The Impulsive Control And Anti-control Of Chaotic Systems With Only One Nonlinear Term

Chaos is a special dynamic phenomenon in nonlinear science, with some fascinating characteristics.It displays irregular movements in the deterministic systems,which are similar to those in stochastic ones.In many areas it has great potential applications.Now it has been becoming research focus in nonlinear science in recent years.Generally speaking, the chaos control is divided into two parts.One is the inhibition of the chaotic behavior, and the other is the creation of or enhancing the chaotic behavior of systems,which is commonly referred to as chaos anti-control. Since 1990s, Ott, Grebogi and Yorke gave an effective chaos control method, known as OGY method, for the first time,it completely changed conservative views that the chaos could not be controlled. Up to now, in order to achieve the control of chaotic systems, many types of control methods have been put forward, including optimal control, identification control,digital control, fuzzy control, adaptive control and so on. However, it should be pointed out that these control methods are mainly applied to some special chaotic system such as Chua system,Lorenz system,Chen system. Very few studies have been devoted to generally chaotic system with only one nonlinear term.Since in chaotic attractors there are the wealth of unstable periodic orbits, which are found to be of great significance in many disciplines,especially in engineering technologies, the anti-control of chaos is an issue worthy of intensive study in the nonlinear science.In 1996,Chen Guan-rong and Lai De-jian for the first time were engaged in research work in this area, and proposed a set of simple and efficient linear iterative methods,mainly applied to discrete chaotic systems. However, it is still difficult to apply to the continuous chaotic systems.In this paper, a class of continuous chaotic systems with only one nonlinear term is to be controlled by impulsive and linear feedback controls.First some definitions of chaos,chaos control and anti-control,basic history, theory and significance of chaos are briefly reviewed. Then we focused on discussing a class of effective chaotic control method--impulse control.It is to steady the chaotic dynamics to the desired stabilization point. The theoretical aspects of verification and numerical simulation are also given. Finally, chaotic systems with only one nonlinear term for anti-control problem are studied. The linear feedback control scheme is adopted and a new chaotic phenomenon is displayed. At the same time,some numerical evidence of chaotic behaviors have been given, including the phase diagrams of system, the sensitive dependence on initial conditions,the bifurcation diagrams, Lyapunov exponents, fractal dimensions,continuous power spectrums.