Bifurcations of a discrete-time neuron model

Abstract

In this paper, we discuss the bifurcations of a discrete-time neuron model. First, we prove that the fast subsystem of the model undergoes fold bifurcation and flip bifurcation. Numerical simulation shows that the subsystem produces chaos as the parameter changes. Next, discussing the qualitative properties of the fixed point of the model, we clarify all non-hyperbolic cases. Then, computing the normal form, we prove the model undergoes supercritical Neimark-Sacker bifurcation and produces a unique stable invariant circle. Furthermore, we prove that the system can produce p : q weak resonances, where $q\ge 7$, from which we simulate numerically a stable 7-periodic orbit on the invariant circle. Finally, applying center manifold theorem, we find that although the non-degeneracy conditions of both the flip bifurcation and the generalized flip bifurcation are not satisfied, the model produces flip bifurcation by the numerical simulation.

Keywords:

• Fold bifurcation
• flip bifurcation
• Neimark-Sacker bifurcation
• invariant circle
• resonance

AMS Subject Classifications:

• 39A10
• 39A28
• 37G10

Acknowledgements

The authors thank two anonymous referees for their constructive comments and suggestions.

Notes

No potential conflict of interest was reported by the author.

Additional information

Funding

This work was supported by the grand FP7-PEOPLE-2012-IRSES [number 316338]; Startup Foundation for Doctors of Lingnan Normal University [grant number ZL1605]; National Natural Science Foundation of China [grant number 11371314]; High-Level Talent Project of Colleges and Universities in Guangdong Province [grant number QBS201501]; Science and Technology Planning Projects of Zhanjiang [grant number 2016B01178], [grant number 2015B01011].