Codimension-two bifurcation analysis in two-dimensional Hindmarsh–Rose model

Abstract

In this paper, we analyze the codimension-2 bifurcations of equilibria of a two-dimensional Hindmarsh-Rose model. By using the bifurcation methods and techniques, we give a rigorous mathematical analysis of Bautin bifurcation. The main result is that no more than two limit cycles can be bifurcated from the equilibrium via Hopf bifurcation; sufficient conditions for the existence of one or two limit cycles are obtained. This paper also shows that the model undergoes a Bogdanov-Takens bifurcation which includes a saddle-node bifurcation, an Andronov-Hopf bifurcation, and a homoclinic bifurcation. In some case, the globally asymptotical stability is discussed.

1 : 3 Resonance and Chaos in a Discrete Hindmarsh-Rose Model

“…Using the Cardan formula (see [28]), we get the following results (see also [8,21] The stability of these fixed points can be found in [21]. In this paper, we focus on the existence and bifurcation analysis of 1 : 3 resonance.…”

“…The Hindmarsh-Rose model is known to reproduce all dynamical behaviors, such as quiescence, spiking, bursting, irregular spiking, and irregular bursting [4,6]. Bifurcation analysis is examined once more in the past, with respect to one or two bifurcation parameters [7][8][9][10][11]. Local bifurcations and global bifurcations are also analysed and these bifurcation phenomena can be used to explain the transitions between the dynamical behaviors.…”

“…For example, the transition between spiking and bursting in the model can be understood by homoclinic bifurcations [12,13]. More information on bifurcation can be found in [1,4,[8][9][10][11][14][15][16][17][18][19][20][21][22].…”

A new unidirectional coupling for global synchronization of two Hindmarsh-Rose neurons

“…The Hindmarsh-Rose (HR) model is a system of three coupled nonlinear differential equations which can describe the action potential generation of a wide range of neuronal cells [1], [2], [3]. The HR neurons reveal various dynamical behaviors, including spikes, bursting, and chaotic attractors [2], [3].…”

Synchronization in Two Coupled Hindmarsh-Rose Neurons

“…Since Pecora and Carroll (1990) [19] studied synchronization of chaotic systems, chaotic synchronization has been widely used in the secure communication, oscillator networks and neural networks during the last 20 years, see for example, [2,3,4,11,12,13,14,15,16,17,20,22,23,24,25,27,28,29,30].…”