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Codimension-two bifurcation analysis in two-dimensional Hindmarsh–Rose model

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Abstract

In this paper, we analyze the codimension-2 bifurcations of equilibria of a two-dimensional Hindmarsh–Rose model. By using the bifurcation methods and techniques, we give a rigorous mathematical analysis of Bautin bifurcation. The main result is that no more than two limit cycles can be bifurcated from the equilibrium via Hopf bifurcation; sufficient conditions for the existence of one or two limit cycles are obtained. This paper also shows that the model undergoes a Bogdanov–Takens bifurcation which includes a saddle-node bifurcation, an Andronov–Hopf bifurcation, and a homoclinic bifurcation. In some case, the globally asymptotical stability is discussed.

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  1. School of Mathematical Sciences, South China University of Technology, Guangzhou, 510640, P.R. China

    Xuanliang Liu & Shenquan Liu

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  1. Xuanliang Liu
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  2. Shenquan Liu
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Liu, X., Liu, S. Codimension-two bifurcation analysis in two-dimensional Hindmarsh–Rose model. Nonlinear Dyn 67, 847–857 (2012). https://doi.org/10.1007/s11071-011-0030-6

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