Subharmonic Analysis Of Second Order Delay Equations

In recent years, a large number of differential equation model and time delay from the field of building structures, circuits, optics, social economics, environmental science, medicine, neural networks, mechanics, etc., and has achieved important goals. At the same time, the time delay control power systems have been used, for example, time-delay feedback control has been a major way to handle the chaos. This is very important to our understanding in such a way to do research in the future.From the professional point of view, in the power system is always inevitable time delay phenomenon, relatively speaking, the phenomenon of time delay in the power system is always there, because for the convenience of discussion, we only ignorea. With the development of technology depth and theoretical studies of the theoretical and practical significance can not be ignored. Delay dynamical systems theory, a very significant issue for us, its direction and content are increasingly being used. Most of the concerns for Systems with Time Delay stability, Hopf bifurcation, chaos and other related issues. The motion of the system imbalance, usually caused by time delay, resulting in various forms of branch.[30] discussed the cycle of external perturbations, the Delay Vandepol equation Subharmonic the existence and stability, which in the dynamical system has a very important significance, caused by a mechanical researchers’ attention. Delay systems under parametric excitation, the existence of the tone of reconciliation and stability, the domestic rare discussion. Known in the literature [1] gives an order delay equation reversed the reconciliation of the existence and stability of a numerical example in the conditions of parametric excitation. Dynamical systems, most affected by the periodic excitation of the model, such as seismic waves, and time delay structure of the elastic foundation are described by a second order delay equation.[33] gives a fully parametric analysis of root distribution of the linear part of the characteristics of such systems. On this basis, given this kind of second order delay equation the existence of downward reconciliation and stability analysis of parametric excitation, and further elaborated by the parametric excitation caused by the motion of the system state.This paper is divided into three chapters. The first chapter is the introduction, brief background, the delay equation, and its recent developments and key issues.Chapter Ⅱ will explain the general procedure of multi-dimensional analysis of nonlinear vibration systems with common u=f(u,u,t,t-τ,s) as example.Chapter three the application of multiscale analysis method, to withstand periodic excitation with time delay of seismic wave and elastic foundation role in the structure of the simplified mathematical model to cycle parameter perturbations was studied. And the existence of solution and approximate expressions of the conclusion...