Dynamic Analysis for a Fractional-Order Autonomous Chaotic System

Abstract

We introduce a discretization process to discretize a modified fractional-order optically injected semiconductor lasers model and investigate its dynamical behaviors. More precisely, a sufficient condition for the existence and uniqueness of the solution is obtained, and the necessary and sufficient conditions of stability of the discrete system are investigated. The results show that the system’s fractional parameter has an effect on the stability of the discrete system, and the system has rich dynamic characteristics such as Hopf bifurcation, attractor crisis, and chaotic attractors.

Variable Order Fractional Permanent Magnet Synchronous Motor: Dynamical Analysis and Numerical Simulation

“…Firstly, we discuss the stability conditions of the static equilibrium point. (12). Then the equilibrium point 0 S is asymptotically stable when 1   and unstable when 1   .…”

“…The equilibrium points of the system (12) are presented in the following theorem: Theorem 2 System (12) has the following equilibrium points: (i) Only the static solution 0 (0, 0, 0) when…”

“…Fractional calculus plays a significant role in describing natural phenomena, and it is considered as a generalization of integral calculus [1]- [5]. For example, the fractional derivative concerns with the history of the considered variable hence it is more suitable for representing memory and hereditary behavior of physical models as in [6]- [12]. Electromagnetic fields with their applications are reformulated using a fractional derivative, and many important enhancements are satisfied in this area.…”

Dynamic Analysis of Complex Synchronization Schemes between Integer Order and Fractional Order Chaotic Systems with Different Dimensions

“…So, it is not surprising that most of the systems we encounter in the real world are nonlinear. And what is interesting is that some of these nonlinear systems can be described by fractional order differential equations which can display a variety of behaviors including chaos and hyperchaos [1][2][3][4][5]. Recently, study on synchronization of fractional order chaotic systems has started to attract increasing attention of many researchers [6][7][8][9][10][11][12], since the synchronization of chaotic systems with integer order is understood well and widely explored [13][14][15].…”

Neimark-Sacker Bifurcation and Chaos Control in a Fractional-Order Plant-Herbivore Model

“…In order to control the chaos due to emergence of Neimark-Sacker bifurcation, pole-placement and hybrid control strategies are implemented on system (16). Similar methods of discretization for fractional-order systems are also used in [18][19][20][21][22].…”