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- Study on Stochastic Quasi-Linear Partial Differential Equations of Evolution

pp. 1-11 | DOI: 10.5890/DNC.2020.03.001

A. Anguraj, K. Ramkumar

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In this article, the existence and uniqueness of local mild solution of a stochastic counterpart of TosioKato’s Quasi-linear partial differential equation with additive cylindrical wiener process in a separable Hilbert space is established using contraction mapping principle.

- Bernoulli Mapping with Holeanda Saddle-Node Scenario of the Birth of Hyperbolic Smale–Williams Attractor

pp. 13-26 | DOI: 10.5890/DNC.2020.03.002

OlgaB. Isaeva, Igor R. Sataev1

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One-dimensional Bernoulli mapping with hole is suggested to describe the regularities of the appearance of a chaotic set under the saddle-node scenario of the birth of the Smale–Williams hyperbolic attractor. In such a mapping, a non-trivial chaotic set (with non-zero Hausdorff dimension) arises in the general case as a result of a cascade of period-adding bifurcations characterized by geometric scaling both in the phase space and in the parameter space. Numerical analysis of the behavior of models demonstrating the saddle-node scenario of birth of a hyperbolic chaotic Smale– Williams attractor shows that these regularities are preserved in the case of multi dimensional systems. Limits of applicability of the approximate 1D model are discussed.

- The Thomas Attractor with and without Delay: Complex Dynamics to Amplitude Death

pp. 27-45 | DOI: 10.5890/DNC.2020.03.003

Brayden McDonald, S. Roy Choudhury

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Bifurcations in the Thomas cyclic system leading from simple dynamics into chaotic regimes are considered. In particular, the existence of only one non-trivial ﬁxed point of the system in parameter space implies that this point attractor may only be destabilized via a Hopf bifurcation as the single system parameter is varied. Saddle-node, transcritical and pitchfork bifurcations are precluded. The periodic orbit immediately following the Hopf bifurcation is constructed analytically by the method of multiple scales, and its stability is determined from the resulting normal form and veriﬁed by numerical simulations. The dynamically rich range of parameters past the Hopf bifurcation is next systematically explored. In particular, the period-doubling sequences there are found to be more complex than noted previously, and include period-three-like windows for instance. As the system parameter is decreased below these period-incrementing bifurcations, various additional features of the subsequent crises are also carefully tracked. Finally, we consider the effect of delay on the system,leading to the suppression of both the Hopf bifurcation as well as all of the subsequent complex dynamics. In modern terminology, this is an example of Amplitude Death,rather than Oscillation Death, as the complex system dynamics is quenched, with all of the variables settling to a ﬁxed pointof the original system.

- Existence of Nonoscillatory Solutions for Mixed Neutral Fractional Differential Equation

pp. 47-61 | DOI: 10.5890/DNC.2020.03.004

Velu Muthulakshmi, Subramani Pavithra

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In this paper, we establish some sufﬁcient conditions for the existence of nonoscillatory solution for a class of mixed neutral fractional differential equations with Liouville fractional derivative of order α ≥0 on the halfaxis. Our results generalize some of the existing results in the literature. Some examples are given to illustrate our results.

- Universal Behavior of the Convergence to the Stationary State for a Tangent Bifurcation in the Logistic Map

pp. 63-70 | DOI: 10.5890/DNC.2020.03.005

Joelson D. V. Hermes, Fl´avio Heleno Graciano, Edson D. Leonel

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The scaling formalism is applied to understand and describe the evolution towards the equilibrium at and near at a tangent bifurcation in the logistic map. At the bifurcation the convergence to the steady state is described by a homogeneous function leading to a set of critical exponents. Near the bifurcation the convergence is rather exponential whose relaxation time is given by a power law. We use two different approaches to obtain the critical exponents: (1) a phenomenological investigation based on three scaling hypotheses leading to a scaling law relating three critical exponents and; (2) an approximation that transforms the recurrence equations in a differential equation which is solved under appropriate conditions given analytically the scaling exponents. The numerical results give support for the theoretical approach.

- Steady-State and Dynamic Characteristics of Water-lubricated Rubber Bearings under Two Sets of Reynolds Boundary Conditions

pp. 71-82 | DOI: 10.5890/DNC.2020.03.006

Gang Liu, Ming Li

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The lubrication characteristics of bearings are signiﬁcantly inﬂuenced by boundary conditions. The double Reynolds boundary conditions (DRBCs) are introduced to analyze the hydrodynamic lubrication characteristics of water-lubricated rubber bearings for turbulent ﬂows, considering the elasticity of rubber liner. Differences in its steady state and dynamic characteristics arising from the DRBCs and the Reynolds boundary conditions (RBCs) are discussed based on the ﬁnite difference method. The results show that the water-ﬁlm reformation boundary is signiﬁcantly different between the two sets of conditions, however,the load capacity of the bearing is only slightly different. The attitude angle and friction are greater for DRBCs than for RBCs. In the horizontal direction, direct stiffness and direct damping coefﬁcients are larger for DRBCs than for RBCs, but in the vertical direction the opposite result holds.

- Effect of Fractional-Order on the Dynamic of two Mutually Coupled van der Pol Oscillators: Hubs, Multistability and its Control

pp. 83-98 | DOI: 10.5890/DNC.2020.03.007

Ngo Mouelas Ad`ele1, Kammogne Soup Tewa Alain, Kengne Romanic, Fotsin Hilaire Bertrand1, EssimbiZobo Bernard

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This paper presents a novel approach to analyze the dynamic effect of the fractional-order derivative of the two mutually coupled van der Pol oscillators. The stability analysis is presented by two complementary phase diagrams: the isospike diagrams and the two Lyapunov exponent spectra. These diagrams reveal precisely the Hubs, spirals bifurcation and chaos when the derivative order is ﬁxed at q = 0.95. In addition, when the fractional-order is set as a control parameter, various methods for detecting chaos including bifurcation diagrams, spectrum of largest Lyapunov exponentare exploited to establish the connection between the system parameters and various complicated dynamics. A transition was also observed between a desynchronized state and a multistability situation. These diagrams displayed the coexistence of four disconnected attractors (two symmetric). We study the basins of attraction of the system in the multi stability regime which thereby reveal the coexistence of attractors in the systems when the fractional-order derivative is taken as a function of initials conditions. Based on the parametric control, we have controlled this striking phenomenon in the system. Finally, the hardware circuit is implemented and the results are found to be in good agreement with the numerical investigations.

- Complex Dynamicsofan Exploited Prey-PredatorModel with NonlinearPrey Refuge

pp. 99-116 | DOI: 10.5890/DNC.2020.03.008

Md. Manarul Haque, Sahabuddin Sarwardi

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In this paper we study the renewable resources of a prey-predator system with prey refuge and nonlinear harvesting. Taxation is imposed to prevent over exploitations to maintain ecological balance. The steady state of the system are determined and various dynamical behavior are discussed in its steady states under certain parametric conditions. The boundedness, feasibility of interior equilibria, bionomic equilibrium have been studied. The main observation is that the taxation plays an important role in regulating the dynamics of the present system. Moreover the variation of the taxation change the system from periodic behaviors to chaos. Some numerical illustration are given in order to support of our analytical and theoretical ﬁndings.

- The Dynamical Behaviorof a Two Degrees of Freedom Oblique Impact System

pp. 117-139 | DOI: 10.5890/DNC.2020.03.009

Xiaowei Tang, Xilin Fu, Xiaohui Sun

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The oblique impact phenomena is quite common in practical engineering. In this paper, by using the theory of discontinuous dynamical systems, we studied the complex dynamics behaviors of a two degrees of freedom oblique impact system. We can see that the dynamics of oblique impact is more complex than that of the direct impact. The occurrence or disappearance conditions of sticking motion and grazing motion on the separation boundaries are given in Section 3. The conditions here are necessary and sufﬁcient, which generate better results than those obtained with only sufﬁcient conditions. The results appropriately interpret the physical phenomenon of this oblique impact system, hence validate our conclusions. As a supplement, we also give the analytic conditions of the existence of periodic motions. Numerical simulations for sticking motion and grazing motion are presented at last.

- Unsteady Magneto hydrodynamic Boundary Layer Flow towards a Heated Porous Stretching Surface with Thermal Radiation and Heat Source/Sink Effects

pp. 141-151 | DOI: 10.5890/DNC.2020.03.010

Santosh Chaudhary, Susheela Chaudhary, Sawai Singh

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Mathematical model of unsteady boundary layer ﬂow and heat transfer is explored for analyzing the study of inﬂuence of thermal radiation on incompressible viscous electrically conducting ﬂuid over continuous stretching surface embedded in a porous medium in the presence of heat source/sink. The scope of inﬂuencing parameters that describing phenomenon are determined and governing time dependent boundary layer equations are transformed to ordinary differential equations by using appropriate similarity transformation. Numerical computation of the problem was carried out by shooting iteration technique together with Runge-Kutta fourth order integration scheme. Effects of unsteadiness parameter, permeability parameter, magnetic parameter, thermal radiation parameter, Prandtl number and heat source/sink parameter on velocity and temperature proﬁles are computed and illustrated graphically, where as local skin friction coefﬁcient and local Nusselt number are represented numerically through tables. In nonmagnetic ﬂow condition the result is found in concordance with earlier investigations.

- Fluid Flow and Solute Transfer in a Permeable Tube with Inﬂuence of Slip Velocity

pp. 153-166 | DOI: 10.5890/DNC.2020.03.011

M. Varunkumar, P. Muthu

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In this paper,the inﬂuence of slip velocity on the ﬂuid ﬂow and solute transfer in a tube with permeable boundary is studied as a mathematical model for blood ﬂow in glomerular capillaries. The viscous incompressible ﬂuid ﬂow across the permeable tube wall, as a result of differences in both hydrostatic and osmotic pressure, is considered. The solutions of differential equations governing the ﬂuid ﬂow and solute transfer are obtained using analyticaland Crank-Nicolsontype numericalmethods. It is observedthat the effect of slip on the hydrostatic and osmotic pressures, velocity proﬁles, concentration proﬁle, solute mass ﬂux and total solute clearance is signiﬁcantand the results are presented graphically.

- Noise-induced Intermittent Oscillation Death in a Synergetic Model

pp. 167-172 | DOI: 10.5890/DNC.2020.03.012

R.Jaimes-Re´ategui, D.A.Magall´on-Garc´ıa1, A.Gallegos, G.Huerta-Cuellar1, J.H.Garc´ıa-L´opez, A. N. Pisarchik

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We study noise-induced intermittency in asynergetic model of two coupled oscillators with asymmetric nonlinear coupling. This model was previously used to simulate visual perception of ambiguous images. We show that additive noise induces preferencefor one of the coexisting unstable steady states. When the noise intensity exceeds a certain threshold value,the oscillations of one of the coupled subsystems are interrupted during some time intervals, resulting in intermittent oscillation death, while another subsystem exhibitsnoisy oscillationsin the vicinity of an unstable ﬁxed point.