Bifurcation analysis of a two-dimensional discrete Hindmarsh–Rose type model

Abstract

Abstract In this paper, bifurcation analysis of a discrete Hindmarsh–Rose model is carried out in the plane. This paper shows that the model undergoes a flip bifurcation, a Neimark–Sacker bifurcation, and 1:2 $1:2$ resonance which includes a pitchfork bifurcation, a Neimark–Sacker bifurcation, and a heteroclinic bifurcation. The sufficient conditions of existence of the fixed points and their stability are first derived. The flip bifurcation and Neimark–Sacker bifurcation are analyzed by using the inner product method and normal form theory. The conditions for the occurrence of 1:2 $1:2$ resonance are also presented. Furthermore, the sufficient conditions of pitchfork, Neimark–Sacker, and heteroclinic bifurcations are derived and expressed by implicit functions. The numerical analysis shows us consistence with the theoretical results and exhibits interesting dynamics, especially symmetric and invariant closed orbits. The dynamics observed in this paper can be used to mimic the dynamical behaviors of one single neuron and design a humanoid locomotion model for applications in bio-engineering and so on.

Keywords

• Hindmarsh–Rose model
• Flip bifurcation
• Pitchfork bifurcation
• Neimark–Sacker bifurcation
• 1 : 2 $1:2$ resonance
• Symmetric orbits