Rich Dynamics in a Hindmarsh–Rose Neuronal Model with Time Delay

Author:

Jiao Xubin1ORCID,Liu Xiuxiang1

Affiliation:

1. School of Mathematical Sciences, South China Normal University, Guangzhou 510631, P. R. China

Abstract

This paper presents the stability and bifurcation of the Hindmarsh–Rose neuronal model with time delay. First, we discuss the existence and stability of the equilibria, revealing that the system can have a maximum of three equilibria which exhibit always stable, always unstable, and stability switching dynamics, depending on the stability of the equilibria when there is no time delay. Furthermore, an explicit formula for the time delay at which Hopf bifurcation occurs is derived, and the direction and stability of the Hopf bifurcation are also given. Finally, numerical simulations are carried out to support our theoretical results. We find that due to the existence of time delay, our system can exhibit many interesting dynamical behaviors depending on the number of equilibria and the increase in time delay. For instance, with increasing time delay, bistability transitions to the coexistence of a periodic solution and a stable equilibrium, followed by the emergence of quasi-periodic motion alongside stable equilibria, and eventually leading to chaotic motion coexisting with a stable equilibrium.

Funder

Guangdong Natural Science Foundation

National Natural Science Foundation of China

Publisher

World Scientific Pub Co Pte Ltd

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