Period-adding bifurcation and chaos in a hybrid Hindmarsh–Rose model

Abstract

Recently, the hybrid neuron models which combine the basic neuron models with impulsive effect(the state reset process) had been proposed, however, the preset value and the reset value of membrane potential were both fixed constants in the known models. In this paper, the Hindmarsh–Rose neuron model with nonlinear reset process is presented where the preset value and the reset value of membrane potential are variable constants. We conduct a qualitative analysis in the vicinity of the equilibrium point or the limit cycle of the proposed system by using the theories of impulsive semi-dynamical systems. Firstly, the more detailed impulsive set and phase set are given, then using the fixed point of Poincaré map, the existences of order-1and order-k(k>=2) period solutions are investigated subsequently. Furthermore, numerical investigations including period-adding bifurcation, multiple attractors coexistence, switch-like behavior are presented to further describe the bifurcation and chaos phenomena. Finally, the obtained results and possible applications of the proposed model are elaborated.