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Difference Equations and Discrete Dynamical Systems: Theory and Applications

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A special issue of Mathematics (ISSN 2227-7390).

Deadline for manuscript submissions: closed (31 October 2019) | Viewed by 14883

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Prof. Dr. Mustafa R.S. Kulenovic

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Department of Mathematics, University of Rhode Island, Kingston, RI 02881-0816, USA
Interests: bifurcation theory; difference equations; discrete dynamical systems; global dynamics; population dynamics
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Dear Colleagues,

A dynamic system is characterized by three major components: phase space, evolution operator(s), and time scale. Discrete dynamic systems are governed by difference equations which may result from discretizing continuous dynamic systems or modeling evolution systems for which the time scale is discrete. The discrete dynamic systems are prevalent in signal processing, population dynamics, numerical analysis and scientific computation, economics, health sciences, and so forth.

In this Special Issue of the journal Mathematics, we focus our attention on the asymptotic behavior of deterministic and stochastic dynamic systems for which the underlying phase space is either a continuum or a discrete set. Special attention will be given to different types of bifurcation in discrete dynamic systems such as the period doubling bifurcation, discrete Hopf bifurcation, flip bifurcation of some classes of difference equations and first-order systems.

Prof. Dr. Mustafa R.S. Kulenovic
Guest Editor

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Keywords

asymptotic
bifurcation
chaos
global dynamics
manifold
stability

Published Papers (6 papers)

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Research

12 pages, 769 KiB

Open AccessFeature PaperArticle

High Convergence Order Iterative Procedures for Solving Equations Originating from Real Life Problems

by Ramandeep Behl, Ioannis K. Argyros and Ali Saleh Alshomrani

Mathematics 2019, 7(9), 855; https://doi.org/10.3390/math7090855 - 17 Sep 2019

Cited by 2 | Viewed by 1836

Abstract

The foremost aim of this paper is to suggest a local study for high order iterative procedures for solving nonlinear problems involving Banach space valued operators. We only deploy suppositions on the first-order derivative of the operator. Our conditions involve the Lipschitz or Hölder case as compared to the earlier ones. Moreover, when we specialize to these cases, they provide us: larger radius of convergence, higher bounds on the distances, more precise information on the solution and smaller Lipschitz or Hölder constants. Hence, we extend the suitability of them. Our new technique can also be used to broaden the usage of existing iterative procedures too. Finally, we check our results on a good number of numerical examples, which demonstrate that they are capable of solving such problems where earlier studies cannot apply. Full article

(This article belongs to the Special Issue Difference Equations and Discrete Dynamical Systems: Theory and Applications)

8 pages, 225 KiB

Open AccessArticle

Extending the Applicability of a Two-Step Chord-Type Method for Non-Differentiable Operators

by Ioannis K. Argyros, Neha Gupta and J. P. Jaiswal

Mathematics 2019, 7(9), 804; https://doi.org/10.3390/math7090804 - 1 Sep 2019

Cited by 2 | Viewed by 1746

Abstract

The semi-local convergence analysis of a well defined and efficient two-step Chord-type method in Banach spaces is presented in this study. The recurrence relation technique is used under some weak assumptions. The pertinency of the assumed method is extended for nonlinear non-differentiable operators. The convergence theorem is also established to show the existence and uniqueness of the approximate solution. A numerical illustration is quoted to certify the theoretical part which shows that earlier studies fail if the function is non-differentiable. Full article

(This article belongs to the Special Issue Difference Equations and Discrete Dynamical Systems: Theory and Applications)

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16 pages, 622 KiB

Open AccessArticle

Kolmogorov-Arnold-Moser Theory and Symmetries for a Polynomial Quadratic Second Order Difference Equation

by Tarek F. Ibrahim and Zehra Nurkanović

Mathematics 2019, 7(9), 790; https://doi.org/10.3390/math7090790 - 30 Aug 2019

Cited by 6 | Viewed by 2877

Abstract

By using the Kolmogorov-Arnold-Moser (KAM) theory, we investigate the stability of two elliptic equilibrium points (zero equilibrium and negative equilibrium) of the difference equation [...] Read more.

By using the Kolmogorov-Arnold-Moser (KAM) theory, we investigate the stability of two elliptic equilibrium points (zero equilibrium and negative equilibrium) of the difference equation

t_{n + 1} = α t_{n} + β t_{n}^{2} - t_{n - 1}, n = 0, 1, 2, \dots,

where are

t_{- 1}

t_{0}

α \in R

α \neq 0

β > 0

. By using the symmetries we find the periodic solutions with some periods. Finally, some numerical examples are given to verify our theoretical results. Full article

(This article belongs to the Special Issue Difference Equations and Discrete Dynamical Systems: Theory and Applications)

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14 pages, 260 KiB

Open AccessArticle

On the Oscillation of Non-Linear Fractional Difference Equations with Damping

by Jehad Alzabut, Velu Muthulakshmi, Abdullah Özbekler and Hakan Adıgüzel

Mathematics 2019, 7(8), 687; https://doi.org/10.3390/math7080687 - 1 Aug 2019

Cited by 11 | Viewed by 2248

Abstract

In studying the Riccati transformation technique, some mathematical inequalities and comparison results, we establish new oscillation criteria for a non-linear fractional difference equation with damping term. Preliminary details including notations, definitions and essential lemmas on discrete fractional calculus are furnished before proceeding to [...] Read more.

(This article belongs to the Special Issue Difference Equations and Discrete Dynamical Systems: Theory and Applications)

15 pages, 939 KiB

Open AccessArticle

Stability and Bifurcation Analysis on a Predator–Prey System with the Weak Allee Effect

by Jianming Zhang, Lijun Zhang and Yuzhen Bai

Mathematics 2019, 7(5), 432; https://doi.org/10.3390/math7050432 - 16 May 2019

Cited by 13 | Viewed by 2730

Abstract

In this paper, the dynamics of a predator-prey system with the weak Allee effect is considered. The sufficient conditions for the existence of Hopf bifurcation and stability switches induced by delay are investigated. By using the theory of normal form and center manifold, an explicit expression, which can be applied to determine the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions, are obtained. Numerical simulations are performed to illustrate the theoretical analysis results. Full article

(This article belongs to the Special Issue Difference Equations and Discrete Dynamical Systems: Theory and Applications)

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17 pages, 367 KiB

Open AccessArticle

Global Dynamics of Leslie-Gower Competitive Systems in the Plane

by Mustafa R. S. Kulenović and David T. McArdle

Mathematics 2019, 7(1), 76; https://doi.org/10.3390/math7010076 - 12 Jan 2019

Cited by 2 | Viewed by 2622

Abstract

Global dynamic results are obtained for families of competitive systems of difference equations of the form [...] Read more.

Global dynamic results are obtained for families of competitive systems of difference equations of the form

x_{n + 1} = \frac{b_{1} x_{n}}{α_{1} + x_{n} + c_{1} y_{n}}, y_{n + 1} = \frac{b_{2} y_{n}}{α_{2} + c_{2} x_{n} + y_{n}} n = 0, 1, \dots,

where the parameters

b_{1}, b_{2}

are positive numbers, and

α_{1}, α_{2}, c_{1},

and

c_{2}

and the initial conditions

x_{0}

and

y_{0}

are arbitrary non-negative numbers, when one or both of

α_{i}, i = 1, 2

equalls 0. We assume that the denominators of both equations are always positive. We will show that the presence of more parameters will create more dynamic scenarios. Full article

(This article belongs to the Special Issue Difference Equations and Discrete Dynamical Systems: Theory and Applications)

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