The Existence Of Quasi-periodic Invariant Tori For Double Hopf Bifurcation Of Generalized Gopalsamy Neural Network Model With Delay

Neural networks have been widely applied to associative memory,Ipattern recognition,combinatorial optimization and so on.These applications depend hcavily on the dynamic behavior of neural network models.Therefore,the loifur-(ation analysis of neural networks is still a.hot.topic in recent.yea.rs.In this thesis,we study the existence of quasi-periodic invariant for double Hopf bifurcation of generalizced Gopalsamy neural inetwork nmocdel by the bifurca-tion tlieory and KAM theorem.First.,we regard the connection weight.b and the delay ? as bifurcation parameters and identify the critical values for double hopf bifurcation(associatcd charactcristic cquation cquatuon has two pairs of purcl imaginary roots).Second,by using the normal form method and the center manifold theo-rem,we obtain the normal forms up to the fifth order near a double Hopf critical point.Then,the parameter conditioins of the existence of quasi-periodie invariant tori for truncated normal form can be obtained by using polar coordinates.Since the original system is not topologically equivalent to the truncation system,we analyze the persistence of quasi-periodic invariant 2-tori for the original system by using the KAM method at the last part of the thesis.In order to use the KAM method,we need coordinate transformations to reduce the original system to some normal form.Then,we can prove that in a sufficiently small neighborhood of the bifurcation point,the full system has quasi-periodic 2-tori for most of the parameter sets where its truncated normal form possesses 2-tori.