Discrete Dynamics in Nature and Society

Volume 2016, Article ID 8712496, 13 pages

http://dx.doi.org/10.1155/2016/8712496

## Dynamic Analysis for a Fractional-Order Autonomous Chaotic System

^{1}School of Mathematics and Physics, Lanzhou Jiaotong University, Lanzhou 730070, China^{2}School of Traffic and Transportation, Lanzhou Jiaotong University, Lanzhou 730070, China

Received 23 January 2016; Accepted 26 June 2016

Academic Editor: Viktor Avrutin

Copyright © 2016 Jiangang Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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.

#### 1. Introduction

The idea of fractional-order calculus (FOC) has been well known since the development of the regular calculus. The significant progress on FOC has been witnessed because the FOC has a wide range of applications in diffusion. In the past decades, chaotic systems have become a focal point of renewed interest for many researchers. And we can find these nonlinear systems in various natural and man-made systems, which are known to have great sensitivity to initial conditions. And we can find chaos phenomenon in various natural and man-made systems, and these chaotic systems have great sensitivity to initial conditions. Because differential equations with fractional order can be applied in many areas of science and engineering, they attracted many researchers’ attention and their complex behaviors have been widely studied in recent years. Nowadays, there is increasing interest in the subject of a fractional model which can give a more realistic interpretation of practical phenomena. Furthermore, many systems in interdisciplinary fields can be described by the fractional differential equations, such as turbulence, electromagnetism, signal processing, and quantum evolution of complex systems. It is also demonstrated that some fractional-order differential systems behave chaotically or hyperchaotically, such as fractional-order Chen system [1–3] and fractional-order Lorenz system [4, 5]. The fractional-order equations are more suitable than integer-order ones to describe the biological, economic, and social systems where memory effects are important [6]. More recently, it has been found that some differential systems with fractional order possess chaotic behavior [7–16]. And many Lorenz-like or Lorenz-based chaotic systems were proposed and investigated. Some classical 3D autonomous chaotic systems have three particular fixed points: one saddle and two unstable saddle foci [17]. And the other 3D chaotic systems have two unstable saddle foci [18, 19]. Yang and Chen [20] found another 3D chaotic system with three fixed points: one saddle and two stable equilibriums. However, an increasing number of three-dimensional chaotic systems have been found these years in many physical and engineering fields. Nonlinear dynamics of a semiconductor laser subject to optical injection is currently a hot research field due to its rich physics and complexity as well as its potential applications in communications [21, 22]. Momani et al. [23] applied the multistep generalized differential transform method to solve the fractional-order multiple chaotic FitzHugh-Nagumo (FHN) neurons model. They illustrated the algorithm by studying the dynamics of three coupled chaotic FHN neurons equations with different gap junctions under external electrical stimulation, and the fractional derivatives are described in the Caputo sense. The numerical simulation results show that only few terms are required to deduce the approximate solutions which are found to be accurate and efficient.

Optically injected semiconductor lasers revealed amazingly rich behaviors like stable locking, coexistence of attractors, quasiperiodicity, instabilities, pulsations, and many routes to chaos such as period-doubling cascades, intermittency, breakup of tori, and homoclinic and heteroclinic tangencies [24–30], which deserve further systematic investigation. Wieczorek et al. [31] studied the dynamics and bifurcations of a single-mode semiconductor laser with optical injection and investigated the dependence of the dynamics on the injected field strength and its detuning from the unperturbed laser resonant frequency and the linewidth enhancement factor which seem to have the most significant influence on the dynamics. Chlouverakis and Adams [32] discussed the dynamics of a semiconductor laser subject to optical injection using the method of the Largest Lyapunov Exponent (LLE) and used this model as one of three examples to investigate the relation between the correlation () and the Kaplan-Yorke dimension () of three-dimensional chaotic flows. Chu et al. [33] studied the dynamical properties of the semiconductor laser subject to optical injection and a new almost-Hamilton chaotic system with very high Lyapunov dimensions is constructed and investigated. The aim of this paper is to investigate the dynamical behavior of a discretization fractional order of a modified optically injected semiconductor lasers model.

The paper is organized as follows. In Section 2, we give some related preliminaries. We present the fractional-order semiconductor lasers model and introduce a discretization process to discretize a modified fractional-order optically injected semiconductor lasers model in Section 3. In Section 4, a sufficient condition for existence and uniqueness of the solution of the fractional-order system is investigated and the equilibrium points and their asymptotic stability in the fractional-order model and its discrete counterpart are also studied. In Section 5, we give some numerical simulations, which not only illustrate our results with the theoretical analysis, but also exhibit the complex dynamical behaviors of the discretization fractional order of a modified optically injected semiconductor lasers model. Finally, conclusions are given in Section 6.

#### 2. Preliminaries

The Caputo definition of fractional derivative [34] is given as follows:where is the least integer which is not less than and is the Riemann-Liouville integral operator of order which is given bywhere is Euler’s Gamma function. The operator is termed “Caputo differential operator of order .” In [35, 36], the geometric and physical interpretations of the fractional derivatives were investigated.

The stability conditions and their applications to fractional-order differential equations were reported in [37–40]. The local stability of the equilibrium points of a linearized fractional-order system can be obtained from the following Matignon result [37]:where , , and are the eigenvalues of the system. Then, we consider the following nonlinear autonomous fractional-order system:where and is a nonlinear vector function in terms of . The Jacobian matrix evaluated at the equilibrium point isHence, we have the following lemma [41].

Lemma 1. *If all the eigenvalues , , and of the equilibrium point of system (4) satisfy Matignon’s condition (3), then is locally asymptotically stable.*

#### 3. Fractional-Order Semiconductor Lasers Model and Its Discretization

Chlouverakis and Adams [32] present a modified optically injected semiconductor lasers model as follows:where the complex electric field and is the normalized population inversion, is the injection strength, is the detuning, and is the linewidth enhancement factor. And the parameters and , is the inverse photon lifetime, is the inverse electron lifetime, and is the angular relaxation oscillation frequency. For simplification, in this paper, we established a dimensionless modified optically injected semiconductor lasers model by dimensionless method:where are the state variables and are real parameters. The bifurcation diagram of versus control parameter , the Lyapunov-exponent spectrum, and the Kaplan-Yorke dimension for specific values set versus the bifurcation parameter on the open interval are shown in Figures 1(a)–1(c), respectively.