Study On Stability And Hopf Bifurcation Of KdV-Burgers-Kuramoto Time Delayed Feedback Control System

Time delay phenomena is widespread in physical circuits,optics,neural networks,biology,economics and many other fields.Research has shown that time delay incorporated in a system can generate complex dynamical phenomena such as instability,Hopf bifurcation and chaos.However,if time delay feedback is added into a system,it can help to control complex dynamical behaviors that the system exhibits.Therefore,it is an important and meaningful research topic to study the time delay feedback control of chaotic systems.In this thesis,we are devoted to the study of time delay feedback control of a class of KdV-Burgers-Kuramoto chaotic system.By introducing one single delay and distributed time delay into the system,we respectively investigate the local stability of the equilibrium point and the properties of Hopf bifurcation.The specific work is as follows:1.One single time delay feedback term is added into the three-dimensional autonomous equation we call the KdV-Burgers-Kuramoto chaotic system,which is derived from KdV-Burgers-Kuramoto travelling wave equation and recently we have found exhibits chaotic dynamical behavior.Firstly,regarding the delay as the parameter,we obtain the conditions ensuring local stability of the equilibrium point and the existence of Hopf bifurcation by analyzing roots distribution of the characteristic equation(transcendental equation).Then,formulae determing the direction,stability and period of Hopf bifurcation are derived by using the normal form theory and the center manifold theorem.Finally,some numerical examples are presented to verify the effectiveness of the theoretical results.2.The distributed delay feedback term is introduced into the KdV-BurgersKuramoto chaotic system.Regarding the mean time delay as the parameter,we study the local stability of the equilibrium point and the existence of Hopf bifurcation by using the Routh-Hurwitz criteria.Formulae determing the direction,stability and period of Hopf bifurcation are obtained by using the normal form theory and the center manifold theorem.The correctness of the theoretical results is verified by numerical examples.