Integral Step Size Makes a Difference to Bifurcations of a Discrete-Time Hindmarsh–Rose Model

Abstract

A three-dimensional discrete-time Hindmarsh–Rose model obtained by the forward Euler scheme is investigated in this paper. When the integral step size is chosen as a bifurcation parameter, conditions of existence for the fold bifurcation, the flip bifurcation, and the Hopf bifurcation are derived by using the center manifold theorem, bifurcation theory and a criterion of Hopf bifurcation. Numerical simulations including time series, bifurcation diagrams, Lyapunov exponents, phase portraits show the consistence with the analytical analysis. Our research results demonstrate that the integral step size makes a difference corresponding to local and global bifurcations of the three-dimensional discrete-time Hindmarsh–Rose model. These results can supply a solid analytical basis to the study of Hindmarsh–Rose model, and it is necessary to illustrate how much the integral step size is adopted in advance when numerical solutions or approximate solutions of the original continuous-time model is concerned.