PD Control at the Hopf Bifurcation Point of a Neuron System with Inertia and Delay

Abstract

In this paper, a two-neuron system with inertia and delay is proposed firstly. a PD controller is then applied to the system for the purpose of improving its dynamical performance. Through the mathematical transformation, we extend the system to a four-dimensional one with only time delays. With the help of the associated characteristic equation of the mathematical model, suffcient conditions for ensuring the system stability are proposed. Furthermore, with the time delay as the bifurcation parameter, relevant requirements for the generation of Hopf bifurcation are derived. Then a series of numerical simulatiosns are carried out to justify the theoretical analysis and it is found that the application of PD control scheme helps to advance the bifurcation point dramatically through a slight adjustment of the controller parameters.

Control of Hopf Bifurcation Type of a Neuron Model Using Washout Filter

“…Wang et al explored the bifurcation mechanisms related to four types of bursters through the analysis of phase plane and calculated the first Lyapunov coefficient of the Hopf bifurcation, which can decide whether it is supercritical or subcritical [15]. Shi et al proposed sufficient conditions for ensuring the system stability and derived relevant requirements for the generation of Hopf bifurcation with the help of the associated characteristic equation of the mathematical model [16].…”