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 , 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.