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- Hamiltonian Perturbation Theory on a Lie Algebra. Application to a non-autonomous Symmetric Top

pp. 347-367 | DOI: 10.5890/DNC.2021.09.001

Lorenzo Valvo, Michel Vittot

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We propose a perturbation algorithm for Hamiltonian systems on a Lie algebra V, so that it can be applied to non-canonical Hamiltonian systems. Given a Hamiltonian system that preserves a subalgebra B of V, when we add a perturbation the subalgebra B will no longer be preserved. We show how to transform the perturbed dynamical system to preserve B up to terms quadratic in the perturbation. We apply this method to study the dynamics of a non-autonomous symmetric Rigid Body. In this example our algebraic transform plays the role of Iterative Lemma in the proof of a KAM-like statement.

- Partial Strong Stabilization of Semi-Linear Systems and Robustness of Optimal Control

pp. 369-380 | DOI: 10.5890/DNC.2021.09.002

A. El Alami1, M. Chqondi, Y. Akdim

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This work addresses the idea of optimal stabilization, namely robustness of optimal stabilization with nonlinearity Lipschitz of distributed semilinear systems using bounded control. This problem is treated under the condition of the unbounded operator, we show that the system is stable once the exact observability assumption is executed together with a Lipschitz property of the nonlinear operator. The concept of bounded control is also investigated in realistic domain. The stabilizing feedback is characterized by the minimization of the problem of cost. We also give different applications to parabolic and hyperbolic equations.

- Analysis on Dynamics of Delayed Intraguild Predation Model with Ratio-Dependent Functional Response

pp. 381-396 | DOI: 10.5890/DNC.2021.09.003

S. Magudeeswaran, K. Sathiyanathan, R. Sivasamy, S.Vinoth, M. Sivabalan

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The aim of this work is to analyze the dynamics of intraguild predation model with ratio-dependent functional response and time-delay. We examine the conditions for local stability and existence of Hopf-bifurcation. Also, the condition for global stability is established by using proper Lyapanov function. Finally numerical simulations are given to verify the proposed theoretical results and the system investigate through graphical illustrations.

- A Note on Existence of Global Solutions for Impulsive Functional Integrodifferential Systems

pp. 397-407 | DOI: 10.5890/DNC.2021.09.004

C. Dineshkumar, R. Udhayakumar

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In our manuscript, we research the existence of global solutions for a class of impulsive abstract functional integrodifferential systems with nonlocal conditions. We proved our outcomes by utilizing the Leray-Schauder’s Alternative fixed point theorem. Lastly, a model is presented for illustration of theory.

- Mathematical Model of HBV/HCV Co-Infection

pp. 409-424 | DOI: 10.5890/DNC.2021.09.005

Nita H Shah, Nisha Sheoran, Ekta Jayswal

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The co-infection of hepatitis B (HBV) and hepatitis C (HCV) virus is a complex clinical entity that has an estimated worldwide prevalence of 1–15%. In this paper HBV/HCV co-infection is modelled mathematically through the set of deterministic non-linear differential equations. This dynamical system has four equilibrium points i.e. disease-free, co-infection free, HCV free and endemic point. Reproduction number is computed for endemic equilibria. Local stability for all the equilibrium point is proved using Routh-Hurwitz criterion. Global stability is also studied for all the equilibria. The sensitivity analysis of relevant parameters in reproduction number is analyzed to see the effect of each parameter in disease spread.

- Description of the Set of Strictly Regular Quadratic Bistochastic Operators and Examples

pp. 425-433 | DOI: 10.5890/DNC.2021.09.006

Mirmukhsin Makhmudov

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The present paper focuses on the dynamical systems of the quadratic bistochastic operators (QBO) on the standard simplex. In the paper, we show the character of connection of the dynamical systems of a bistochastic operator with the dynamical systems of the extreme bistochastic operators. In addition, we prove that almost all quadratic bistochastic operators are strictly regular and give a description of the strictly regular quadratic bistochastic operators in the convex polytope QBO. Furthermore, the density of the set of strictly regular QBO in the set of QBO is proved and nontrivial examples of strictly regular bistochastic operators are given.

- Dynamics of Two Delays Differential Equations Model of HIV Pathogenesis with Absorption and Saturation Incidence

pp. 435-444 | DOI: 10.5890/DNC.2021.09.007

Vinoth Sivakumar, Jayakumar Thippan, Prasantha Bharathi Dhandapani

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In this paper, we proposed and analyzed time delays HIV pathogenesis model with absorption and saturation incidence. We derived the basic reproductive number R0 which is used to show the stability of the disease-free and infected steady states. Further, we studied the effect of the time delay of the infected steady state. In addition, we examined the existence of Hopf bifurcation on infected steady-state and the model exhibits Hopf bifurcation by using delay as a bifurcation parameter. Numerical simulations are provided to illustrate the corresponding theoretical results.

- Vibrations in a Growing Nonlinear Chain

pp. 445-459 | DOI: 10.5890/DNC.2021.09.008

S.A. Surulere, M.Y. Shatalov, A.V. Mkolesia, I.A. Fedotov

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A one-dimensional chain describing the linear statistical increment of growing homogeneous atoms was arbitrarily built and investigated using an energy potential function. The analytic form of the considered potential has two exponential terms which describes chaotic behavior when the chain was excited. In order to investigate the dynamics of statistical attachment of individual atoms in the slender gold chain, the total energy of the entire system was changed by increasing the kinetic energy upon increment of homogeneous atoms in the chain. This resulted in a corresponding increase of the total energy in the system. On the other hand, the potential energy of the system on increment of homogeneous atoms equals zero, because the distance between corresponding atoms equals to the molecular distance (minimum potential distance). We considered the dynamical system with linear damping and without linear damping.

Different initial points were investigated to obtain trends of vibration that includes chaotic and regular oscillations. At some initial point(s), the attached atom experiences an infinite jump which means it falls off the nonlinear slender chain and the chain was broken. The interpretation of this phenomenon means the gold chain will result into an unstable nanostructure. We compared the numerical simulation of the system with different built-in ordinary differential equation solvers of various computer algebra software. Numerical simulation were carried out by plotting the system of growing atoms’ displacement against time. The system of linearly attached atoms were numerically simulated and inferences were stated from the study. In all cases considered, we inferred that amplitude of oscillation significantly increased at the end of the chain (terminal point) as compared to the initial point the oscillation started.

- Oscillatory Criteria for Some non Conformable Differential Equation with Damping

pp. 461-469 | DOI: 10.5890/DNC.2021.09.009

Juan E. N´apoles Valdes

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In this paper we present some criteria on the oscillation of solutions of a Non Conformable Differential Equations of $\alpha+\alpha$ order, under natural considerations. The local derivative considered was defined by the author in a previous work and a change of variables is used to transform the generalized differential equation into an ordinary differential equation of second order and using a Generalized Riccatti Transformation, together with known integration techniques, we obtain the desired results.

- Complex Dynamics of an Epidemic Model with Optimal Vaccination and Treatment in the Presence of Population Dispersal

pp. 471-497 | DOI: 10.5890/DNC.2021.09.010

Manotosh Mandal, Soovoojeet Jana, Swapan Kumar Nandi, T.K. Kar

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In this article, we describe an SEIR type epidemic model with a transportrelated infection between two cities in the presence of vaccination and treatment control. The epidemiological threshold, commonly known as basic reproduction number is derived and its impact on the dynamical behaviour of the disease has been established. The optimal control problem is constructed with the objective of minimizing the effect of infection in the systemand then it is solved. We compare the result of the model to a real-world problem to establish that our model can be used in some practical cases if the parameters are known.

- A Study on Effects of Biotic Resources on a Prey-Predator Population

pp. 499-522 | DOI: 10.5890/DNC.2021.09.011

Manotosh Mandal, Soovoojeet Jana, Swapan Kumar Nandi, T.K. Kar

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The environmental carrying capacities for the prey population and the predator population are restricted by their availability of foods. In this article, we introduce a prey-predator type ecological model in which the prey and predator have different biotic resources for food. Therefore the suggested predator-prey model depends on the ratio-dependent ecological modelwhich can be applied in the study of food chains. The details dynamical behaviour of the proposed model has been carried out. The different bifurcations and numerical analyzes are demonstrated to illustrate the dynamical behavior of our proposed model system.

- Inverse Problems of the Holling-Tanner Model Identification with Incomplete Information

pp. 523-534 | DOI: 10.5890/DNC.2021.09.012

A.A. Adeniji, M.Y. Shatalov, I. Fedotov, A.C. Mkolesia

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In this paper we present a novel method for numerical parameter identification of the Holling-Tanner model with incomplete information. It means that information about predator or prey is unavailable, or only particular information about these species is available. The proposed method is based on elimination of variable characterizing unknown population from the original system of equations and obtaining a new nonlinear ordinary differential equation. In this equation, the dependent variable characterizes dynamics of the known population and new set of parameters functionally depends on the original unknown parameters. In the case of the Holling-Tanner model, the number of new parameters is higher than the number of original unknown parameters. Hence, there exist several constraints between new unknown parameters, which must be taken into consideration in the process of the parameter identification. The conventional method, based on the Lagrange constraint minimization of a goal function gives a nonlinear system of equations where the number of equations is equal to the sum of new unknown parameters and constraints. In this novel method, proposed in this paper, the number of equation is exactly equal to the number of constraints which substantially simplifies solution of the problem.

- Existence, Uniqueness and Stability Results of Fractional Volterra-Fredholm Integro Differential Equations of $\psi$-Hilfer Type

pp. 535-545 | DOI: 10.5890/DNC.2021.09.013

Ahmed A. Hamoud, Abdulrahman A. Sharif, Kirtiwant P. Ghadle

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In this paper, we establish some new conditions for the existence and uniqueness of solutions for a class of nonlinearψ-Hilfer fractionalVolterra-Fredholm integro differential equations with boundary conditions. In addition, the Ulam-Hyers stability for solutions of the given problem are also discussed. The desired results are proved by using generalized Gronwall inequality, aid of fixed point theorems due to Banach and Schauder in weighted spaces.

- The Switching Function Projective Synchronization Dynamics of two Distinct Van der Pol-Duffing Oscillators with a Memristor-Duffing Oscillator

pp. 547-570 | DOI: 10.5890/DNC.2021.09.014

Fuhong Min, Chen Wei, Hanyuan Ma, Yiping Dou, Chunbiao Li

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In this paper, through the discontinuous dynamical system theory, the system interactions of two distinct Van der Pol-Duffing oscillators and a Memristor-Duffing oscillator is discussed under a switching nonlinear controller with symbolic functions. The interaction conditions of three chaotic systems are treated as separation boundaries which is time-varying. Thus the corresponding motion domains are constrained by the boundaries and studied, and the analytical conditions for function project synchronization of three nonautonomous system via the switch-ability and attractivity of edge flows are developed. The control parameter maps with different invariant sets are also studied under the analytical conditions. The partial and full function projective synchronization are carried out via numerical simulations, and the interactions of the control parameters on the synchronization has been investigated. The switching projective synchronization are experimentally realized via analog circuit, and the experimental results validate the theoretical analysis.

- Steiner 4-diameter, maximum degree and size of a graph

pp. 571-584 | DOI: 10.5890/DNC.2021.09.015

He Li, Shumin Zhang, Bo Zhu, Chengfu Ye

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The Steiner $k$-diameter $sdiam_k(G)$ of a graph $G$, introduced by Chartrand, Oellermann, Tian and Zou in 1989, is a natural generalization of the concept of classical diameter. When $k = 2$, $sdiam_2(G) = diam(G)$ is the classical diameter. Let $d, \ell$ and $n$ be natural numbers and $d < n, \ell < n$. Denote by $H_k(n,\ell,d)$ the set of all graphs of order $n$ with maximum degree $\ell$ and $sdiam_k(G) \le d$. Let $e_k(n,\ell,d)=min\{e(G):G \in H_k(n,\ell,d)\}$. In this paper, we focus on the case $k = 4$, and study the exact value of $e_4(n,\ell,d)$.