboev V.A,

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Abstract

The basic aim of the given paper is presentation of the basic principles of the new method defined as Non-orthogonal Amplitude Frequency Analysis of the Smoothed Signals (NAFASS). The new method is based on linear principle for the strongly-correlated variables and presentation of nonlinear signals. We demonstrate the possibilities of the NAFASS approach on analysis of real data related to dynamics and the fine structure of the Solar activity

Analysis of a Fractional-Order Nonlinear System with Hysteresis Nonlinearity via Describing Function

pp. 81–89 | DOI: 10.5890/JAND.2015.03.007

Ramiro S. Barbosa, Isabel S. Jesus, and J.A. Tenreiro Machado

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Abstract

The describing function(DF) is one method often used for the analysis of nonlinear systems and the prediction of limit-cycles. In this study, we explore the DF using frequency response methods in order to analyze the effectiveness of this technique in a fractional-order nonlinear system. Since it is common to find different types of nonlinearities in real systems, the DF method may reveal of great practical interest. In this perspective, we investigate the limit-cycle prediction and frequency response analysis of a fractional-orderplant model with hysteresis nonlinearity. The results presented may give some guidelines for the design of linear and nonlinear controllers of arbitrary order

Nonlinear Self-adjointness for a Generalized Fisher Equation in Cylindrical Coordinates

pp. 91–100 | DOI: 10.5890/JAND.2015.03.008

M.L. Gandarias, M.S. Bruzon, and M. Rosa

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Abstract

In this work we study a generalization of the well known Fisher equation in cylindrical coordinates. We determine the subclasses of these equations which are nonlinear self-adjoint. By using a general theorem on conservation laws proved by Nail Ibragimov and the symmetry generators we find conservation laws for these partial differential equations without classical Lagrangians.

JAND Volume 4, Issue 2

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Study of the Effect of the Coupling in a Dispersion-managed Dual Core Optical Fiber Using the Collective Variables Approach

pp.101–116 | DOI: 10.5890/JAND.2015.06.001

Roger Bertin Djob, Aurelien Kenfack-Jiotsa, and Timoleon Crepin Kofane

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Abstract

This paper highlights the interaction between two gaussian pulses propagating inside a dual core optical fiber by the mean of collective variables (CVs) approach. The main result is that energies of the signals being propagated in such fiber with linear coupling always end up being compensated whatever their amplitudes at the entry. It also appears convergence or divergence of the temporal positions of the neighboring solitary waves.

Unknown Input Observer Design for Linear Fractional-Order Time-Delay Systems

pp. 117–130 | DOI: 10.5890/JAND.2015.06.002

Y. Boukal, M. Darouach, M. Zasadzinski and N.E. Radhy

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Abstract

This paper considers unknown input functional fractional order observer design for fractional-order linear time-invariant (LTI) systems with a constant time delay. After given the existence conditions of such observer, based on the fractional order Lyapunov stability approach, a sufficient condition for the asymptotic stability of the estimation error is given in a linear matrix inequality (LMI) formulation. The obtained results are illustrated by a numerical example.

A Model of Evolutionary Dynamics with Quasiperiodic Forcing

pp. 131–140 | DOI: 10.5890/JAND.2015.06.003

Elizabeth Wesson and Richard Rand

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Abstract

Evolutionary dynamics combines game theory and nonlinear dynamics to model competition in biological and social situations. The replicator equation is a standard paradigm in evolutionary dynamics. The growth rate of each strategy is its excess fitness: the deviation of its fitness from the average. The game-theoretic aspect of the model lies in the choice of fitness function, which is determined by a payoff matrix. Previous work by Ruelas and Rand investigated the Rock- Paper-Scissors replicator dynamics problem with periodic forcing of the payoff coefficients. This work extends the previous to consider the case of quasiperiodic forcing. This model may find applications in biological or social systems where competition is affected by cyclical processes on different scales, such as days/years or weeks/years. We study the quasiperiodically forced Rock-Paper-Scissors problem using numerical simulation, and Floquet theory and harmonic balance. We investigate the linear stability of the interior equilibrium point; we find that the region of stability in frequency space has fractal boundary.

Stagnation Point Solution Due to a Continuously Stretching Surface with Applied Magnetic Field Using HAM

pp. 141–152 | DOI: 10.5890/JAND.2015.06.004

Rajeswari Seshadri and Shankar Rao Munjam

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Abstract

In the present study, we consider the steady two-dimensional stagnation point flow due to a stretching surface in an ambient fluid. The fluid is viscous, incompressible and electrically conducting near the stagnation region on a stretching surface. A uniform magnetic field of strength B is applied in the positive y.direction normal to the stretching surface. The equations governing the flow are solved using Homotopy Analysis Method (HAM). The flow variables are computed in the form of a series with its coefficients containing the parameters such as the magnetic field and the ratio of stretching velocity so that the effect of these parameters on the flow variables can be computed without much computational effort. The optimal values of the convergence control parameters and the the averaged squared residual errors are computed for the flow variables. The software Mathematica is used to perform all the semi-analytical calculations. It was observed that the effect of ratio of stretching velocity V0 on flow velocity is very significant in the sense that zero skin friction occurs at V0 = 0.5. and the trend in the variation of velocity profiles as well as skin friction coefficients are just opposite for V0 < 0.5 and V0 > 0.5.

Longitudinal Dimensions of Polygon-shaped Planetary Waves

pp. 153–167 | DOI: 10.5890/JAND.2015.06.005

Ranis N. Ibragimov1†and Guang Lin2

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Abstract

Conservation Laws of a Gardner Equation with Time-dependent Coefficients

pp. 169–180 | DOI: 10.5890/JAND.2015.06.006

M.S. Bruzon, M.L. Gandarias, and R. de la Rosa

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Abstract

A study of a variable-coefficient Gardner equation is carried out. The subclasses of the equation which are nonlinear self-adjoint have been determined. Conservation laws have also been obtained using two different methods: the direct method of the multipliers and Ibragimov’s theorem based on nonlinear self-adjointness of the equation. It has shown that for this equation conservation laws obtained by using both methods are equivalent.

Influence of Diffusion on Bio-chemical Reaction of the Morphogenesis Process

pp. 181–195 | DOI: 10.5890/JAND.2015.06.007

M. Sambath and K. Balachandran

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Abstract

This paper presents a mathematical strategy for exploring the dynamical behavior of bio-chemical model in temporal morphogenesis which is a generalization of the model studied by Gierer-Meinhardt. We here study the spatially homogeneous and non-homogeneous periodic solutions through all parameters of the system are spatially homogeneous. In order to verify our theoretical results, some numerical simulations are also carried out.

Recognition and Threat Level Estimation of FOD Based on Image Content Features and Experiment Analysis

pp. 197–213 | DOI: 10.5890/JAND.2015603.008

Gang Xiao, Yu Li, Xiao Yun, and Jinhua Xie

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Abstract

Foreign Object Debris (FOD) is debris or article alien, which may have fallen onto a runway or taxiway, would potentially cause damage to aircraft. FOD surveillance system is used to FOD surveillance system as a significant technology. However, FOD is a kind of relatively random and diverse target that makes it difficult to analyze and recognize. In the reality, the primary goal of FOD recognition cannot recognize FOD accurately, because airport staffs do not care much about what exactly the FOD is but how great the FOD could threat to airplane. Therefore, based on FOD image features definition and threat level model, we proposed a simple and efficient method to estimate the FOD threat level. By comparison with BP neural network model, the error of new FOD threats coefficient model are no more than 15%. There are five sections discussed in the paper. First, the FOD surveillance system was introduced briefly. Second, the features of FOD including color, texture and shape and related extraction algorithm had been discussed. The HSV color model, texture model and normal shape feature like area, perimeter and eccentricity which are all used to describe FOD targets. Fourteen typical FOD targets such as stone fragment, metal chip, piece of rubber, pieces of paper, plastic and small plant are selected as experiment objects. Thirdly, a new FOD threat coefficient model was proposed, in terms of the values of features based on the huge amount of experiment results and human subjective judgment. Next, the threat level classifier was constructed by the BP neural network. The classifier was trained by some experiment FOD targets with given threat level. The others FOD targets were tested. Finally, the recognition and estimation results indicated that the proposed threat level model were effective and efficient, satisfied the robustness and real-time requirement.

JAND Volume 4, Issue 3

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Crises in Chaotic Pendulum with Fuzzy Uncertainty

pp.215–221 | DOI: 10.5890/JAND.2015.09.001

Ling Hong, Jun Jiang, Jian-Qiao Sun

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Abstract

Crises in chaotic pendulum in the presence of fuzzy uncertainty are observed by means of the fuzzy generalized cell mapping method. A fuzzy chaotic attractor is characterized by its topology and membership distribution function. A fuzzy crisis implies a simultaneous sudden change both in the topology of a fuzzy chaotic attractor and in its membership distribution. It happens when a fuzzy chaotic attractor collides with a regular or a chaotic saddle. Two types of fuzzy crises are specified, namely, boundary and interior crises. In the case of a fuzzy boundary crisis, a fuzzy chaotic attractor disappears after a collision with a regular saddle on the basin boundary. In the case of a fuzzy interior crisis, a fuzzy chaotic attractor suddenly changes in its size after a collision with a chaotic saddle in the basin interior.

On a Class of Generalized Hydrodynamic Type Systems of Equations

pp. 223–228 | DOI: 10.5890/JAND.2015.09.002

V.E. Fedorov, P.N. Davydov

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Abstract

By means of the degenerate semigroups theory methods the local existence of a unique solution is proved for initial-boundary value problems to a class of partial differential equations systems of generalized hydrodynamics type. General results are illustrated by examples of a system with the nonlinear viscosity and a weighted system. .

Influence of Systematic Coupling Stiffness Parameter on Coupling Duffing System Lag Self-synchronization Characteristic

pp. 229–237 | DOI: 10.5890/JAND.2015.09.003

Zhao-Hui Ren1, Yu-Hang Xu1, Yan-Long Han2, Nan Zhang1, Bang-Chun Wen1

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Abstract

Self-synchronization, compound synchronization and intelligent control synchronization widely exist in the mechanical system engineering, while lag self-synchronization movement is a special form of cooperation movement. Based on coupling Duffing system, this paper studies lag self-synchronization problem, analyses general change law of the system co-rotating synchronization frequency, antisynchronization frequency and lag phase angle by analytic analysis and numerical quantitative analysis, studies coupling parameter influences on systematic lag self-synchronization, and analyses the cause of lag self-synchronization. The results show that the root cause of lag self-synchronization is systematic stiffness namely systematic natural characteristic; that co-rotating synchronization vibration frequency and phase difference depend on coupling stiffness parameter; and that frequency and phase difference of anti-synchronization vibration are independent of coupling stiffness parameter; and when coupling stiffness parameter is larger, phase difference of two oscillators in the two kinds of synchronization is nonzero constant value.

Tribo-dynamics Analysis of Satellite-bone Multi-axis Linkage System

pp. 239–250 | DOI: 10.5890/JAND.2015.09.004

Jimin Xu1, Honglun Hong1, Xiaoyang Yuan1, Zhiming Zhao2

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Abstract

Friction torque is the most important factor that influences stability and precision of multi-axis linkage system during low-speed running. It is primarily related to shafting structure, external load, motion state and other factors, and presents highly nonlinear characteristics. In this paper, satellite-bone multi-axis linkage system in microgravity environment is taken as the research object to focus on the coupling tribo-dynamics issues of influencing system’s precision. The shafting structural features of two degrees of freedom (2-DOF) vertical-axis turntable, source of friction torque and coating antifriction technology are studied. The corresponding tribo-dynamics model is established. The model shows motion process of multi-axis linkage system is a coupling process of tribology and dynamics. In order to eliminate the effect of friction torque fluctuations, friction compensation based on LuGre friction model is introduced. Then tracking precision of the visual axis is about 40 (large movement range). For the imperfect situations with friction compensation, local actuator is introduced to combine with wide-range basic multi-axis system to realize the accurate movement within small range. Then the tracking precision of visual axis is expected to reach about 2 to 4 (small movement rang).

Stability and Bifurcation of a Nonlinear Aero-thermo-elastic Panel in Supersonic Flow

pp. 251–257 | DOI: 10.5890/JAND.2015.09.005

Wei Kang1, Yang Tang1, Min Xu1, Jia-Zhong Zhang2

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Abstract

Stability and bifurcation of a nonlinear supersonic panel under aerothermal loads are analyzed numerically in the present study. In the structural model, von Karman’s large deformation theory is taken into account for the geometric nonlinearity of the panel. In light of Hamilton’s principle, the governing equation of motion of a twodimensional aero-thermo-elastic panel is established. Coupling with the panel vibration, aerodynamic pressure is evaluated by first order supersonic piston theory and aerothermal load is approximated by quasi-steady theory of thermal stress. By transforming the partial differential equation to a series of ordinary differential equations via Galerkin method, fixed points and their stabilities of the system are studied using nonlinear dynamic theory. The complex dynamic responses regions are discussed with temperature loads as a bifurcation parameter. The results show that the thermal stress has a significant influence in the stability of the panel. The panel system undergoes Hopf bifurcation, period doubling, quasi-period and chaos with the increase of the temperature.

Nonlinear Effects of Dusty Plasmas using Homogenous Nonequilibrium Molecular Dynamics Simulations

pp. 259–265 | DOI: 10.5890/JAND.2015.09.006

Aamir Shahzad1,2, Mao-Gang He1

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Abstract

Three-dimensional strongly coupled complex (dusty) plasma (SCCDP) is modeled using homogenous nonequilibrium molecular dynamics (HNEMD) simulations. The thermal conductivity (&lambda 0) and the effects of external force field (F.) strength on the &lambda 0 of SCCDP are calculated at higher screening strengths (&kappa ) from generalized Evan’s algorithm. It has been shown that the presented investigations exhibit a non-Newtonian effect that the &lambda 0(&Gamma) increases with increasing force field strength that represents interaction contributions in Yukawa conductivity. It is also verified that the results obtained with different external force filed strengths are in satisfactory agreement with earlier numerical results and with reference set of data showed deviations within less than ± 10% for most of the present data point. Our very recently computed thermal conductivity at lower &kappa is validated by comparing the results of &lambda 0(&Gamma) at higher &kappa that also extended the range of force field strength (0.001 &le F. &le 0.1) which explains the nature of nonlinearity of SCCDP.

The Adaptive Synchronization of the Stochastic Fractional-order Complex Lorenz System

pp. 267–279 | DOI: 10.5890/JAND.2015.09.007

Xiaojun Liu, Ling Hong?

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Abstract

In this paper, the adaptive synchronization of a stochastic fractionalorder complex Lorenz system is analyzed. Firstly, the Laguerre polynomial approximation method is applied to investigate the fractionalorder system with a random parameter which obeys an exponential distribution. Based on this method, the stochastic system is reduced into the equivalent deterministic one. Besides, based on the stability theory of fractional-order systems, the adaptive synchronization for the deterministic system with unknown parameters is realized by designing appropriate synchronization controllers and estimation laws for uncertain parameters. Numerical simulations are used to demonstrate the effectiveness and feasibility of the proposed scheme.

Nonlinear Dynamic Characteristic Analysis of Planetary Gear Transmission System for the Wind Turbine

pp. 281–294 | DOI: 10.5890/JAND.20156.09.008

Zhao-Hui Ren1, Liang Fu1, Ying-Juan Liu2, Shi-Hua Zhou1, Bang-Chun Wen1

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Abstract

Foreign Object Debris (FOD) is debris or article alien, which may A compared with the translational-torsional dynamic model of planetary gear transmission system (PGTS) used in wind turbine is established in order to analyze the dynamic characteristics of the PGTS more accurately. The influences of the meshing stiffness, input and output torques, meshing error, nonlinear characteristics of the support bearing and gravity are considered in the model. Based on previous model, the vibration differential equations of the drive-train are obtained through the Lagrange’s equation. The dynamic response characteristics are investigated using the Runge–Kutta numerical method, and the factors of the above proposed excitation are analyzed. The results show clearly that the vibration responses have different characteristics due to the different speeds of each component in the PGTS. The nonlinear behaviour of support bearing causes the dynamic response more complicated In addition, under the internal and external excitations, more frequency multiplication and frequency combination components appear. The vibration frequency components of the system are mainly concentrated in the frequency range below 200Hz. The results of the study in this paper can provide necessary theoretical basis for natural characteristics study, dynamic response and optimization design method of the MW wind turbine PGTS.

Real-time 2D Concentration Measurement of CH4 in Oscillating Flames Using CT Tunable Diode Laser Absorption Spectroscopy

pp. 295–303 | DOI: 10.5890/JAND.20156.09.009

Takahiro Kamimoto1, Yoshihiro Deguchi1, Ning Zhang1,2, Ryosuke Nakao1, Taku Takagi1, Jia-Zhong Zhang2

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Abstract

Foreign Object Debris (FOD) is debris or article alien, which may One of the major problems of gas turbine combustors is the combustion oscillation. The combustion oscillation of gas turbines has many complex causes such as pressure fluctuations, combustion instabilities, and mechanical designs of the combustion chamber. Although the significant research efforts have been dedicated to this topic, the combustion oscillation problems have not yet been solved because of its complexity and nonlinearity. In this study, the theoretical and experimental research has been conducted in order to develop the noncontact and fast response 2D CH4 distribution measurement method to elucidate nonlinear combustion oscillation problems. The method is based on a computed tomography (CT) method using tunable diode laser absorption spectroscopy (TDLAS). The CT-TDLAS method was applied to oscillating flames and the time resolved 2D CH4 concentration distributions were successfully measured using 16 path CT-TDLAS measurement cell. CT-TDLAS has the kHz response time and the method enables the real-time 2D species concentration measurement to be applicable to the nonlinear phenomena of combustion oscillation problems in gas turbines.

Fluid-Structure Coupling Effects on the Aerodynamic Performance of Airfoil with a Local Flexible Structure at Low Reynolds Number

pp. 305–312 | DOI: 10.5890/JAND.20156.09.010

Wei Kang1, Min Xu1, Jia-Zhong Zhang2

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Abstract

Foreign Object Debris (FOD) is debris or article alien, which may A fluid structure interaction method for an airfoil with a local flexible structure is presented for flow control of Micro Air Vehicles. An improved ALE-CBS scheme is developed for unsteady viscous flow coupling in combination with the theory of shallow arch with large deformation. After the verification of the presented algorithm, the method is used to study the interaction between flow and airfoil with a local flexible structure with different elastic stiffness. The momentum and energy exchange are investigated to reveal the unsteady coupling effects on aerodynamic performance. The results show that the coupling between fluid and structure enhances the momentum and energy exchange from main flow into the boundary layer. It induces the separation bubble moving downstream, and decreases the negative pressure in the separation zone on the upper surface, which leads to lift enhancement. The utilization of local flexible structure can be considered as an effective flow control technique for enhancement of the aerodynamic performance of Micro Air Vehicles.

Modal Analyses of a Thin Shell with Constrained Layer Damping (CLD) Based on Rayleigh-Ritz Method

pp. 313–327 | DOI: 10.5890/JAND.20156.09.011

Xu-Yuan Song, Hong-Jun Ren, Jing-Yu Zhai, Qing-Kai Han

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Abstract

Foreign Object Debris (FOD) is debris or article alien, which may This paper presents analytical results of natural frequencies and loss factors of node-diameter and node-circumferential modes of a thin shell treated with constrained layer damping (CLD) based on Rayleigh-Ritz method. General differential equations of motion of the thin CLD shell are derived firstly by following the Donnell-Mushtari shell theory. By taking the beam characteristic functions as the admissible functions, the Rayleigh-Ritz method is employed to deduce the higher degree equations of natural frequencies of the thin CLD shell under different boundary conditions. It is confirmed that the present method is accurate and convenient so that it is applicable to the thin CLD shell compared with classic analytic method or transfer matrix method. Several examples are achieved and compared to illustrate the effects of the viscoelastic material (VEM) and constrain layer’s thickness ratios on the natural frequencies and modal loss factors, besides the effect of boundary conditions. The results show that the thickness ratio of VEM affects sensitively the modal frequencies and the total damping capacity of the thin CLD shell.

The Method of High Order Fatigue Test of Thin Plate Composite Structure With Hard Coating

pp. 329–337 | DOI: 10.5890/JAND.20156.09.012

Hui Li†, Wei Sun, Zhong Luo, Bengchun Wen

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Abstract

Foreign Object Debris (FOD) is debris or article alien, which may In order to solve the high order fatigue test problem of thin plate composite structure with hard coating, a new test method and its technological process is proposed based on the summarizing the experience of massive experiments. Besides, by considering technical difficulties of the high order fatigue test of the hard coating composite structure, several key techniques are described in details, such as how to measure dynamic strain without damaging the hard coating, how to predict excitation amplitude required by the high order fatigue test, and how to avoid the interference resulted from the strain softening of hard coating and to measure high order nature frequencies accurately. Finally, an experimental test is done to verify the practicability and reliability of this method.

JAND Volume 4, Issue 4

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Several Fractional Differences and Their Applications to Discrete Maps

pp.339–348 | DOI: 10.5890/JAND.2015.11.001

Guo-Cheng Wu, Dumitru Baleanu, Sheng-Da Zeng

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Abstract

Several definitions of fractional differences are discussed. Their applications to fractional maps are compared. As an example, the logistic equation of integer order is discretized by these fractional difference methods. The comparative results show that the discrete fractional calculus is an efficient tool and the maps derived in this way have simpler forms but hold rich dynamical behaviors.

Initial-Boundary Value Problems for Local Fractional Laplace Equation Arising in Fractal Electrostatics

pp. 349–356 | DOI: 10.5890/JAND.2015.11.002

Xiao-Jun Yang, H. M. Srivastava and Dumitru Baleanu

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Abstract

The initial-boundary value problems for the local fractional Laplace equation, which arises in fractal electrostatics, are investigated in this article. The non-differentiable solutions with different initial and boundary conditions are obtained by using the local fractional series expansion method.

Stability and Bifurcations Analysis for 2-DOF Vibroimpact System by Parameter Continuation Method. Part I: Loading Curve

pp. 357–370 | DOI: 10.5890/JAND.2015.11.003

V.A. Bazhenov, P.P. Lizunov, O.S. Pogorelova, T.G. Postnikova, V.V. Otrashevskaia

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Abstract

Vibroimpact system dynamic behaviour is studied by numerical parametric continuation technique combined with shooting and Newton- Raphson’s methods. The technique is adapted to two-body twodegree- of-freedom vibroimpact system under periodic excitation. Impact is simulated by nonlinear contact interaction force based on Hertz’s contact theory. Stability or instability of obtained periodic solutions is determined by monodromy matrix eigenvalues based on Floquet’s theory. Analysis of dynamic behaviour for specific vibroimpact system was performed. The instability zones, different oscillatory regimes and bifurcation points were found. Poincare sections were also constructed.

Multiple Moving Force Identification Based on Bridge Bending Moment Influence Lines

pp. 371–378 | DOI: 10.5890/JAND.2015.11.004

C.Z. Qian, C.P. Chen, L.M. Dai

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Abstract

Identification of the moving forces on a bridge is essential to bridge design and management. A new method for identifying the time varying axle loads is presented in this research using the bending moment influence line. Based on the theorem of modal superposition and taking the damping force into consideration, several modal accelerations can be obtained from measured accelerations of the bridge at several sections. Based on the d’Alembertian theory, the inertia force of the bridge is expressed as a distributed load. Using the bending moment influence lines of the bridge, the moving force contributing to the moment and the inertia force are obtained. The formulation about flexural moment and moving force is obtained correspondingly. An optimization method is used to find the solution of the equations, and then the moving forces can be obtained at any given time. Examples show that the method has a high accuracy in identifying moving forces even though there are more than one time varying forces. By applying the proposed method, information regarding the moving forces can be obtained without solving the dynamic equation, resulting in an efficient model for applications in engineering.

A Matrix-Based Computational Scheme of Generalized Harmonic BalanceMethod for Periodic Solutions of Nonlinear Vibratory Systems

pp. 379–389 | DOI: 10.5890/JAND.2015.11.005

Yuefang Wang, Zhiwei Liu

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Abstract

A matrix-based computational scheme is developed based on the Generalized Harmonic Balance method for periodic solutions of nonlinear dynamical systems. The nonlinear external loading is expanded into a Taylor’s series as a function of displacement and velocity, and is then expressed as a combination of Fourier harmonics through the Generalized Harmonic Balance method. Using the Newton-Raphson’s approach, an iteration scheme is formulated to obtain the solution of harmonic coefficients for the displacement. The present scheme is a general purpose realization of the Generalized Harmonic Balance method in the sense that it does not need an analytical Fourier expansion of loadings, and all of the coefficient matrices involved with the scheme are created in a standard way. An example of a periodically forced Duffing oscillator is provided to demonstrate the performance of the present scheme. Numerical solutions of period-1 motion from the present scheme are compared with numerical results given by the Runge-Kutta method. The numerical results agree well with analytical predictions by Luo et al.

Vibrational Resonance in the Duffing Oscillator with Distributed Time-Delayed Feedback

pp. 391–404 | DOI: 10.5890/JAND.2015.11.006

C. Jeevarathinam, S. Rajasekar, and M.A.F. Sanjuan

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Abstract

We analyze the vibrational resonance in the Duffing oscillator system in the presence of (i) a gamma distributed time-delayed feedback and (ii) integrative time-delayed (uniformly distributed time delays over a finite interval) feedback. Particularly, applying a theoretical procedure we obtain an expression for the response amplitude Q at the low-frequency of the driving biharmonic force. For both double-well potential and single-well potential cases we are able to identify the regions in parameter space where either (i) two resonances, (ii) a single resonance or (iii) no resonance occur. Theoretically predicted values of Q and the values of a control parameter at which resonance occurs are in good agreement with our numerical simulation. The analysis shows a strong influence of both types of time-delayed feedback on vibrational resonance.

Analysis of Stability in High Speed Milling Machining by Means of Spectral Decomposition

pp. 405–424 | DOI: 10.5890/JAND.2015.11.007

Antonio Carminelli, Giuseppe Catania

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Abstract

A technique to study the critical chatter conditions arising in the high speed milling machining process is proposed, by means of evaluating system eigenvalue stability and eigenvalue sensitivity with respect to system parameters. Starting from the equations of motion of a general machine tool system, a set of linear, ordinary, time dependent parameter, Periodic Delay Differential Equations (PDDEs) may be the result. Three approaches (multi-step, full-discretization and semi-discretization) based on the extended Floquet theory and time discretization techniques are analyzed. The semi-Discretization (SD) method is considered in order to derive an eigenvalue sensitivity formula and to show the limits of this approach. A different approach not based on the Floquet theory is proposed. A set of linear, ordinary, constant parameter, PDDEs is obtained by applying a spectral decomposition and a generalized harmonic balance technique. The stability of the solution is evaluated by solving a non standard eigenvalue problem, making it possible to predict the occurrence of chatter vibration during milling. The proposed approach makes it possible to obtain a straightforward formula for the sensitivity of eigenvalues, and of the critical chatter condition, with respect to the variation of a system parameter. A numerical example is presented in order to test the proposed approach. Finally, strengths and limits of the proposed technique are critically discussed and a comparison with the SD method is carried out.

ApproximateWeakly NonlinearModel of gas Dynamics EquationsWith Utilized Korobeinikov’s Chemical Reaction

pp. 425–437 | DOI: 10.5890/JAND.2015.11.008

Ranis N. Ibragimov, Sayavur Bakhtiyarov

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Abstract

The main purpose is to develop an analytic approach for investigating the wave front propagation on a surface of a cylinder that can be used a descriptor of a detonation engine. The analysis is based on a weakly nonlinear approximated gas dynamic equations with incorporated approximation of the Korobeinkov’s chemical reaction model that are used to describe the two-dimensional detonation field on a surface of a two-dimensional cylindrical chamber without thickness. We found that the wave fronts can be expressed analytically in explicit form for special classes of flow (e.g. isentropic gas flow). In more general cases, the dynamics of wave fronts can still be determined explicitly provided that the wave front is known at initial time and the exact solution, or its approximation is known a priori, e.g. from experimental or numerical analysis.

JAND Volume 5, Issue 1

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Nonlinear Dynamics Analysis of a Continuum Rotor through Generalized Harmonic Balance Method

pp.1–31 | DOI: 10.5890/JAND.2016.03.001

Haiyang Luo, Yuefang Wang

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Abstract

The dynamics of a continuum rotor system excited by nonlinear oilfilm force and electromagnetic force is investigated through the Generalized Harmonic Balance Method. The governing differential equation of the continuum rotor system is reduced through the Galerkin’s approximation method. Firstly, the motion in Y-direction is considered as the response of a planar continuum rotor. The analytical periodic motion is obtained and the bifurcation and stability is determined. Secondly, the dynamical model of a spatial continuum rotor is studied. The stability and bifurcation of the analytical periodic motion is obtained and compared to numerical results to show the accuracy of the Generalized Harmonic Balance Method.

Application of the Generalized Mean Value Function for Detection of Defects in Metal Cylindrical Slugs

pp. 33–41 | DOI: 10.5890/JAND.2016.03.002

Raoul R. Nigmatullin, Sergey I. Osokin, Victor M. Larionov, Yuriy V. Vankov, Evgeniya V. Izmajlova, Wei Zhang

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Abstract

In this paper we present a free-oscillation method for acoustic detection of defects in metal rods. For detection of normal rods from rods with defects we use a treatment procedure based on the statistics of the fractional moments. This procedure allows to extract the quanti-tative information from the acoustic signals that are propagated in cylindrical slugs after mechanical strike and having different defects. This information allows separating normal rods (without defects) from the rods having different cuts and differentiating the saw-cut defects with different depth from each other. The analysis of data obtained from this research shows that the proposed method of the fractional moments can be applied for quantitative ”reading” of envelopes of different acoustic signals that are obtained by means of free-oscillation method and propagated in dense medium having local defects.

Generalized Combination-combination Synchronization of Chaos in Identical Orders Chaotic Systems

pp. 43–58 | DOI: 10.5890/JAND.2016.03.003

K.S. Ojo, A.N. Njah, O.I. Olusola

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Abstract

This paper proposes a generalized combination-combination synchronization scheme (GCCS) for identical order chaotic systems. Using the active backstepping technique. Based on active backstepping technique, suitable controllers are designed to achieve GCCS for any desired scaling factor among four chaotic Josephson junctions (JJs) evolving from different initial conditions. Numerical simulations are carried out to verify the feasibility and effectiveness of the proposed GCCS via active backstepping technique. The result shows that the proposed GCCS could be used to vary the JJ signal to any desired level and provide higher security in information transmission due its complex dynamical formulation.

Paralysis Mechanism of α-Conotoxin SI from Molecular Dynamics Simulations and Free Energy Calculations

pp. 59–64 | DOI: 10.5890/JAND.2016.03.004

Onur Tuna, Serdar Kuyucak, Turgut Ba¸stug

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Abstract

Peptide toxins offer new avenues for development of novel drugs. α- Conotoxin SI, which binds to neuromuscular Nicotinic Acetylcholine Receptor, is such an analgesic drug candidate. Understanding the mechanism of action is crucial for improving the properties of drug candidates. Here, we use docking and molecular dynamics simulations to propose a model for binding of α-Conotoxin SI to Nicotinic Acetylcholine Receptor and discuss its paralysis effect.

Numerical Solution of Energy Transmission Lines Equivalent Circuit Equations with Adomian Decomposition Method

pp. 65–71 | DOI: 10.5890/JAND.2016.03.005

N.F.O. Serteller, D. Ustundag

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Abstract

In this paper, analysis for a mathematical model of an equivalent circuit to provide solutions of electrical energy power transmission lines (ETL) with Adomian Decomposition Method (ADM) has been proposed. By using Mathematica program, partial differential equations as a function of voltage (current) forming the model are solved and compared with the finite difference method (FDM). The results of some special examples obtained from ADM and FDM illustrate very good synchronism and show the simplicity and the efficiency of the method.

Stability and Hopf Bifurcation of a Predator-Prey Model with Discrete and Distributed Delays

pp. 73–91 | DOI: 10.5890/JAND.2016.03.006

Canan Celik , Emine Degirmenci

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Abstract

In this study, a delayed ratio dependent predator-prey model with both discrete and distributed delays is investigated. First, the local stability of a positive equilibrium is studied and then the existence of Hopf bifurcations is established. By using the normal form theory and center manifold theorem, the explicit algorithm determining the stability, direction of the bifurcating periodic solutions are derived. Finally, we perform the numerical simulations for justifying the theoretical results.

Global Dynamics of a Three Species Predator-Prey Competition Model with Holling type II Functional Response on a Circular Domain

pp. 93–104 | DOI: 10.5890/JAND.2016.03.007

Walid Abid, R. Yafia, M.A. Aziz-Alaoui, H. Bouhafa and A. Abichou

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Abstract

This paper is devoted to the study of a three species ecosystem model consisting of a prey, a predator and a top predator. This model is given by a reaction diffusion system defined on a circular spatial domain and incorporates the Holling type II and a modified Leslie- Gower functional response. The aim of this paper is to investigate theoretically and numerically the asymptotic behavior of the interior equilibrium of the model. The conditions of boundedness, existence of a positively invariant and attracting set are proved. Sufficient conditions of local/global stability of the positive steady state are established. In the end, we present a numerical evidence of time evolution of the pattern formation.

A Bayesian Random Walk Approach for Mapping Dynamic Quantitative Trait

pp. 105–115 | DOI: 10.5890/JAND.2016.03.008

Burak Karacoren

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Abstract

Analyzing longitudinal observations with an appropriate methodology is important for detecting significant single nucleotide polymorphisms (SNPs) for complex traits in genome wide association studies (GWAS). Here we assumed that genomic signal over time could be traced by a Bayesian random walk- Kalman filter model in state space form to obtain longitudinal residuals. The Kalman filter removes the requirement for collecting the whole data set before making estimations, thus estimates are immediately available. For example, with the functional mapping approach, whole sets of observations must to be collected in advance. This may include months (if not years) of waiting time depending on the nature of the phenotypes and the experimental design. Whereas, because of the longitudinal residuals used in association mapping, biological reasoning can be easily applied given that the signal is genuine. The advantages of our method are demonstrated by both simulated and real datasets.

Numerical Modeling of Photon Migration in Human Neck Based on the Radiative Transport Equation

pp. 117–125 | DOI: 10.5890/JAND.2016.03.009

Hiroyuki Fujii, Shinpei Okawa, Ken Nadamoto, Eiji Okada, Yukio Yamada, Yoko Hoshi, and Masao Watanabe

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Abstract

Biomedical optical imaging has a possibility of a comprehensive diagnosis of thyroid cancer in conjunction with ultrasound imaging. For improvement of the optical imaging, this study develops a higher order scheme for solving the time-dependent radiative transport equation (RTE) by use of the finite-difference and discrete-ordinate methods. The accuracy and efficiency of the developed scheme are examined by comparison with the analytical solutions of the RTE in homogeneous media. Then, the developed scheme is applied to describing photon migration in the human neck model. The numerical simulations show complex behaviors of photon migration in the human neck model due to multiple diffusive reflection near the trachea.

JAND Volume 5, Issue 2

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Global Dynamics Analysis of a Continuum Rotor through G-Function and LFunction Method

pp.127–146 | DOI: 10.5890/JAND.2016.06.001

Haiyang Luo1,2, Yuefang Wang1,2

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Abstract

The global dynamics of a continuum rotor excited by nonlinear oilfilm force and electromagnetic force is investigated. The governing equation of the system is obtained using the Galerkin’s method. The dynamical systems are reduced into two 2-dimensional systems in Yand Z- directions. The analytical conditions for global transversality and tangency to the separatrix are presented. The global and local dynamics of the system is determined through the G-function and G(1)-function and is compared to numerical solution. The periodicity of periodic flow of the rotor is determined by the L-function. The complexity of the nonlinear continuum rotor system is demonstrated through global transversality and tangency of the periodic motions to separatrix.

Relative Controllability of Nonlinear Neutral Fractional Volterra Integrodifferential Systems with Multiple Delays in Control

pp. 147–160 | DOI: 10.5890/JAND.2016.06.002

K. Balachandran, S. Divya,

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Abstract

In this work, we investigate the global relative controllability of nonlinear neutral fractional Volterra integrodifferential systems with multiple delays. Controllability criteria for nonlinear fractional order systems are established. The results are obtained by using the Schauder fixed point theorem. Moreover, some numerical examples are provided to illustrate the effectiveness and applicability of the proposed criteria.

Coupled Systems with Hyperchaos and Quasiperiodicity

pp. 161–167 | DOI: 10.5890/JAND.2016.06.003

A.P. Kuznetsov, Yu.V. Sedova

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Abstract

A model with hyperchaos is studied by means of Lyapunov twoparameter analysis. The regions of chaos and hyperchaos, as well as autonomous quasiperiodicity are identified. We discuss the picture of domains of different regimes in the parameter plane of coupled systems, corresponding to the cases of interaction of quasiperiodic and hyperchaotic subsystems.

Mitigating Grazing Bifurcation and Vibro-Impact Instability in Time-Frequency Domain

pp. 169–184 | DOI: 10.5890/JAND.2016.06.004

Chi-Wei Kuo, C. Steve Suh+

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Abstract

Impact oscillators are found in many applications. It is common for these applications to undergo the inadvertent state of grazing bifurcation. Vibro-impact incited grazing and route-to-chaos are difficult to control. The Newtonian model of a vibro-impact system rich of complex nonlinear behaviors is considered for the mitigation of impact induced instability and grazing. A novel concept capable of simultaneous control of vibration amplitude in the time-domain and spectral response in the frequency-domain is adopted to formulate a viable control solution. The concept has been demonstrated to be feasible for the control of dynamic instability including bifurcation and route-to-chaos in many nonlinear systems. The developed controller explores wavelet adaptive filters and filtered-x least mean square algorithm to the successful moderation of the grazing and dynamic instability of the non-smooth system. The qualitative behavior of the controlled impact oscillator follows a definitive fractal topology before settling into a stable manifold. The controlled response is categorically quasi-periodic and of the prescribed vibration amplitude and frequency spectrum.

The Dynamical System Generated by the Floor Function λ x

pp. 185–191 | DOI: 10.5890/JAND.2016.06.005

U.A. Rozikov†, I.A. Sattarov, J.B. Usmonov

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Abstract

We investigate the dynamical system generated by the floor function λ x defined on R and with a parameter λ ∈ R. For each given m ∈ N we show that there exists a region of values of λ, where the floor function has exactly m fixed points (which are non-negative integers), also there is another region for λ , where there are exactly m+1 fixed points (which are non-positive integers). Moreover the full set Z of integer numbers is the set of fixed points iff λ = 1. We show that depending on λ and on the initial point x the limit of the forward orbit of the dynamical system may be one of the following possibilities: (i) a fixed point, (ii) a two-periodic orbit or (iii) ±∞.

Theorem on the Bifurcations of the Slow Invariant Manifold of a System of Two Linear Oscillators Coupled to a k-order Nonlinear Oscillator

pp. 193–197 | DOI: 10.5890/JAND.2016.06.006

Jamal-Odysseas Maaita†

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Abstract

we study a system of two linear oscillators coupled to a k-order nonlinear oscillator with a mass much smaller than the mass of the linear oscillators. We prove that the Slow Invariant Manifold of the system may bifurcate only when the order of the nonlinear oscillator is an odd number.

Consequence of Prey Refuge in a Tri-trophic Prey-dependent Food Chain Model with Intra-specific Competition

pp. 199–219 | DOI: 10.5890/JAND.2016.06.007

Nijamuddin Ali1, Santabrata Chakravarty2 †

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Abstract

The present article deals with the influence of a constant proportion of prey refuge in presence of intra-specific competition among predator population of a prey-dependent three species food chain model. The behaviour of the system near the biologically feasible equilibria is thoroughly analyzed. The preliminary results such as boundedness and dissipativeness of the system are established. Stability analysis including local and global stability of the equilibria has been carried out in order to examine the behaviour of the system. The present system experiences Hopf-Andronov bifurcation for suitable choice of the parameter values. The influences of the prey refuge parameters on the dynamical behaviour of the system are exhibited through several plots and discussed at some equilibrium positions. It is worth-noting that prey refuge has stabilization effect in some selected situations and bears the potential to control chaotic dynamics of the system. Hence, prey refuge may be of some use for biological control mechanism. Numerical simulations are performed to validate the applicability of the model under consideration.

Investigation on Rattle of Gears under Random Loading

pp. 221–230 | DOI: 10.5890/JAND.2016.06.008

Yubing Wen, Jianming Yang†

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Abstract

This paper investigates the random rattle motion of a single stage spur gear pair subjected to deterministic and random excitation. The motion is treated separately as free flight between the boundaries and the instant impact at the two boundaries. Numerical path integration method is applied to obtain the probability density of the response during free flight. Monte Carlo simulation is also conducted to verify the efficiency and accuracy of path integration in solving this strongly nonlinear dynamic problem. Comparison between results from path integration and Monte Carlo simulation is made and good agreement is achieved.

Initial-boundary Value Problems to the One-dimensional Compressible Navier- Stokes-Poisson system with Large Amplitude

pp. 231–241 | DOI: 10.5890/JAND.2016.06.009

Li Wang, Lei Jin †

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Abstract

Magnetic fluid is a new type of functional material. The motion of com-pressible, viscous self-gravitating fluids can be expressed by Navier-Stokes-Poisson equations. This study demonstrates the global, non-vacuum solutions with large amplitude to the initialboundary value problem of the one-dimensional compressible Navier- Stokes-Poisson system with degenerate dependent viscosity coefficients and density and temperature dependent heat conductivity coefficients. The main constituent of the detail analysis is to derive the positive lower and upper bounds on the specific volume and the absolute temperature.

JAND Volume 5, Issue 3

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Cell Cycle Dynamics in a Response/Signaling Feedback System With Overlap

pp.243–267 | DOI: 10.5890/JAND.2016.09.001

Gregory Moses†, Denise Scalfano

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Abstract

We continue our study of the dynamics of the yeast cell division cycle, in particular a feedback model where cells in one fixed part of the cycle, the Signaling region, affect the growth of cells in another part of the cycle, the Responsive region. This causes cells to cluster together as they pass through their cell division cycles, a known biological phenomenon that had been previously unexplained. In previous work, these regions were assumed to have disjoint interiors. We consider the dynamics when oscillators are coupled directly to themselves by allowing the responsive and signaling regions to overlap. We see that although the dynamics in an important subspace are largely preserved, the dynamics of the system in the full phase space exhibit less clearly quantifiable behavior. While in previously studied models, stability in the clustered subspace implies stability in the full phase space, here this is not the case, and the clustered subspace only unreliably predicts the behavior in the full phase space.

Numerical Bifurcation Analysis of Discontinuous 2-DOF Vibroimpact System. Part 2: Frequency-Amplitude Response

pp. 269–281 | DOI: 10.5890/JAND.2016.09.002

V.A. Bazhenov, P.P. Lizunov, O.S. Pogorelova, T.G. Postnikova†

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Abstract

This paper discusses bifurcations in nonlinear vibroimpact system. It is a discontinuous dynamical system. We were studying stability and bifurcations in specific two-body two-degree-of-freedom vibroimpact system by numerical parameter continuation method. We adapted parameter continuation technique for this system. Theoretical basis for dynamic characteristics composition was presented in [1]. The instability zones and bifurcation points for loading curves were determined in [1] under excitation amplitude variation. In this paper we investigate the instability zones and bifurcation points under variation of excitation frequency when we consider the frequency-amplitude response. We have observed phenomena unique for nonsmooth systems with discontinuous right-hand side: discontinuous bifurcation points where set-valued Floquet multipliers cross the unit circle by jump. At these points monodromy matrix is changed by jump too. We also have observed chattering regimes leading to chaos.

Phase Portraits, Hopf Bifurcations and Limit Cycles of the Ratio Dependent Holling-Tanner Models for Predator-prey Interactions

pp. 283–304 | DOI: 10.5890/JAND.2016.09.003

M. Sivakumar†, K. Balachandran

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Abstract

In this paper we consider a ratio dependent Holling Tanner predator prey model with type II functional responses. We analyzed the local stability, phase portraits, existence and uniqueness of stable limit cycles and Hopf bifurcation. The ranges of the parameter involved are provided under which the unique interior equilibrium can be determined for a stable (or an unstable) node or focus without diffusion. Furthermore the Turing instability analysis of the system with diffusion are studied. Numerical simulations using MATLAB are carried out to demonstrate the theoretical results obtained.

Modelling a Thermal Diffusion Interface for Robust Fractional Control

pp. 305–324 | DOI: 10.5890/JAND.2016.09.004

Riad Assaf†, Roy Abi Zeid Daou, Xavier Moreau

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Abstract

The objective of this work is to review and elaborate an approximation method of the transfer functions of both semi-infinite and finite homogeneous plane thermal diffusive interfaces, in view to use in robust control applications and to obtain accurate time responses. Novel models are proposed based on the fractional order integration found naturally when modelling diffusive interfaces. The models are verified by checking with the existing exact responses. The results obtained in this part of the work show a very good approximation of the transfer functions of the systems when using a rational representation in both frequency and time domains for several inputs such as the impulse and the step.

Synchronization of the Mobile Robot to a Chaotic Trajectory

pp. 325–335 | DOI: 10.5890/JAND.2016.09.005

Elvira Rafikova, Marat Rafikov†, Guilherme Rinaldo

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Abstract

In this paper we propose a new approach that allows a mobile robot to follow a chaotic trajectory, which is a projection of the strange attractor of any deterministic chaotic system, on the xoy plane. This approach is reminiscent of the master-slave synchronization. The desired chaotic curve is expressed in terms of a virtual reference robot. The suboptimal State Dependent Riccati Equation (SDRE) control is applied to achieve synchronization between the real and virtual robots. We demonstrate two examples of this approach, leading the robot system to a synchronization with Rossler and Lorenz strange attractor’s xoy projections.

Delay-Coupled Mathieu Equations in Synchrotron Dynamics

pp. 337–348 | DOI: 10.5890/JAND.2016.09.006

Alexander Bernstein1†, Richard Rand2

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Abstract

This paper investigates the dynamics of two coupled Mathieu equations. The coupling functions involve both delayed and nondelayed terms. We use a perturbation method to obtain a slow flow which is then studied using Routh-Hurwitz stability criterion. Analytic results are shown to compare favorably with numerical integration. The numerical integrator, DDE23, is shown to inadvertently add damping. It is found that the nondelayed coupling parameter plays a significant role in the system dynamics. We note that our interest in this problem comes from an application to the design of nuclear accelerators.

Validation of Blended Potential Flow Model for Lifting Rotors with Wake Contraction

pp. 349–371 | DOI: 10.5890/JAND.2016.09.007

Jianzhe Huang†, David A. Peters

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Abstract

Dynamic wake models have been evolved from the earliest, threedegree- of-freedom models (derived from momentum theory) to full finite-state models derived from potential flow theory by a formal Galerkin method. These models are widely used in industry, but still have some drawbacks that need to be remedied. These drawbacks include: 1.) lack of good convergence both on the disk and off the disk (one can use one or the other but not both), 2.) poor results downstream in the limit of shallow skew angles, 3.) poor convergence inside of the rotor wake, 4.) lack of the effect of wake contraction. In this paper, a blended model adopted applications of adjoint theorem, a special change of variables to overcome these obstacles. The resultant model is well behaved in all regimes and is applicable to use in realistic problems of flight simulation.

JAND Volume 5, Issue 4

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Study of Local Correlations of the Simultaneous wind Speed-irradiance Measurements Using the Time Dependent Intrinsic Correlation Method

pp.373–390 | DOI: 10.5890/JAND.2016.12.001

Rudy Calif1†, Fran¸cois Schmitt2, Yongxiang Huang3

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Abstract

Renewable resources such as atmospheric wind speed and global horizontal irradiance, possess huge fluctuations over a large range of spatial and temporal scales, indicating their nonlinear and nonstationary properties. In this study, the multiple scale dynamics and the correlations between simultaneous time series are analyzed using EMD (Empirical Mode Decomposition) based methods, particularly appropriate for such time series. We consider simultaneous wind speed-global horizontal irradiance measurements, sampled at 1 hour over a period of three years, from 2010 to 2013, at Guadeloupean Archipelago (French West Indies) located at 16o15N latitude and 60o30W longitude. After EMD decomposition of both time series, power laws are observed in the Fourier and Hilbert spaces over a broad range of frequencies. Furthermore, we investigate their local correlations using the Time Dependent Intrinsic Correlation method (TDIC). The time evolution and the scale dependence of their correlation are determined at different time scales and for different intrinsic modes functions. The estimation of TDIC have highlighted strong correlations for all the time scales, particularly strong negative correlations between both time series for mean periods 2h Tm < 273 days indicating a complementarity between wind speed and global horizontal irradiance for these time scales.

Equations and Stable Modes of Parametron

pp. 391–398 | DOI: 10.5890/JAND.2016.12.002

O.V. Privalova, L.V. Shtukin, D.Yu. Skubov†

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Abstract

This work is devoted to investigation of nonlinear electric circuit—the resonance scheme with nonlinear reactive element. The role of this non-linearity plays the saturation of magnetic transformed coupling. The stationary mode in this scheme is supported by generator of alternating current. The idea of work of parametron is an opportunity of two stationary modes of current corresponded to borders of the main zone of parametrical resonance. These modes are used as binary logical states with high-frequency ability of switching. The efficiency of this scheme increases with growth of frequency and simultaneously decreasing of scale. The aim of this article is a summarizing of the method of study for scheme of this type for this and some analogous devices.

Nonlinear Analysis of Two-layer Fluid Sloshing in a Rectangular Tank Subjected to Width Direction Excitation Open Access

pp. 399–421 | DOI: 10.5890/JAND.2016.12.003

Fumitaka Yoshizumi†

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Abstract

This paper describes a nonlinear theory to describe oscillations of two liquid layers formed in a rectangular tank. Nonlinear equations based on the variational principle and the Galerkin method are used, which are written in a form of direct expanding in eigenmodes. Analysis using the equations is described under the experimental conditions reported in a previous study. Both the experiment and the analysis demonstrate a “peculiar oscillation” in which the interface of the two liquids oscillates at 1/5 to 1/7 of the excitation frequency in a particular excitation frequency range. By observing the nonlinear force time series, four eigenmodes are identified to be mainly relevant to the oscillation. An amplitude equation analysis was applied to the four relevant eigenmodes. It was found that this “peculiar oscillation” is one of summed and differential harmonic oscillations (combination oscillation) in which one asymmetric and one symmetric eigenmode excite each other through the mediation of other two asymmetric eigenmodes.

Energy Harvesting with a Piezoelectric Thunder

pp. 423–439 | DOI: 10.5890/JAND.2016.12.004

Fengxia Wang1†, WeiWu2, Mahmoudiandehkordi Soroush1, and Amin Abedini1

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Abstract

In this work the energy harvesting performance of a piezoelectric curved energy generator (THUNDER) is studied via experimental and analytical methods. The analytical model of the THUNDER is created based on the linear mechanical electrical constitutive law of the piezoelectric material, the linear elastic constitutive law of the substrate, and the Euler-Bernoulli beam theory. With these linear modal functions, the Rayleigh-Ritz approach was used to then obtain the reduced mechanical electrical coupled modulation equations. With above analytical model, two types of energy harvest circuit are proposed and compared: 1) directly charging mode at low level excitation, and 2) memory stored optimal duty cycle step-down converter mode at high level excitation. The value of the optimal duty cycle is determined based on the characteristics of the vibration signals of the ambient vibration source. To reduce the energy consumption of the microcontroller, the optimum duty cycle values are stored in the microprocessor instead of doing onsite computing during energy harvesting process. For the purpose of designing a low power cost mechanical switch to control the operation of both modes, the threshold voltage between these two operation modes is converted into threshold displacement.

Study on Dynamical System with Time-delay

pp. 441–456 | DOI: 10.5890/JAND.2016.12.005

Amit Mondal1,†, Nurul Islam2

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Abstract

In this paper, stability analysis of the nonlinear time-delayed Sprott system is made by applying a small perturbation near the critical point. Next, we will discuss a scheme for delay synchronization. To achieve our claim, we will make the chaos synchronization between the coupled Sprott system with delay parameters. Here, we will discuss five distinct cases. Numerical simulation is done to verify our claim.

Global Existence and Blowup of Solutions of Two Species Chemotaxis Model

pp. 457–469 | DOI: 10.5890/JAND.2016.12.006

V. Bhuvaneswari†, K. Balachandran

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Abstract

This paper is devoted to obtain the global existence and blow up of solutions for two species chemotaxis system by the ratio of two solutions method. Our main concern is to show that the blow up properties of solutions depend only on the first eigenvalue.

Delay Terms in the Slow Flow

pp. 471–484 | DOI: 10.5890/JAND.2016.12.007

Si Mohamed Sah1†, Richard H. Rand2

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Abstract

This work concerns the dynamics of nonlinear systems that are subjected to delayed self-feedback. Perturbation methods applied to such systems give rise to slow flows which characteristically contain delayed variables. We consider two approaches to analyzing Hopf bifurcations in such slow flows. In one approach, which we refer to as approach I, we follow many researchers in replacing the delayed variables in the slow flow with non-delayed variables, thereby reducing the DDE slow flow to an ODE. In a second approach, which we refer to as approach II, we keep the delayed variables in the slow flow. By comparing these two approaches we are able to assess the accuracy of making the simplifying assumption which replaces the DDE slow flow by an ODE. We apply this comparison to two examples, Duffing and van der Pol equations with self-feedback.

Controllability of Fractional Nonlinear Systems in Banach Spaces

pp. 485–494 | DOI: 10.5890/JAND.2016.12.008

R. Joice Nirmala†, K. Balachandran

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Abstract

This paper investigates the solution of fractional dynamical systems with the inverse operator method and Mittag-Leffler function. Controllability of linear fractional dynamical systems is studied by obtaining the Grammian operator. Sufficient conditions for both nonlinear and integrodifferential systems are established by the contraction principle. Some examples are provided to illustrate the theory.

The Closed-Form Steady-State Probability Density Function of van der Pol Oscillator under Random Excitations

pp. 495–502 | DOI: 10.5890/JAND.2016.12.009

Lincong Chen, Jian-Qiao Sun†

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Abstract

The strongly nonlinear van der Pol oscillator represents a special challenge, which has prevented many methods from obtaining the closed-form solutions of the steady-state probability density functions (PDFs) in the literature. In this paper, we apply our recently developed method called the iterative method of weighted residue to analytically construct steady-state PDFs of the van der Pol oscillator under external Gaussian white noise excitation. The steady-state PDF is assumed to be an exponential function of polynomials in the state variables. The iterative method of weighted residue is used to compute the PDF. The iterative procedure that makes use of the obtained closed-form solutions of steady-state PDFs as the weighting function for the method improves the accuracy of the solution and the convergence of the solution process. The closed-form steady-state PDFs of strongly nonlinear van der Pol oscillator are presented in this paper, which were not available in the literature before, and are compared with those from the Monte Carlo simulations. The analytical and simulation results are in excellent agreement over a wide range of damping coefficients.

JAND Volume 6, Issue 1

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Large Deviations for Nonlinear Ito Type Stochastic Integrodifferential Equations

pp.1–15 | DOI: 10.5890/JAND.2017.03.001

M. Suvinthra†, K. Balachandran

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Abstract

In this work, we consider a nonlinear Itˆo type stochastic integrodifferential equation and study the Freidlin-Wentzell type large deviation principle for its solution processes. The weak convergence approach is employed to establish the Laplace principle which in turn is equivalent to the large deviation principle. The compactness criterion is verified by means of sequential compactness of solutions of the associated controlled equation. The weak convergence result is asserted via solutions of the controlled equation with stochastic perturbation. Finally, examples are included to illustrate the theory.

Approximate Analytical Solutions of A Nonlinear Oscillator Equation Modeling A Constrained Mechanical System

pp. 17–26 | DOI: 10.5890/JAND.2017.03.002

Serge Bruno Yamgoue1, Bonaventure Nana1, Francois Beceau Pelap2

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Abstract

In this paper, we consider a class of nonlinear oscillators whose equations of motion are in the form of that of a cubic Duffing oscillator extended by a term which is a quadratic monomial in the velocity and whose coefficient is a rational function of the position. We apply a combination of harmonic balance and Newton method to seek analytical approximations to the periodic solutions to the equation. The analysis can be applied directly to the equation in its "natural" rational form or after reducing it to the same denominator and considering only the numerator. The advantages and drawback of these two usages of the method are also discussed.

Nonlinear Throughflow Effects on Thermally Modulated Rotating Porous Medium

pp. 27–44 | DOI: 10.5890/JAND.2017.03.003

Palle Kiran1, B.S. Bhadauria2, Y Narasimhulu1

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Abstract

The effect of throughflow and temperature modulation on a rotating porous medium is investigated. The generalized Darcy model is used for the momentum equation. Heat transfer analysis is based on weakly nonlinear thermal instability. It is computed numerically in terms of the Nusselt number, which is governed by a non-autonomous complex Ginzburg-Landau equation. Both concepts, rotation and throughflow are used as an external mechanism to regulate heat transfer. The effect of amplitude and frequency of modulation on heat transport is discussed and presented graphically. The effect of throughflow has duel by nature on heat transfer, the outflow enhances and inflow diminishes the heat transfer. It is found that, high rotational rates promotes heat transfer than low rotational rates. Further, the effect of modulation on mean Nusselt number depends on both the phase difference and frequency rather than on only the choice of the frequency of small amplitude modulation.

The Fractional Hamilton-Jacobi-Bellman Equation

pp. 45–56 | DOI: 10.5890/JAND.2017.03.004

M. Veretennikova1†, V. Kolokoltsov2

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Abstract

In this paper we initiate the rigorous analysis of controlled Continuous Time Random Walks (CTRWs) and their scaling limits, which paves the way to the real application of the research on CTRWs, anomalous diffusion and related processes. For the first time the convergence is proved for payoff functions of controlled scaled CTRWs and their position dependent extensions to the solution of a new pseudodifferential equation which may be called the fractional Hamilton- Jacobi-Bellman equation.

Krylov Bogoliubov Type Analysis of Variants of the Mathieu Equation

pp. 57–77 | DOI: 10.5890/JAND.2017.03.005

B. Shayak1,3†, Pranav Vyas2†

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Abstract

In this work we show that a Krylov-Bogoliubov type analysis is a powerful method for analysing variants of the Mathieu equation. We first demonstrate the technique by rederiving the results obtained by prior authors using different techniques and then apply it to a case where the system has a quasiperiodic drive (inhomogeneity) in addition to a quasiperiodic parametric term. A realistic system where such a forcing is present is an induction motor, so we adopt that as our model system to show the details of the method.

On Identically Distributed non-Volterra Cubic Stochastic Operator

pp. 79–90 | DOI: 10.5890/JAND.2017.03.006

U. U. Jamilov1, M. Ladra2

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Abstract

We introduce the notion of identically distributed strictly non- Volterra cubic stochastic operator. We show that any identically distributed strictly non-Volterra cubic stochastic operator has a unique fixed point and that such operator has the property of being regular.

Nonlinear Dynamical Modeling and Vibration Responses of An L-Shaped Beam-Mass Structure

pp. 91–104 | DOI: 10.5890/JAND.2017.03.007

Jin Wei, Dengqing Cao, Yang Yang, Wenhu Huang

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Abstract

The global modal approach is employed to obtain a set of ordinary differential equations of motion describing the nonlinear dynamics of an L-shaped beam structure in this paper. Firstly, the Lagrangian of nonlinear dynamics for the whole system is formulated. The linear partial differential equations of transverse motion are derived for each beam, along with their boundary and matching conditions. Consequently, the characteristic equation is formulated for the whole system. The natural frequencies and global mode shapes of the system are determined, and orthogonality relations of the global mode shapes are established. Then, the Lagrange's procedure is employed to obtain the nonlinear ODEs of motion for the structure with multiple- DoF. A comparison between the natural frequencies obtained by the proposed method and those from finite element method is given to illustrate the validity of our approach. Through the nonlinear ODEs presented in this article, a study on the variation of dynamic responses for the systems with different number of global modes is performed to give a suggestion of how many modes should be taken for vibration analysis of the structure.

On Stabilization and State Estimation of Impulsive Singularly Perturbed Systems via Sliding Mode Control

pp. 105–119 | DOI: 10.5890/JAND.2017.03.008

Mohamad S. Alwan†

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Abstract

This paper addresses the problems of designing a nonlinear sliding mode control (SMC) and nonlinear sliding mode observer (SMO) for a class of linear time杋nvariant (LTI) singularly perturbed systems (SPS) subject to impulsive effects. The continuous states are viewed as an interconnected (or composite) system with two-time scale (slow and fast) subsystems. The main goal is to design a SMC law through the slow reduced order subsystems to achieve closed-loop stability of the full order system. This approach in turn results in lessening some unnecessary sufficient conditions on the fast subsystem. Then, assuming that partial output measurement of the slow subsystem is available, a similar control design is adopted to estimate the states of full order SPS, where a sliding mode modification of a Luenberger observer is used.

Variation of Response Amplitude in Parametrically Driven Single Duffing Oscillator and Unidirectionally Coupled Duffing Oscillators

pp. 121–129 | DOI: 10.5890/JAND.2017.03.009

S. Rajamani† , S. Rajasekar†

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Abstract

We present our investigation on the effect of parametric force on the response amplitude in the single Duffing oscillator and unidirectionally coupled n Duffing oscillators. In the single oscillator parametric perturbation is of the form f xsinωt. Parametric perturbation induced oscillatory motion is found for values of f above a critical value. In the oscillatory motion the dominant frequency is found to be ω/2. A(ω/2), the amplitude of oscillation at the frequency ω/2, is found to vary linearly with ω. We consider unidirectionally coupled n oscillators with first oscillator alone driven by a parametric force and the other oscillators are nonlinearly or linearly coupled but one-way only. Depending upon the values of the coupling strength δ the oscillators, after first several oscillators, exhibit damped or undamped signal propagation. In the nonlinearly coupled oscillators the dominant frequency of oscillation is ω/2. In the linearly coupled system the frequency ω/2 is absent. The oscillators other than the first oscillator exhibiting oscillatory motions have frequencies ω or 2ω or both depending upon the values of the coupling strength.

JAND Volume 6, Issue 2

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Fractional Calculus Applications in Modeling and Design of Control Systems

pp.131–134 | DOI: 10.5890/JAND.2017.06.001

Cristina I. Muresan1, Piotr Ostalczyk2†, Manuel D. Ortigueira3

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Abstract

Fractional calculus represents the generalization of integration and differentiation to an arbitrary order. Since the very first occurrence of fractional differentiation more than 300 years ago, fractional calculus and research related to its possible application have deserved ever-growing attention and interest. The research community has managed to bring forward ideas and concepts that justify the importance of fractional calculus for future engineering and science discoveries. What has begun as a means to describe abnormal behaviours in viscoelasticity or diffusion, power law phenomena, long range processes or fractal structures has spread to almost all engineering fields and applied sciences. Nowadays, its use in control engineering has been gaining more and more popularity in both modeling and identification, as well as in the controller tuning.

Experimental Veriﬁcation of the Time-Fractional Diﬀusion of Methanol in Silica

pp. 135–151 | DOI: 10.5890/JAND.2017.06.002

Alexey A. Zhokh†, Peter E. Strizhak

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Abstract

Experimental study of the mass transfer kinetics for methanol in mesoporous silica is presented. Analysis of the experimental data shows that there is no good correspondence between them and corresponding solutions found according to the second Fick’s law for various pores geometries of the silica. Contrary, we show a good fit of the experimental data by a solution of the time-fractional diffusion equation with proper boundary conditions that correspond to experiment. Our results support that mass transfer in silica, which is a geometrically restricted media, may exhibit anomalous features, due to the geometrical constraints associated with randomly porous structure of a solid.

A Method for the Hankel-Norm Approximation of Fractional-Order Systems

pp. 153–171 | DOI: 10.5890/JAND.2017.06.003

Jay L. Adams†, Robert J. Veillette , Tom T. Hartley

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Abstract

A model-reduction methodology for fractional-order systems based on the Hankel-norm is presented. The methodology involves the truncation of a Laurent series associated with the fractional-order system in a transformed domain. The truncated Laurent series coefficients are used to construct a finite-order transfer function to approximate the original system. Standard model-reduction techniques are then applied to obtain a final low-order approximation. The Hankel norm of the approximation error can be specified a priori. The approximation method is applied to several fractional-order and other infinite-order systems. It is shown to be more generally applicable than standard finite-order modeling techniques.

On the Solutions of Some Boundary Value Problems for Integro-diﬀerential Inclusions of Fractional Order

pp. 173–179 | DOI: 10.5890/JAND.2017.06.004

Aurelian Cernea†

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Abstract

We study the existence of solutions for fractional integro-differential inclusions with nonlocal boundary conditions and with multi-order fractional integral conditions. We establish Filippov type existence results in the case of nonconvex set-valued maps.

Fractional Order Image Processing of Medical Images

pp. 181–191 | DOI: 10.5890/JAND.2017.06.005

Tiago Bento , Duarte Val´erio†, Pedro Teodoro, Jorge Martins

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Abstract

To perform a robot-assisted surgery of a prosthesis implantation on a patient’s femur, we may need to get the femoral head-neck orientation for the application. We can extract that information from Computed Tomography scans, using image processing. In image processing, edge detection often makes use of integer-order differentiation operators (e.g. Canny and LoG operators). This paper shows that introducing non-integer (fractional) differentiation to edge detectors (Fractional Canny, Fractional LoG, Fractional Derivative operators) can improve automatic edge detection results.

Voltage Sychronization in Arrays of Fractional-Order Energy Storage Elements

pp. 193–223 | DOI: 10.5890/JAND.2017.06.006

Tom T. Hartley†

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Abstract

In this paper, the balancing of a collection of electric energy storage devices is considered. Specific devices discussed are capacitors, supercapacitors, and batteries. The balancing problem is formulated as a complex network with dynamic nodes. The connection problem is approached from a graph theory viewpoint and solved by using homogeneous or entangled graphs. The utility of this approach is demonstrated with several examples.

Quadratic Spline Function for the Approximate Solution of an Intermediate Space-Fractional Advection Diﬀusion Equation

pp. 225–236 | DOI: 10.5890/JAND.2017.06.007

E. A. Abdel-Rehim†, M. G. Brikaa

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Abstract

The space fractional advection equation is a linear partial pseudodifferential equation with spatial fractional derivatives in space and is used to model transport at the earth surface. This equation arises when velocity variations are heavy tailed. Space fractional diffusion equation mathematically models the solutes that move through fractal media. In this paper, we are interested in finding the approximation solution of an intermediate fractional advection diffusion equation by using the quadratic spline function. The approximation solution is proved to be conditionally stable. Finally, some numerical examples are given based on this method.

The Lane - Emden Fractional Homogenous Diﬀerential Equation

pp. 237–242 | DOI: 10.5890/JAND.2017.06.008

Constantin Milici1†,Gheorghe Draganescu2

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Abstract

In this paper we introduce a nonlinear fractional differential equation of Lane-Emden type. We establish a solution which satisfies the M¨untz-Sz´asz theorem conditions in terms of power series. Particular solutions are established for different values of the parameters. A validation of our method is based on a case verified with the aid of a Maple program.

Generalization of the equations of Hermite, Legendre and Bessel for the Fractional Case

pp. 243–249 | DOI: 10.5890/JAND.2017.06.009

Constantin Milici1†, Gheorghe Draganescu2

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Abstract

In this paper we introduce and establish the solutions for the fractional Hermite, Legendre and Bessel equations. The construction of the solution is established on the basis of Muntz - Szasz theorem [19]. These new equations open new applications in the field of fractional quantum models, or to new applications in engineering.

Fractional-order State Observers for Integer-Order Linear Systems

pp. 251–264 | DOI: 10.5890/JAND.2017.06.010

Carolina Pacheco1,2 , Manuel A. Duarte-Mermoud1,2†, Norelys Aguila-Camacho1,2 , Rafael CastroLinares3

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Abstract

This paper addresses the state observation problem for linear, time-invariant, single-input single-output, integer-order systems with known parameters, using non-integer-order (fractional-order) observers. This study is aimed to establish if this is possible and if so what would it be the advantages. A novel fractional-order observer is proposed, for which the convergence of the estimation error is theoretically guaranteed, and its performance is compared to classic integer-order Luenberger state observer through simulations. According to simulation results, it is possible to establish some advantages of the proposed fractional-order observer over the classic integer-order Luenberger observer, mainly in the error convergence speed and in the attenuation of the effects of high-frequency disturbances.

Two Cases of Digraph Structures Corresponding to Minimal Positive Realisation of Fractional Continuous-Time Linear Systems of Commensurate Order

pp. 265–282 | DOI: 10.5890/JAND.2017.06.011

Konrad Andrzej MARKOWSKII†

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Abstract

The positive and minimal realisation problem for fractional continuous-time linear single-input and single-output (SISO) systems is formulated. Method based on the one-dimensional digraph for finding a positive and minimal realisation of a given proper transfer function is proposed. Two special cases of the digraph structure are given. Sufficient conditions for the existence of a positive minimal realisation of a given proper transfer function systems are established. The algorithm for computation of a positive minimal realisation is proposed and illustrated with a numerical example. The algorithm is based on a parallel computing method to gain needed speed and computational power for such a solution.

Arrows of Times, Non Integer Operators, Self-Similar Structures, Zeta Functions and Riemann Hypothesis: a Synthetic Categorical Approach

pp. 283–301 | DOI: 10.5890/JAND.2017.06.012

Alain Le Mehaute†, Philippe Riot

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Abstract

The authors have previously reported the existence of a morphism between the Riemann zeta function and the “Cole and Cole” canonical transfer functions observed in dielectric relaxation, electrochemistry, mechanics and electromagnetism. The link with self-similar structures has been addressed for a long time and likewise the discovered of the incompleteness which may be attached to any dynamics controlled by non-integer derivative operators. Furthermore it was already shown that the Riemann Hypothesis can be associated with a transition of an order parameter given by the geometric phase attached to the fractional operators. The aim of this note is to show that all these properties have a generic basis in category theory. The highlighting of the incompleteness of non-integer operators considered as critical by some authors is relevant, but the use of the morphism with zeta function reduces the operational impact of this issue without limited its epistemological consequences.

Derivation of Analytical Inverse Laplace Transform for Fractional Order Integrator

pp. 303–314 | DOI: 10.5890/JAND.2017.06.013

Ali Yuce†, Nusret Tan

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Abstract

There is considerable interest in the study of fractional order derivative integrator but obtaining analytical impulse and step responses is a difficult problem. Therefore all methods reported on to date use approximations for the fractional derivative/integrator both for analytical based computations and more relevantly in simulation studies. In this paper, an analytical formula is first derived for the inverse Laplace transform of fractional order integrator, $1/s^\alpha$ where $\alpha \in R$ and $0 < \alpha < 1$ using Stirling’s formula and Gamma function. Then, the analytical step response of fractional integrator has been computed from the derived impulse response of $1/s^\alpha$ . The obtained analytical formulas for impulse and step responses of fractional order integrator are exact results except the very small error due to the neglected terms of Stirling’s series. The results are compared with some well known integer order approximation methods and Grunwald-Letnikov (GL) approximation technique. It has been shown via numerical examples that the presented method is very successful according to other methods.

JAND Volume 6, Issue 3

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Chaotic Dynamics of Colpitts Oscillator Under Control of MEMS Feedback

pp. 315-332 | DOI:10.5890/JAND.2017.09.001

Saumitra Mishra, R. D. S. Yadava†

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Abstract

The nonlinear dynamics of Colpitts oscillator under control of MEMS varactor in feedback connectivity has been analyzed with objectives for generation and control of high frequency chaotic signals. The feedback signal derived from the capacitive divider in the standard Colpitts oscillator is modified by the MEMS varactor response mirrored by a voltage-controlled current multiplier. The latter implements MEMS capacitance multiplication and serves as a control parameter. The effects of voltage nonlinearity of the MEMS capacitance and the capacitance multiplication factor () have been analyzed by employing Lyapunov exponent, bifurcation diagram, phase portrait and Fourier transform methods. The modified feedback network facilitates high frequency chaos generation due to frequency doubling and high pass filtering effects of the MEMS capacitance. The latter emphasizes high frequency generation and attenuates lower frequencies. The variation of capacitance multiplication factor allows systematic changes in the qualitative nature of oscillator dynamics from a stable low frequency noisy state to Hopf bifurcation to period doubling/ tripling to chaos generation. The analysis suggests new MEMS based tuning and control of chaotic Colpitts oscillations.

Controllability of Nonlinear Neutral Fractional Integrodiﬀerential Systems with Inﬁnite Delay

pp. 333-344 | DOI:10.5890/JAND.2017.09.002

K. Balachandran , S. Divya†

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Abstract

In this paper, we establish sufficient conditions for the controllability of neutral fractional integrodifferential systems with infinite delay and infinite neutral fractional systems with implicit derivative. Fixed point approaches are employed for achieving the required results. Examples are provided to illustrate the efficiency of the results.

Nonlinear Dynamics of Laminar-Turbulent Transition in Generalized 3D Kolmogorov Problem for Incompressible Viscous Fluid at Symmetric Solution Subset

pp. 345-353 | DOI:10.5890/JAND.2017.09.003

Nikolai M. Evstigneev , Nikolai A. Magnitskii†

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Abstract

A three dimensional Kolmogorov problem with extended forcing term for Navier-Stokes equations is considered. The Galerkin-Fourier method is applied and the symmetry preserving subset of solutions is considered. The bifurcation patterns are revealed through the numerical analysis of eigenvalues of the linearized perturbed system from the analytical main stationary solution and through the analysis of phase space trajectories that the system generates. It was found that the initial stage of laminar-turbulent transition undergoes pitchfork bifurcation, through which the system can either go through the series of cycle cascades or through continuous tori bifurcations in accordance with the FShM scenario.

On Some Chaotic Aspects and Center Manifold Reduction of ACT Nonlinear System

pp. 355-367 | DOI:10.5890/JAND.2017.09.004

A. Roy Chowdhury†, A. Ray, P. Saha

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Abstract

Chaotic properties of a new nonlinear dynamical system, namely ACT system, are analyzed through a detailed analysis of its bifurcation diagram, attractor formation, bi-parametric Lyapunov plots. Due to the presence of many parameters in the system it shows a very rich structure in all respects. Details of stability analysis and its relation to the corresponding center manifold reduction are also studied.

Steering Control for a Rigid Body with two Torque Actuators using Adaptive Back Stepping

pp. 369-377 | DOI:10.5890/JAND.2017.09.005

Abdul Baseer Satti†

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Abstract

This paper presents a simple steering control algorithm for a rigid body model, which is a famous example of non-holonomic control systems with drift. The controllability Lie Algebra of a rigid body model contains Lie brackets of depth two. We propose a back-stepping-based adaptive controller design under the strict-feedback form. We analyze two cases for continuous steering. In the first case, the parameters of the model are assumed to be known while in the second case these are estimated by considering them unknown. This approach does not necessitate the conversion of the system model into a “chained form”, and thus does not rely on any special transformation techniques. The practical effectiveness of the controller is illustrated by numerical simulations and graceful stabilization.

Chaos Synchronization of the Fractional Rucklidge System based on New Adomian Polynomials

pp. 379-385 | DOI:10.5890/JAND.2017.09.006

Guo-Cheng Wu1 , Dumitru Baleanu2,3†, Lan-Lan Huang4

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The fractional Rucklidge system is a new kind of chaotic models which hold the feature of memory effects and can depict the long history interactions. A numerical formula is proposed by use of the fast Adomian polynomials. Chaotic behavior are discussed and the Poincare sections are given for various fractional cases. It’s also applied in chaos synchronization of the fractional system.

Fourth Order Runge-Kutta Method for Solving First-order Fully Fuzzy Diﬀerential Equations Under Strongly Generalized H-diﬀerentiability

pp. 387-406 | DOI:10.5890/JAND.2017.09.007

D. Vivek†, K. Kanagarajan , S. Indirakumar

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Abstract

In this paper we use fourth order Runge-Kutta method for solving fully fuzzy differential equations of the form y_(t) = a⊗y(t), y(0)= y0, t ∈ [0,T] under strongly generalized H-differentiability. The algorithm used here are based on cross product of two fuzzy numbers. Using cross product we can divide fully fuzzy differential equation (FFDE) into four different cases. We apply the results to a particular case of FFDE. The Convergence of this method is discussed and numerical examples are given to verify the reliability of this method.

Blow-up of Solutions to Reaction-diﬀusion System with Nonstandard Growth Conditions

pp. 407-425 | DOI:10.5890/JAND.2017.09.008

Arumugam Gurusamy1†, Krishnan Balachandran2,†

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Abstract

This paper is concerned with the existence and blow-up of solutions of reaction diffusion system with p(x)− growth conditions. The existence of weak solution is proved by using the Galerkin method. The blow-up of solutions is established by applying the method of comparison with suitable blow-up of self-similar subsolutions. Finally the theoretical results are illustrated by numerical examples.

"Universal" Fitting Function for Complex Systems: Case of the Short Samplings

pp. 427-443 | DOI:10.5890/JAND.2017.09.009

Raoul R. Nigmatullin†, Wei Zhang*, Domenico Striccoli+

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Abstract

The authors suggest an effective scheme for quantitative description of complex systems, when the number of measurements is relatively small. It has a great importance for quantitative description of expensive and rare experiments when the volume of the sampling is small. They proposed a simple theory that is based on the previous results associated with conception of the intermediate model (IM). The previous results can be generalized and applicable for description of complex systems with short samplings when the influence of the uncontrollable factors becomes significant. As an example, we consider the description of acoustic signals recorded from turbine bearings. It can be proved that the real signals have self-similar (fractal) properties. It helps to compress the length of the initial files (number of data points N = 44100) at least in 88 times and reduced essentially the number of the fitting parameters. The obtained results can be used for diagnosis of different defects during the process of technical exploitation. Each failure has own acoustic “picture” i.e. the amplitude-frequency response (AFR) expressed in terms of the generalized Prony spectrum (GPS). This AFR can be used as a “specific” fingerprint for identification of the unexpected failure and preventing a possible breakdown.

JAND Volume 6, Issue 4

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A Class of Nonlocal Fractional Evolution Equations and Optimal Controls

pp. 445-463 | DOI:10.5890/JAND.2017.12.001

Ravi P. Agarwal1†, Asma2, Vasile Lupulescu3, Donal O’Regan4

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In this paper we study the existence of solutions for a class of semilinear fractional differential equations with nonlocal conditions and involving abstract Volterra operators. The existence of an optimal solution for a class of fractional control problem involving Caputo fractional derivatives is obtained. An example is presented to illustrate our main result.

Boundary Controllability of Delay Differential Systems of Fractional Order with Nonlocal Condition

pp. 465-472 | DOI:10.5890/JAND.2017.12.002

Kamalendra Kumar1†, Rakesh Kumar2

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Abstract

Sufficient conditions for boundary controllability of time varying delay differential systems of fractional order with nonlocal condition in Banach space are established. The results are obtained by using fixed point theorems. An example is provided to illustrate our results.

Conservation Laws by using the Multiplier Method for a Fifth-Order Kdv Equation with Time-Dependent Coefficients and Linear Damping

pp. 473-478 | DOI:10.5890/JAND.2017.12.003

M.S. Bruzon†, M.L. Gandarias, R. de la Rosa

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Abstract

In this paper we consider a family of fifth-order Korteweg-de Vries equations with time-dependent coefficients and linear damping term. By using the multiplier method of Anco and Bluman we determine all the low order conservation laws.

Existence of Solutions for Impulsive Fractional q-difference Equations with Nonlocal Condition

pp. 479-486 | DOI:10.5890/JAND.2017.12.004

D. Vivek†, K. Kanagarajan, S. Harikrishnan

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Abstract

This paper is devoted to proving the existence of solutions to frac- tional impulsive q-difference equations. An approach based on the Schaefer’s fixed point theorem to prove existence of the solution is presented. There is almost no work on the existence results for im- pulsive fractional q-difference equations. The main aim of this paper is to close this gap.

Weakly Nonlinear and Nonlinear Magneto-convection under Thermal Modulation

pp. 487-508 | DOI:10.5890/JAND.2017.12.005

Palle Kiran1†, B.S. Bhadauria2, Y. Narasimhulu1

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Both oscillatory and chaotic convection are studied using weakly non- linear and nonlinear theories. A weakly nonlinear analysis was em- ployed to derive Complex Ginzburg-Landau amplitude equation. The time dependent temperatures of the plates are considered in three ways, out of phase, lower plate and in phase modulation. The first two temperature profiles show impact on heat and mass transfer and the dynamics of the problem. It is also found that in-phase tempera- ture modulation has negligible effect; while out of phase modulation and only lower plate modulation have significant effects on heat and mass transport. Heat mass transfer is measured in the system in terms of the Nusselt and Sherwood numbers. Heat mass transfer be- comes rapid on either increasing Rs,Pr, &lamda, &delta or decreasing Q, &Gamma, &epsilon, &Omega. Further, the Lorentz model has been simplified under modulation ef- fect, and it is observed that, the chaotic nature of the system may altered with modulation. Unstable solution for OPM, stable solu- tions for IPM, LBMO is found depending on the suitable values of modulation parameters.

Influence of Sampling Rate and Discretization Methods in the Parameter Identification of Systems with Hysteresis

pp. 509-520 | DOI:10.5890/JAND.2017.12.006

W.R. Lacerda Junior†, S.A.M. Martins, E.G. Nepomuceno

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Abstract

Hysteresis is a nonlinear behaviour, which has been considered very hard to model. It is commonly found in actuators and sensors, involving quasi-static memory effects between input and output variables. Usually, continuous time models are used to model this feature. However, polynomial NARX model has come up as an alternative to model this behaviour. Since NARX models are discrete-time models, it is important to verify how the sampling rate interfere in obtaining the mathematical model. Further, frequently continuous-time models are used as a bench test, to generate data for identification of several nonlinear behaviour, including hysteresis. This paper investigates how the sampling rate and discretization methods affects the parameter identification of a NARX model for a system with hysteresis. Improved Euler and fourth order Runge-Kutta methods are applied in a Bouc-Wen model for a magneto-rheological damper, which is used as a system to be identified by a NARX model, considering the above mentioned scenario. Least-square based technique is used in this work to estimate model parameters.

A Complex Variable Method to Predict a Range of Arbitrary Shape Ballistic Projectiles

pp. 521-530 | DOI:10.5890/JAND.2017.12.007

Jimmie C. Oxley1, James L. Smith1, Sayavur I. Bakhtiyarov2†, Philipp M. Baldovi2

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Abstract

Ballistic projectiles (fragments, missiles) thrown due to the explosions can be a greater hazard than the blast wave. Safe distances from the blast can be readily calculated since it falls off as the cube root of distance. Far more complex is predicting how far fragments may be thrown. This work develops an engineering method to predict a range of explosive ballistic projectiles (EBPs) of arbitrary shapes. Incorporating the numerical solution of the equations of the dynamic motion of projectile with a complex variable method (“linearization of single-bonded area”) the velocities and the trajectories of arbitrary shape EBPs have been determined. The results are compared to previously developed model predictions and the fragment recovery tests results.

Nonlinear analysis of the yaw motion of a mariner vehicle under PD$^μ$ control

pp. 531-545 | DOI:10.5890/JAND.2019.12.008

C. A. Kitio Kwuimy, C. Nataraj†

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Abstract

This paper considers a mariner under PD$^μ$ controller and analyses the effects of controller parameters on the yaw rate by using the Nomoto model. The Nomoto model describing the time evolution of the yaw rate of the steering dynamics of a mariner is reduced to an asymmetric Duffing oscillator with fractional order derivative. Under the approximation of calm water, the steady behavior of the mariner shows an “imperfect” supercritical pitchfork bifurcation. Region of safe behavior is identified and strategy to reduce the yaw rate by an appropriated selection of controller parameters are discussed. The frequency analysis of the mariner shows the prominence of hysteresis is reduced for small order of the fractional derivative as well as the amplitude of the yaw rate. Evidence of chaotic response is illustrated using robust chaotic indicators such as the Lyapunov exponent and the fast Fourier transform.

Reaction-diffusion Dynamics and Biological Pattern Formation

pp. 547-564 | DOI:10.5890/JAND.2017.12.009

Kishore Dutta†

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Abstract

The spontaneous formation of a wide variety of natural patterns with different shapes and symmetries in many physical and biological systems is one of the deep mysteries in science. This article describes the physical principles underlying the formation of various intriguing spatio-temporal patterns in Nature with special emphasis on some biological structures. We discuss how the spontaneous symmetry breaking due to diffusion driven instability in the reaction dynamics lead to the emergence of such complicated natural patterns. The mechanism of the formation of various animal coat patterns is explained via the Turing-type reaction-diffusion models.

JAND Volume 7, Issue 1

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Explicit Solutions and Conservation Laws of a (2+1)-dimensional KP-Joseph-Egri Equation with Power Law Nonlinearity

pp. 1-9 | DOI: 10.5890/JAND.2018.03.001

Chaudry Masood Khalique, Khadijo Rashid Adem

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Abstract

This paper obtains solutions of the (2+1)-dimensional Kadomtsov Petviashivilli-Joseph-Egriequation with power law nonlinearity. This equation is the Joseph-Egri equation formulated in the KP sense. The Lie group analysis and the Exp-function method are used to carry out the integration of this equation. The solutions obtained are solitary waves. Moreover, the conservation laws are constructed by using the multiplier method.

Synchronization Dynamics of Modified Relay-coupled Chaotic Systems

pp. 11-24 | DOI: 10.5890/JAND.2018.03.002

Patrick Louodop, Elie B. Megam Ngouonkadi, Paulsamy Muruganandam, Hilda A. Cerdeira

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Abstract

In this manuscript, we study the dynamics of a modiﬁed relay-coupled chaotic systems. The modiﬁcation consists on the fact that the relay unit is modeled to lead the entire network to a desired dynamics. Then we achieve ﬁnite-time synchronization indirectly through a linear combination of the three systems. Further, we consider the existence of a switch on time of the coupling from the relay unit to the outer systems. It appears some interesting behaviors such as bifurcations, alternation of crisis and phases transitions when varied the switch on time. An open result is also found. In our scheme and for the selected changeable initial conditions, it seems that the appearance or disappearance of coexistence of attractors is linked to the type of synchronization we are dealing with. Mathematical demonstrations are given to sustain our theory while numerical simulations show its eﬀectiveness.

Detection of Nodal Snap-through Instability in Reticulated Shell Structures Using Tilt Sensing of Members

pp. 25-44 | DOI: 10.5890/JAND.2018.03.003

Guirong Yan, Qiuhua Duan, Tiantian Li, Genda Chen, Xugang Hua, Ruoqiang Feng

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Abstract

Reticulated shell structures are usually built as roofs for venues where hundreds or even thousands of people assemble. Failure of this type of structure may endanger the safety of many people. This type of structure can fail due to either material failure or loss of stability (instability). Previous research on structural health monitoring has been focused on the detection of material failure. This study is to timely detect one type of instability in reticulated shell structures, nodal snap-through (NST) instability, to prevent a local NST instability from progressing into an overall structural failure. When an NST instability occurs, a joint and its connected members snap through to their new equilibrium positions, leading the geometric shape in the local area to change signiﬁcantly. If the displacements at joints are able to be measured, instability can be easily detected from the deformed shape. However, it is very diﬃcult to measure displacements for such large structures located at a high elevation, if not impossible. In this study, an approach to detecting the NST instability will be developed by identifying the change in tilting angles of members, which can be easily measured and essentially reﬂects the change in geometric shape caused by instability. By applying this approach, a local NST instability can be detected at an early stage, and measures can be taken to prevent a local instability from progressing into a catastrophic structural failure. The eﬀectiveness of this approach has been validated by numerical simulations on a large-scale reticulated shell structure.

Detecting Chaos and Nonlinear Dynamics in Sao Paulo Stock Exchange Index Returns (IBOVESPA)

pp. 45-58 | DOI: 10.5890/JAND.2018.03.004

Ferrouhi El Mehdi

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Abstract

The market eﬃciency theory states that stock returns on eﬃcient markets are random, and then, we cannot predict accurately their future evolutions. Chaos theory is considered as a remedy for this insuﬃciency since the evolution of chaotic systems appears random but follows precise rules. After the application of detection’s tools of deterministic chaos to IBOVESPA returns, we obtained a fractal dimension equal to 2.5, the convergence of Lyapunov exponent towards positive values and ﬁrst Lyapunov exponent characterized by its positivity. Results obtained after the application of the Nearest Neighbors method allows us to conclude IBOVESPA returns are characterized by sensitivity to initial conditions and are therefore chaotic.

Numerical Approach for the Controllability of Composite Fractional Dynamical Systems

pp. 59-72 | DOI: 10.5890/JAND.2018.03.005

Venkatesan Govindaraj, Krishnan Balachandran, Raju K George

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Abstract

This paper deals with the numerical computations for the controllability of linear and nonlinear composite fractional dynamical systems. The goal is to compute a control state that drives the system from a prescribed initial state to described ﬁnal state in a large enough controllability time. We address the controllability results using Grammian matrix and iterative technique. Some examples are provided to illustrate the results.

Correction to the Unknown Input Observer Design for Linear Fractional-Order Time-Delay Systems & a New Enhanced LMI Condition

pp. 73-79 | DOI: 10.5890/JAND.2018.03.006

Y. Boukal, M. Darouach, M.Zasadzinski, N.E. Radhy

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Abstract

This note considers the work entitled “Unknown Input Observer Design for Linear Fractional-Order Time-Delay Systems”. In the above paper [1], the authors gave the existence conditions of such observer, then based on the fractional order Lyapunov stability approach, a suﬃcient condition for the asymptotic stability of the estimation error have given in a linear matrix inequality (LMI) formulation which is incorrect. In this note, we give the correction of Theorem 7. The new proposed Theorem can be applied to a large kind of delayed fractional-order-system when the delay is time varying or constant, while the above mentioned paper consider only a constant time delay. The proof is based on the diﬀusive representation of the fractionalorder derivative and the indirect Lyapunov approach proposed in [2].

Models of Bone Metastases and Therapy using Fractional Derivatives

pp. 81-94 | DOI: 10.5890/JAND.2018.03.007

Luiz Filipe Christ, Duarte Val´erio, Rui Moura Coelho, Susana Vinga

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Abstract

The adult skeleton is a highly specialised organ that undergoes constant remodelling over time. Bone metastasis aﬀect the dynamic of this process. For this reason, the study of this dynamic model is essential to the development of better therapies for the disease. Anomalous diﬀusion phenomena are often found in biological systems, and can be modelled using fractional order derivatives. Consequently, this paper modiﬁes models presented in the literature, consisting of diﬀerential equations of order one, checking how their behaviour changes when fractional derivatives are used instead. Results for both local and one-dimensional models of healthy bone tissue and of tumourous bone tissue have the characteristics expected from the literature, with higher fractional orders leading to a faster, more oscillatory system whereas lower orders have a more damped behaviour.

Mathematical Models of Nonlinear Uniform Consensus II

pp. 95-104 | DOI: 10.5890/JAND.2018.03.008

Mansoor Saburov, Khikmat Saburov

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Abstract

This paper is a continuation of our previous studies on nonlinear consensus. We have considered a nonlinear protocol for a structured time-invariant and synchronous multi-agent system. We present an opinion sharing dynamics of the multi-agent system as a trajectory of a polynomial stochastic operator associated with a stochastic multidimensional hyper-matrix. We provide a criterion for a uniform consensus in the multi-agent system. Particularly, the uniform consensus is achieved in the multi-agent system if all entries of the stochastic multidimensional hyper-matrix are positive. Some numerical results are also presented to support our theoretical results.

Dynamics of Three and Four Non-identical Josephson Junctions

pp. 105-110 | DOI: 10.5890/JAND.2018.03.009

A.P. Kuznetsov, I.R. Sataev, Yu.V. Sedova

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Abstract

Dynamics of chains of three and four coupled non-identical Josephson junctions is considered. Synchronization eﬀects are discussed including resonance Arnold web formation on the base of tori of diﬀerent dimensions.