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Discrete Dynamics in Nature and Society
Volume 2006, Article ID 67908, 10 pages
http://dx.doi.org/10.1155/DDNS/2006/67908

Horseshoe in a class of planar mappings

1Department of Mathematics, Huazhong University of Science and Technology, Wuhan 430074, China
2Department of Control Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, China

Received 6 March 2005; Accepted 2 June 2005

Copyright © 2006 Yan Huang and Xiao-Song Yang. 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

Bursting dynamics of mappings is investigated in this paper. We first present stability analysis of the mappings' equilibria with various parameters. Then for three mappings P, P¯, and P^ with different parameters, we study their powers P4, P¯6, and P^4. We show that the mappings thus obtained are chaotic by giving a rigorous verification of existence of horseshoes in these mappings. Precisely, we prove that the mapping P¯6 is semiconjugate to the 3-shift mapping; the mappings P4 and P^4 are semiconjugate to the 4-shift mapping.