Dynamics and Double Hopf Bifurcations of the Rose–hindmarsh Model With Time Delay

Abstract

In this paper, we are concerned with the Rose–Hindmarsh model with time delay. By applying the generalized Sturm criterion, a number of imaginary roots of the characteristic equation are classified. The absolutely stable regions for any value of time delay are detected. By the continuous software DDE-Biftool, both the Hopf bifurcation curves and double Hopf bifurcation points are illustrated in parametric spaces. The normal form and universal unfolding at double Hopf bifurcation points are considered by the center manifold method. Some examples also indicate that the corresponding unique attractor near each double Hopf point is asymptotically stable.

Desynchronization in synchronous multi-coupled chaotic neurons by mix-adaptive feedback control

“…In our previous work [16], we studied the stability by classifying neighbourhood regimes near the Fold-Hopf points and illustrated the complex bursting-spiking firing modes associated with FoldHopf bifurcations by numerical simulations. Here we take a = 1.0, b = 3.0, c = 1.0, d = 5.0, s = 4.0,x = −1.60, r = 0.005 and choose I x as a control parameter.…”

Delay-induced Hopf bifurcation in a noise-driven excitable neuron model

“…A central problem in the study of ODEs with delay is to identify the existence, direction and stability of Hopf bifurcation in the system and to understand how the behaviour changes as delay parameter changes. In the past, there has been a lot of work [14,15,17,19,20,22,27] addressing how delayed systems develop Hopf bifurcation from a steady state. What is more, Engelborghs et al [2] developed a numerical analysis tool (DDE-BIFTOOL) which can be used to analyse the bifurcation of delay differential equations.…”

Synthesizing attractors of Hindmarsh–Rose neuronal systems

“…Based on the theory of dynamical systems, Hindmarsh and Rose proposed the phenomenological neuron model, which may be seen either as a generalization of the Fitzhugh equations or as a simplification of the physiologically realistic model proposed by Hodgkin and Huxley [1,2]. The Hindmarsh-Rose (HR) model of neuronal activity is aimed to study the spiking-bursting behavior of the membrane potential observed in experiments of a cell in the brain of the pond snail [1].…”

“…The Hindmarsh-Rose (HR) model of neuronal activity is aimed to study the spiking-bursting behavior of the membrane potential observed in experiments of a cell in the brain of the pond snail [1]. The dynamics of a single HR neuron has been studied well, and it is illustrated that it can exhibit complex dynamical behavior including periodic and chaotic spiking (bursting) motion when certain control parameters of nervous cell models are varied [1][2][3].…”