The Bifurcated Periodic Solution For Several Classes Of Generalized Van Der Pol Systems With Delays

Time-delay is universal in real system, and can influence the dynamics of systemintensively. On one hand, the time-delay often makes system lose its stability, and lea-ds to various bifurcations and complex dynamics, on the other hand, proper time-dela-y can be benefit to system under certain conditions, for examples, time-delay can be u-sed to improve local stability, and to stable the unstable periodic oscillations embed inthe strange attractor of chaos. In present paper, the local stability and bifurcated in per-iodic solution of the generalized van der Pol systems with time-delays are studied, thetime-delay effects on the local stability and bifurcated periodic solution are focused on.Firstly, local stability around equilibrium point of the systems is analyzed through sta-bility switches, and the stable region and stability switching points of the systems areobtained. At the stability switching points, the systems will lose their stability and un-dergo periodic oscillations resulted from Hopf bifurcation. The methods of multiple t-ime-scales is used to calculate the bifurcated periodic solutions, and the direction of t-he bifurcation, the kind of the bifurcation and the stability of the bifurcated periodic s-olutions are determined. The methods of multiple scales is with easier calculation, mo-re accurate solution and more convenient to realize in computer than other methods, s-uch as the method of center manifold reduction.The generalized van der Pol system with high order nonlinear terms and delays isstudied, it is found that the time delay can affect the stability of the system, and lead tomultiple stability switches. When the system undergoes periodic oscillations resultedfrom Hopf bifurcation, the time delay and the times of the nonlinear term can influen-ce the amplitude of periodic solution, the amplitude is increasing with increase of thetime delay and decrease of the times of the nonlinear term. Delayed displacement fee-dback itself as damping for the generalized van der Pol system with nonlinear delayterms is studied, it is shown proper time delay can improve the stability of the system,however, when the time delay gets large, the system will lose stability through stabilit-y switches.The studies of present paper indicate the effect of the time delay on the dynamicsof the generalized van der Pol system is dual. On one hand, the time delay often make-s the system lose its stability, and leads to a series of stability switches; on the other h-and, proper time delay can improve the stability of system under certain conditions.W- hen the system undergoes periodic oscillations resulted from Hopf bifurcation, the ti-m e delay and the times of the nonlinear term can affect the characteristics of the peri-odic solution. The results obtained in present paper enrich the content of nonlinear sci-ence, and is a guidance for the application of the time delay feedback control on engi-neering.