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Journal of Applied Mathematics
Volume 2014, Article ID 896478, 10 pages
http://dx.doi.org/10.1155/2014/896478
Research Article

1 : 3 Resonance and Chaos in a Discrete Hindmarsh-Rose Model

School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, China

Received 10 July 2014; Accepted 30 November 2014; Published 17 December 2014

Academic Editor: Zhidong Teng

Copyright © 2014 Bo Li and Zhimin He. 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

1 : 3 resonance of a two-dimensional discrete Hindmarsh-Rose model is discussed by normal form method and bifurcation theory. Numerical simulations are presented to illustrate the theoretical analysis, which predict the occurrence of a closed invariant circle, period-three saddle cycle, and homoclinic structure. Furthermore, it also displays the complex dynamical behaviors, especially the transitions between three main dynamical behaviors, namely, quiescence, spiking, and bursting.