Local and global Hopf bifurcation analysis on simplified bidirectional associative memory neural networks with multiple delays

Abstract

In this paper, a class of simplified bidirectional associative memory (BAM) neural networks with multiple delays are considered. By analyzing the associated characteristic transcendental equation, their linear stability is investigated and Hopf bifurcation is demonstrated. By applying Nyquist criterion, the length of delay which preserves the stability of the zero equilibrium is estimated. Some explicit results are derived for stability and direction of the bifurcating periodic orbit by using the normal form theory and center manifold arguments. Global existence of periodic orbits is also established by using a global Hopf bifurcation theorem for functional differential equations (FDE) and a Bendixson’s criterion for high-dimensional ordinary differential equations (ODE) due to Li and Muldowney. Finally, numerical simulations supporting the theoretical analysis are carried out.