Time Domain And Frequency Domain Analysis For Hopf Bifurcation In Delayed Differential Equations

This Ph.D.thesis is divided into six chapters and main contents are as follows:In Chapter 1, we give a survey to the developments of the theory of Hopf bifurcation for delayed differential equations. Then we introduce the background of problems, the main results of this dissertation.In Chapter 2, we briefly introduce the stability theory of delayed differential equation, Hopf bifurcation theory in the time domain, the normal form theory and center manifold theory, global Hopf bifurcation theory and Hopf bifurcation theory in the frequency domain. Some important preliminaries are also summarized.In Chapter 3, we consider a five dimensional BAM neural network model with two discrete delays. Some sufficient conditions to ensure that the equilibrium of system is asymptotically stable and the Hopf bifurcation exists near the equilibrium are derived. Some explicit formulae for determining the stability and the direction of the Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations and periods are obtained by using the normal form theory and center manifold theory in the time domain. Finally, numerical simulations supporting the theoretical analysis are given.In Chapter 4, we consider a six dimensional BAM neural network model with three discrete delays. Some sufficient conditions to ensure that the equilibrium of system is asymptotically stable and the Hopf bifurcation exists near the equilibrium are derived. Some explicit formulae for determining the stability and the direction of the Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations and periods are obtained by using the normal form theory and center manifold theory in the time domain. Using the global Hopf bifurcation theorem for functional differential equation (FDE) and Bendixson' criterion for high-dimensional ordinary differential equation(ODE), we obtain the global existence of periodic solutions. Finally, numerical simulations supporting the theoretical analysis are given.In Chapter 5, we consider a class of stage-structure predator-prey model with time delay and delays dependent parameters. By analyzing the associated char-acteristic transcendental equation, its linear stability is investigated. Using the Hopf bifurcation theorem in the time domain, we investigate the existence of Hopf bifurcation of the model. Some explicit formulae for determining the stability and the direction of the Hopf bifurcation periodic solutions bifurcating from Hopf bi-furcations and periods are obtained by using the normal form theory and center manifold theory. Finally, numerical simulations supporting the theoretical analysis are carried out.In Chapter 6, a class of simplified tri-neuron BAM network model with two delays is considered. By applying the frequency domain approach and analyzing the associated characteristic equation, the existence of bifurcation parameter point is determined. If the sumτof delayτ1 andτ2 is chosen as a bifurcation parameter, it is found that Hopf bifurcation occurs when the sumτpass through a series of critical values. The direction and the stability of Hopf bifurcation periodic solu-tions are determined by the Nyquist criterion and the graphical Hopf bifurcation theorem. Some numerical simulation for justifying the theoretical analysis are also provided. Finally, main conclusions are given.Finally, the research work of this paper is summarized. Some problems which are expected to resolve are put forward and the directions in the near future are included.