Hopf Bifurcation Analysis for a Four-Dimensional Recurrent Neural Network with Two Delays

Zizhen Zhang,Huizhong Yang

doi:10.1155/2013/436254

Abstract

A four-dimensional recurrent neural network with two delays is considered. The main result is given in terms of local stability and Hopf bifurcation. Sufficient conditions for local stability of the zero equilibrium and existence of the Hopf bifurcation with respect to both delays are obtained by analyzing the distribution of the roots of the associated characteristic equation. In particular, explicit formulae for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are established by using the normal form theory and center manifold theory. Some numerical examples are also presented to verify the theoretical analysis.

Highlights

In recent years, neural networks have attracted many scholars’ attention all over the world and have been applied in different areas such as signal processing [1], pattern recognition [2,3,4], optimization [5], and automatic control [6,7,8]
It is shown that a Hopf bifurcation takes place in the delayed system as the mean delay passes a critical value where a family of periodic solutions bifurcate from the equilibrium
We have investigated a four-dimensional recurrent neural network with two discrete delays

Summary

Introduction

Neural networks have attracted many scholars’ attention all over the world and have been applied in different areas such as signal processing [1], pattern recognition [2,3,4], optimization [5], and automatic control [6,7,8]. It is well known that time delays can play a complicated role on neural networks They can be the source of instabilities and bifurcation in neural networks. Journal of Applied Mathematics can reflect the really large neural networks more closely, we consider the following four-dimensional recurrent neural network with two discrete delays that occur in the interaction between the neurons: x1̇ (t) = − x1 (t) + f (x2 (t − τ1)) , x2̇ (t) = − x2 (t) + f (x3 (t − τ1)) , x3̇ (t) = − x3 (t) + f (x4 (t − τ1)) ,.

Stability of the Zero Equilibrium and Local Hopf Bifurcation

Stability of Bifurcated Periodic Solutions

Numerical Simulation

Conclusion