The Persistence Of Quasi-periodic Invariant 2-tori For Double Hopf Bifurcation Of BAM Neural Network Models With Delays

In this thesis,we mainly study the persistence of quasi-periodic invariant 2-tori for double Hopf bifurcation of BAM neural network models with delays.In chapter 1,we mainly introduce the origin,historical background and sig-nificance of BAM neural network models with delays.Meanwhile,we give the main results and significance of this thesis.In chapter 2,we mainly introduce the normal form theory of delay cdifferen-tial equations with parameters mand a KAM theorem which will be applied to our models.In chapter 3,firstly,we translate the delays in the model into a single delay.Secondly,discusse the existence of double Hopf bifurcation of the system.Thirdly,by the center manifold theory and the normal form theory we obtain the normal form of the system near the bifurcation point.Finally,the normal form is ex-pressed in the polar coordinates the polar coordinates,so that we easily discuss the existence of two-dimensional tori.Chapter 4 is devoted to the persistence of quasi-periodie invariant 2-tori of the system.First,we study the existence and quasi-periodic invariant 2-tori of the truncated normal form.Then,by a KAM theorem,it is proven that the persistence of quasi-periodic invariant 2-tori for double Hopf bifurcation of the system.