Journal of Applied Mathematics
Volume 2013 (2013), Article ID 287123, 5 pages
Research Article

The Related Extension and Application of the Ši'lnikov Theorem

Department of Applied Mathematics, Guangdong University of Technology, Guangzhou 510006, China

Received 13 November 2012; Revised 21 January 2013; Accepted 1 February 2013

Academic Editor: Jin L. Kuang

Copyright © 2013 Baoying Chen. 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.


The traditional Ši'lnikov theorems provide analytic criteria for proving the existence of chaos in high-dimensional autonomous systems. We have established one extended version of the Ši'lnikov homoclinic theorem and have given a set of sufficient conditions under which the system generates chaos in the sense of Smale horseshoes. In this paper, the extension questions of the Ši'lnikov homoclinic theorem and its applications are still discussed. We establish another extended version of the Ši'lnikov homoclinic theorem. In addition, we construct a new three-dimensional chaotic system which meets all the conditions in this extended Ši'lnikov homoclinic theorem. Finally, we list all well-known three-dimensional autonomous quadratic chaotic systems and classify them in the light of the Ši'lnikov theorems.