Study On Bifurcation And Chaos System With Impulsive Control

In the nonlinear science field, bifurcation and chaos analysis, control and anti-control have become front research topics, extremely challenging. The research purposes is to de-sign a controller to suppress or reduce some harmful bifurcation and chaotic behavior of a given nonlinear system (control bifurcation and chaos) or to created, maintain, or enhance the healthy and useful behaviors of bifurcation and chaotic for a given nonlinear system (anti-control bifurcation and chaos), thereby achieving some desirable dynamical behav-iors, so far, its concerned work is very little. As one of the effective control methods, im-pulsive control can be directly or indirectly used for eliminating bifurcation or chaos from a complex dynamical system. Utilize the pulse to contr or anti-control the bifurcation or chaotic system are our research objects.Through a comprehensive analysis and summary of the research history and actuality of bifurcation and chaos, we have conducted a systematic and thorough investigation into the fundamental theory and application of the bifurcation or chaos control by using the nonlinear dynamics theory, and the chaos and bifurcation theory and so on modern nonlinear science analysis methods and the impulse control theory.The main achievements and conclusions in this dissertation are obtained as follows:The stability of discrete impulsive control system was discussed. Based on the theory of impulsive differential, the stability of discrete impulsive control system was studied by using Lyapunov method and some lemmas. Utilizing Lyapunov function proved asymp-totic stability of discrete impulsive system and the essential conditions to this kind of sta-bility have been researched. The theoretical results were verified by an example.The method of impulsive control bifurcation in a nonlinear system was discussed. The stability of the system was proved by using Lyapunov function. The stability condi-tions have been derived. The simulation results confirmed this method validity.The bifurcation and chaos behavior of small-world networks model has been investi-gated. The parameter conditions of bifurcation and chaos producing was deduced. An im-pulsive hybrid control method was proposed to control the bifurcations and stabilize the period orbits embedded in the chaotic attractor of a small-world network. Theoretical cal- culations and simulation results showed that the period doubling bifurcations can be de-layed or eliminated completely. The period doubling route to chaos is therefore at least delayed to greater parameter values.The unstable period orbits embedded can be stabilized and a p-period orbit can be controlled to desired orbits by adjusting control parameters.The problem of analysis and control of bifurcations and chaos has been studied in a TCP-RED congestion control system model. We study the bifurcation and chaotic behav-ior of the TCP-RED system which may cause heavy oscillation of average queue length and induce network instability. We propose an impulsive control method for controlling bifurcations and chaos in the TCP-RED system, which is a simple approach improving the performance of the RED router queue management mechanism by adjusting the control parameters. The simulation experiments show that this method can obtain the stable aver-age queue length without sacrificing the advantages of RED.The method of anti-control bifurcation via impulsive control has been researched. The bifurcation producing condition and anti-control of bifurcations method were dis-cussed. The simulation results showed the complex dynamics under the periodically im-pulsive control alters the bifurcations and chaos orbits of the system. The existence of a new multiple periods doubling bifurcation route to chaos and considerably modified bi-furcation structures and ranges of chaotic behavior are demonstrated by periodic impulses control. Anti-control bifurcations can be viewed as to design bifurcations into a system via impulsive control when such dynamical behaviors are desirable.The problem of anti-control chaos in discrete system has been investigated. An anti-control of chaos strategy was proposed to drive a periodic motion of a discrete system into a chaotic motion by using an impulsive control method. The simulation results showed the method is effective and successfully realized to anti-control of chaos in the discrete system by the method. Some different ways confirmed nonlinear systems can be divered from stable period motion to random chaotic motion by the control method.Finally, a summary has been done for all discussion in the dissertation. The research works in further study are presented.