Bifurcations and chaos in a two-dimensional discrete Hindmarsh–Rose model

• DOI: 10.1007/s11071-013-1161-8
• Publication Date: 2013-12-02T12:29:30Z
• Authors: Bo Li; Zhimin He
• Keywords: Center manifold; Bogdanov–Takens bifurcation; Rose (mathematics); Homoclinic bifurcation; Pitchfork bifurcation; Transcritical bifurcation

In this paper, the dynamics of a two-dimensional discrete Hindmarsh–Rose model is discussed. It is shown that the system undergoes flip bifurcation, Neimark–Sacker bifurcation, and 1:1 resonance by using a center manifold theorem and bifurcation theory. Furthermore, we present the numerical simulations not only to illustrate our results with the theoretical analysis, but also to exhibit the complex dynamical behaviors, including orbits of period 3, 6, 15, cascades of period-doubling bifurcation in orbits of period 2, 4, 8, 16, quasiperiodic orbits, and chaotic sets. These results obtained in this paper show far richer dynamics of the discrete Hindmarsh–Rose model compared with the corresponding continuous model.