Change in critically of synchronous Hopf bifurcation in a multiple-delayed neural system. (English) Zbl 1162.92301

Ncube, Israel; Campbell, Sue Ann; Wu, Jianhong

Ruan, Shigui (ed.) et al., Dynamical systems and their applications in biology. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3163-1/hbk). Fields Inst. Commun. 36, 179-193 (2003).

Summary: We consider a network of three identical neurons with multiple signal transmission delays. The model for such a network is a system of delay differential equations. With the aid of the symbolic computation language MAPLE, we derive the corresponding system of ordinary differential equations describing the semiflow on the centre manifold. It is shown that two cases of a single Hopf bifurcation may occur at the trivial fixed point of the full nonlinear system of delay equations, primarily as a consequence of the structure of the associated characteristic equation. These are (1) the simple root Hopf, and (ii) the double root Hopf. This paper focusses on the first case, paying particular attention to possible change of the criticality of the bifurcations.
For the entire collection see [Zbl 1011.00043].

MSC:

92B20	Neural networks for/in biological studies, artificial life and related topics
34K18	Bifurcation theory of functional-differential equations
34K60	Qualitative investigation and simulation of models involving functional-differential equations
34C60	Qualitative investigation and simulation of ordinary differential equation models

Software:

Maple