Special Issue "Difference Equations and Discrete Dynamical Systems: Theory and Applications"

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

Deadline for manuscript submissions: closed (31 October 2019).

Special Issue Editor

Prof. Dr. Mustafa R.S. Kulenovic
Website
Guest Editor
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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Special Issue Information

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

Manuscript Submission Information

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Keywords

  • asymptotic
  • bifurcation
  • chaos
  • global dynamics
  • manifold
  • stability

Published Papers (6 papers)

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Research

Open AccessFeature PaperArticle
High Convergence Order Iterative Procedures for Solving Equations Originating from Real Life Problems
Mathematics 2019, 7(9), 855; https://doi.org/10.3390/math7090855 - 17 Sep 2019
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 [...] Read more.
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
Open AccessArticle
Extending the Applicability of a Two-Step Chord-Type Method for Non-Differentiable Operators
Mathematics 2019, 7(9), 804; https://doi.org/10.3390/math7090804 - 01 Sep 2019
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. [...] Read more.
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
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Open AccessArticle
Kolmogorov-Arnold-Moser Theory and Symmetries for a Polynomial Quadratic Second Order Difference Equation
Mathematics 2019, 7(9), 790; https://doi.org/10.3390/math7090790 - 30 Aug 2019
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 tn+1=αtn+βtn2tn1, [...] 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 , , where are t 1 , t 0 , α R , α 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
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Open AccessArticle
On the Oscillation of Non-Linear Fractional Difference Equations with Damping
Mathematics 2019, 7(8), 687; https://doi.org/10.3390/math7080687 - 01 Aug 2019
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.
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 the main results. The consistency of the proposed results is demonstrated by presenting some numerical examples. We end the paper with a concluding remark. Full article
Open AccessArticle
Stability and Bifurcation Analysis on a Predator–Prey System with the Weak Allee Effect
Mathematics 2019, 7(5), 432; https://doi.org/10.3390/math7050432 - 16 May 2019
Cited by 3
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, [...] Read more.
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
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Open AccessArticle
Global Dynamics of Leslie-Gower Competitive Systems in the Plane
Mathematics 2019, 7(1), 76; https://doi.org/10.3390/math7010076 - 12 Jan 2019
Abstract
Global dynamic results are obtained for families of competitive systems of difference equations of the form xn+1=b1xnα1+xn+c1yn,yn+1=b2 [...] Read more.
Global dynamic results are obtained for families of competitive systems of difference equations of the form x n + 1 = b 1 x n α 1 + x n + c 1 y n , y n + 1 = b 2 y n α 2 + c 2 x n + y n n = 0 , 1 , , 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
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