How to Generate Chaos from Switching System: A Saddle Focus of Index 1 and Heteroclinic Loop‐Based Approach

Fang Bao,Simin Yu

doi:10.1155/2011/756462

Abstract

There exist two different types of equilibrium points in 3‐D autonomous systems, named as saddle foci of index 1 and index 2, which are crucial for chaos generation. Although saddle foci of index 2 have been usually applied for creating double‐scroll or double‐wing chaotic attractors, saddle foci of index 1 are further considered for chaos generation in this paper. A novel approach for constructing chaotic systems is investigated by applying the switching control strategy and yielding a heteroclinic loop which connects two saddle foci of index 1. A basic 3‐D linear system with an arbitrary normal direction of the eigenplane, possessing a saddle focus of index 1 whose corresponding eigenvalues satisfy the Shil′nikov inequality, is first introduced. Then a heteroclinic loop connecting two saddle foci of index 1 will be formed by applying the switching control strategy to the basic 3‐D linear system. The heteroclinic loop consists of an unstable manifold, a stable manifold, and a heteroclinic point. Under the necessary conditions for forming the heteroclinic loop, the intended two‐segmented piecewise linear system which exhibits the chaotic behavior in the sense of the Smale horseshoe can be finally constructed. An illustrative example is given, confirming the effectiveness of the proposed method.

Highlights

It is well known that saddle foci of index 2 are crucial for chaos generation in 3-D autonomous systems, where each saddle focus of index 2 creates one corresponding scroll or wing
Based on the heteroclinic Shil’nikov theorem and the switching control strategy, this paper proposes a novel approach for constructing piecewise linear chaotic systems by employing saddle foci of index 1
X1, y1, z1 T and P2 x2, y2, z2 T are the equilibrium points. Both saddle foci of index 1 and saddle foci of index 2 play a crucial role in chaos generation in 3-D autonomous systems, whenever they are smooth continuous chaotic systems or piecewise linear chaotic ones

Summary

Introduction

It is well known that saddle foci of index 2 are crucial for chaos generation in 3-D autonomous systems, where each saddle focus of index 2 creates one corresponding scroll or wing. Li and Chen present an approach for constructing a piecewise linear chaotic system based on the heteroclinic Shil’nikov theorem, starting from a linear system with a saddle focus of index 2 10. In 11 , a piecewise linear chaotic system is constructed based on heteroclinic Shil’nikov theorem and the switching control strategy, by selecting two linear systems with two saddle foci of index 2, the eigenplanes of which are supposed to be symmetric. Based on the heteroclinic Shil’nikov theorem and the switching control strategy, this paper proposes a novel approach for constructing piecewise linear chaotic systems by employing saddle foci of index 1.

A Basic 3-D Linear Nominal System with a Saddle Focus of Index 1

Design of a Basic 3-D Linear Nominal System with a Saddle Focus of Index 1

Constructing a Switching Chaotic System with Saddle Foci of Index 1

Dissipation and Lyapunov Exponents

The Solution of the State Equations

Conclusions