Stability And Hopf Bifurcation Of Nonlinear System With Time Delay

Time delays are frequently encountered in many practical systems and very often are the main cause for poor performance and instability of system. In view of this, the issue of time-delay system is a topic of great practical importance, which has attracted a great deal of interest for several decades. In this paper, the features of dynamics are investigated in nonlinear systems with time delay, such as the local stability of equilibria, Hopf bifurcation, bifurcating periodic solutions and chaos control. Impact of time delay on the features of dynamic system is analyzed especially.First of all, the van der Pol equation with two time delays is considered. The local stability of the zero solution of the equation is investigated. One of the delays is regarded as a bifurcation parameter. It is found that Hopf bifurcation occurs when the delay passes through a sequence of critical value. The stability of bifurcating periodic solutions, the direction and other properties of Hopf bifurcation are determined by applying the normal form theory and the center manifold theorem. Some numerical examples are given to demonstrate the effectiveness of the theoretical analysis.Secondly, Lüsystem with time-delayed feedback is considered. Time-delayed feedback has been introduced as a powerful tool for control of unstable periodic orbits or control of unstable steady states. Regarding the time delay as a bifurcation parameter, the condition in which Hopf bifurcation occurs, the stability, direction and other properties of bifurcating periodic solutions are analyzed. Several numerical simulations are given to demonstrate that chaotic oscillation is converted into a stable steady state or a stable periodic orbit when the delay passes through certain critical values,Then, the small-world network model with time delay is considered. Regarding the time delay as a bifurcation parameter, it is found that Hopf bifurcation occurs in one-dimensional continuous system when the time delay passes through a critical value. By using the normal form theory and the center manifold theorem, the stability, direction and other properties of bifurcating periodic solutions are discussed. A numerical example is given to verify the theoretical analysis.At last, we get conclusions on nonlinear systems with time delay and the prospect of research topics in the future.