Double Hopf Bifurcation and the Existence of Quasi-Periodic Invariant Tori in a Generalized Gopalsamy Delayed Neural Network Model

Abstract

In this paper, we discuss the double Hopf bifurcation and the existence of quasi-periodic invariant tori in a generalized Gopalsamy neural network model. Regarding the connection weight and the delay as bifurcation parameters of the double Hopf bifurcation, we derive the normal form up to the fifth order near the critical point by using the center manifold theorem and the normal form method, and obtain sufficient conditions on the existence of invariant 2-tori for the truncated normal form. Moreover, we investigate the effect of higher-order terms on these 2-tori by a KAM theorem. It is proved that in a sufficiently small neighborhood of the bifurcation point, the neural network model has quasi-periodic invariant 2-tori for most of the parameter set where its truncated normal form possesses invariant 2-tori. We give a numerical example to verify the conditions on all results in remarks.