Journal of Engineering

Volume 2015, Article ID 615187, 9 pages

http://dx.doi.org/10.1155/2015/615187

## Generation Method of Multipiecewise Linear Chaotic Systems Based on the Heteroclinic Shil’nikov Theorem and Switching Control

^{1}School of Automation, Guangdong University of Technology, Guangzhou 510006, China^{2}Department of Photoelectric Engineering, Binzhou University, Binzhou 256603, China^{3}School of Electronics and Information, Hangzhou Dianzi University, Hangzhou 310018, China

Received 31 March 2015; Accepted 3 June 2015

Academic Editor: René Yamapi

Copyright © 2015 Chunyan Han 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

Based on the heteroclinic Shil’nikov theorem and switching control, a kind of multipiecewise linear chaotic system is constructed in this paper. Firstly, two fundamental linear systems are constructed via linearization of a chaotic system at its two equilibrium points. Secondly, a two-piecewise linear chaotic system which satisfies the Shil’nikov theorem is generated by constructing heteroclinic loop between equilibrium points of the two fundamental systems by switching control. Finally, another multipiecewise linear chaotic system that also satisfies the Shil’nikov theorem is obtained via alternate translation of the two fundamental linear systems and heteroclinic loop construction of adjacent equilibria for the multipiecewise linear system. Some basic dynamical characteristics, including divergence, Lyapunov exponents, and bifurcation diagrams of the constructed systems, are analyzed. Meanwhile, computer simulation and circuit design are used for the proposed chaotic systems, and they are demonstrated to be effective for the method of chaos anticontrol.

#### 1. Introduction

In 1994, Schiff et al. first put forward the concept of chaos anticontrol [1], which involves discrete-time and continuous-time system for the chaos anticontrol. In the study of chaos anticontrol for continuous-time system, a lot of progress has been made and some methods of generating chaotic and hyperchaotic systems were found, such as the methods of time-delay feedback to chaos [2–4], topological conjugate to chaos [5], pulse control to chaos [6], parameter perturbation control to hyperchaos [7–9], and state feedback control to hyperchaos [10–13]. However, these methods of chaos anticontrol, which are based on the parameter “try,” numerical simulation, and the Lyapunov index calculation, rely on the experience, lacking more theoretical basis. Therefore, it is still a challenge to construct a general approach of generating chaotic system.

Shil’nikov theorem is a beneficial tool to analyze the chaotic behavior of a nonlinear system [14], and it is also one of the most common decision theorems to determine whether chaos exists or not.

In 2004, Zhou et al. firstly constructed a three-dimensional autonomous chaotic system which has only one equilibrium point [15] by using the homoclinic orbit Shil’nikov theorem. Then, Li and Chen proposed a method to construct a two-piecewise linear chaotic system by using the heteroclinic Shil’nikov theorem [16, 17]. Recently, Yu et al. also proposed a method for constructing multipiecewise linear chaotic systems [18, 19]. However, the above studies lack further theoretical analysis and appropriate experimental verification.

In this paper, we use the heteroclinic Shil’nikov theorem to construct a kind of chaotic systems with both nonlinearities of two-piecewise linear and multipiecewise linear based on switching control method. Some basic dynamical characteristics of the constructed systems are analyzed theoretically and verified experimentally. Results of simulations and circuit experiments demonstrate the validity for the method of chaos anticontrol.

#### 2. Review of the Shil’nikov Theorem

Consider the following three-dimensional autonomous dynamical system:where belongs to and is a hyperbolic saddle focus; that is, the eigenvalues of Jacobian are and , where , , , , and are all real constants.

Theorem 1 (the heteroclinic Shil’nikov theorem) (see [14]). *Given the three-dimensional autonomous system shown in (1), let and be two distinct equilibrium points for (1). Suppose the following:*(1)*Both and are saddle foci that satisfy the Shil’nikov inequality with the further constraint or .*(2)*There is a heteroclinic loop, , in which it joins to and is made up of two heteroclinic orbits .*

*
Both the original system (1) and its perturbed varieties exhibit the Smale horseshoe chaos.*

*In the following study, we will use the switching controller to construct piecewise linear chaotic systems based on this theorem.*

*3. Two Fundamental Linear Systems*

*For simplicity, we use the well-known Lorenz system to generate fundamental linear systems which are used to construct the piecewise linear chaotic systems.*

*The Lorenz system is described bywhere , , and . The equilibrium points of (2) are given as*

*From (3), we can see that the equilibrium points and have certain symmetry. By linearizing (2) at the equilibrium points and , the linear equations are obtained, respectively, as follows: *

*Equations (4) and (5) are the required fundamental linear systems which are used to construct a piecewise linear chaotic system. The equilibrium points of the two linear systems are and , and the corresponding eigenvalues are: and . It can be seen that the equilibrium points and are all the saddle foci of index 2 and satisfy . According to the heteroclinic Shil’nikov theorem, there is chaos in the sense of Smale horseshoe as long as the equilibrium points of system (4) and system (5) are connected with the heteroclinic loop via a certain way.*

*The eigenvectors corresponding to the equilibrium points and of systems (4) and (5) are, respectively,*

*At point , the mathematical expressions of the one-dimensional stable manifold and the two-dimensional unstable manifold are given aswhere the direction vectors of the stable manifold are , , and and the direction vectors of the unstable manifold are given by , , and .*

*Similarly, the space straight line and the space plane , which, respectively, correspond to the one-dimensional stable manifolds and the two-dimensional unstable manifolds of equilibrium are described aswhere the direction vectors of the stable manifold are , , and and those of unstable manifold are , , and .*

*4. Construction of a Two-Piecewise Linear Chaotic System*

*Next, we will study the existence conditions of the heteroclinic loop connecting the equilibrium points of system (4) and system (5), choosing the controller as and the switching plane as .*

*After translational transform for equilibrium points and , systems (4) and (5) can be changed into the following form:where is the switching controller whose concrete form is determined by the conditions of forming heteroclinic loops for system (9).*

*Let the equilibrium points of system (9) be and . At the equilibria and , the space straight lines, and , and the space plans, and , are shown in Figure 1.*