| Neural networks have been widely applied to associative memory,pattern recognition,sequence recognition,medical diagnosis,optimization,signal process-ing and so on.The delayed unidirectional coupling neural network model,along with the delay and connection weight within a single network,produces a series of complex dynamic behaviors,such as zero bifurcation,Hopf bifurcation,double Hopf bifurcation,invariant tori,etc.In this thesis,the bifurcation theory and KAM theorem are used to study the double Hopf bifurcation and its persistence near the bifurcation points of the delayed unidirectional coupling neural network model.Chapter 1 introduces the research background and practical significance of neural network models.In chapter 2,the connection weight(612and delayare selected as the bifurcation parameters to verify the existence of the double Hopf bifurcation.The normal forml of the system near the critical point is reduced to the fifth order by using the center manifold theorem and the normal forml method.In order to analyze the existence of invariant tori of truncated system,the normal forml is trans-formed into polar coordinates.Chapter 3 analyzes the existence of two and three dimensional tori of the truncated system.Since the higher order term of normal forml is ignored near the double Hopf bifurcation point,the truncated system cannot be equivalent to the original system.We apply a KAM theorem to dis-cussing whether there is still a quasi-periodic invariant torus after the truncated system is added with higher-order terms.Since the coordinate transformation of the normalization process is reversible,we prove that the original system has quasi-periodic invariant 2-dimension and 3-dimension tori for most parameters in an allowed domain. |
