Multiple Bifurcations and Chaos Control in a Coupled Network of Discrete Fractional Order Predator-Prey System

Discrete-time dynamical system exhibits richer dynamical behaviors such as chaos rather than continuous-time dynamical systems. In order to describe chaos in two dimensional fractional order Lesli-Gower predator-prey systems, we need to transition from fractional continuous-time dynamical systems to the discrete-time version. One of the practical ways to achieve this transition is to use piecewise constant arguments in the model. After the discretization procedure based on the use of piecewise constant arguments in the interval t is an element of[nh,(n+1)h)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t\in [nh, (n+1)h)$$\end{document}, we obtain a new two dimensional system of difference equations. Necessary and sufficient conditions for the stability of the equilibrium points are given by using Schur-Cohn criterions. It is also investigated the existence of possible bifurcation types about the positive equilibrium point of the discrete system. Theoretical analysis shows that the system undergoes Neimark-Sacker and flip bifurcations with respect to parameter q. In addition, OGY feedback control method is implemented in order to control chaos in discrete model. Bifurcations in a coupled network of the discrete predator-prey system are also examined. Numerical simulations show that when the coupling strength parameter arrives the critical value, chaotic behavior is formed in the complex dynamical networks. All of the theoretical results dealing with the stability, bifurcation and transition chaos in the coupled network are stimulated by numerical simulations.