Dynamics And Control For Nonlinear System With Delays

With the rapid developing of adaptive control, time delay in dynamic systems has drawn much attention in some scientific fields such as applied mechanics, automatic control, engineering and neural networks. Time delay systems have become important research objects in many scientific fields. Many scholars have done a lot of works for time-delay systems, and they have achieved many important results, especially in the fields of electronics and circuits, optics, neural networks, biological environment and medicine, large-scale structure, machinery. The research of time delay systems is not only meaningful, but also is a challenging direction.This paper gives a detailed survey on the research progress of the time delay systems from the view of dynamics,First, this paper introduced the research background, the development and applications of the time delay systems in recent years. Hence the research methods of the time delay systems have been analyzed. It shows that the Phase space and Solution control of the time delay systems are infinite, and thus Taylor expansion methods or overlook the small time-delay is unreliable. The research on the Dynamics and control of these delayed systems mainly concentrated on the stability analysis, bifurcation analysis, chaos, time delay feedback control, and numerical simulation methods. The eigenvalue method and Lyapunov method are the main method to study the stability analysis for time delayed systems. Eigenvalue method does the local stability analysis, while Lyapunov method needs lots of inequality skills and it is hard to get Lyapunov function. Bifurcation analysis usually focuses on the static bifurcation and Hopf bifurcation, etc. It uses the method which parallels with the ordinary differential equations, so the main tools for analyzing these systems are Hopf bifurcation theorem, center popular theorem, normative theory, Poincare mapping, etc. An effective method to control system dynamic behavior is time delay feedback control method. It is a good method to control system produce complex motion and eliminate harmful motion.Based on these studies, we investigate the dynamics and control of a negative damping time-delay systems which is negative damping and cubic non-linear. First we used the eigenvalue method to analyze the stability of trivial equilibrium of this system. We found that trivial solution of system was unstable for any given positive time delay. At the same time we studied linear time delay feedback control of this model, obtained the stable interval for this controlled system and derived the largest time delay which determined by linear time delay feedback model. In addition, applying Hopf bifurcation theorem, center popular theorem, normative theory, we studied the conditions for system admitting a Hopf bifurcation and the stability of periodic orbits which bifurcated from trivial equilibrium. We also studied the complex dynamic motion of the systems by numerical methods, including the periodicity and the multi-stabilities. Our research has also shown that, time delay state feedback is a good method to control system produce complex motion and eliminate harmful motion. The above studies lay an important foundation for continuing to study time delay system dynamics.