A study of fixed points and hopf bifurcation of hindmarsh-rose model

Abstract

In this article, a class of Hindmarsh-Rose model is studied. First, all necessary conditions for the parameters of system are found in order to have one stable fixed point which presents the resting state for this famous model. After that, using the Hopf’s theorem proofs analytically the existence of a Hopf bifurcation, which is a critical point where a system’s stability switches and a periodic solution arises. More precisely, it is a local bifurcation in which a fixed point of a dynamical system loses stability, as a pair of complex conjugate eigenvalues cross the complex plane imaginary axis. Moreover, with the suitable assumptions for the dynamical system, a small-amplitude limit cycle branches from the fixed point