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Mathematical Problems in Engineering
Volume 2011, Article ID 756462, 10 pages
Research Article

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

College of Automation, Guangdong University of Technology, Guangzhou 510006, China

Received 16 July 2011; Accepted 12 September 2011

Academic Editor: Reza Jazar

Copyright © 2011 Fang Bao and Simin Yu. 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.


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.