Influence of high order nonlinearity on chaotic bursting structure in slow–fast dynamics

Abstract

Nonlinear truncation based on Taylor expansion has widely been used for the analysis of a real model, while the order of truncation may lead to different behaviors. This paper devotes to investigate the influence of the cubic and fifth order nonlinearity on the bursting oscillations in a relatively simple slow–fast chaotic model. To reveal the characteristics of spiking oscillations, we propose a new type of cross-section based on the excitation, which can be used to compute the projections of Poincaré map conveniently. Higher order nonlinear term may result in more fine structures in a chaotic bursting attractor, implying the trajectory for spiking state alternates between more types of regular oscillations and chaos in turn. Since there exist two choices when the trajectory moving along an equilibrium branches to a pitchfork bifurcation point, it needs two neighboring periods of excitation for the trajectory to finish one cycle of the quiescent and spiking state.