Research On Control And Synchronization Of Chaos In The Nonlinear Vibration

Chaos is a special phenomenon which reflects the irregular and highly complex structures in time and in space that follow deterministic laws and equations. It has been studied extensively in many areas in recent years.Chaos control refers to controlling a chaotic motion to a desired deterministic one by some methods. The research works on it was firstly reported in 1990. The control scheme presented includes OGY method, delay feedback method and proportionally impulsive control method. In this paper, some research works is carried out on the topic. For the autonomous dynamic system, e.g., Lorenz system, Chen system, based on the Lyapunov stability theory, a control method is presented to control their chaotic responses to stabilize on the equilibrium point. Simultaneously a method is tried to control their chaos to a stable periodic trajectory. In addition, by using its characteristic of approaching a nonlinear function by a very high accuracy, a method based on artificial neural network is presented to control the chaotic response of autonomous and nonautonomous system to a stable equilibrium point or a prospective input function. The function can be even non-continuous, e.g., swath wave or jump function.In chaos control, there is an important area called chaos synchronization. Chaos synchronization refers to the identical motion between the original chaos system and the controlled system. Since the chaos is sensitive to the initial condition, it is impossible to predict the chaotic response. It is difficult to make a system, called derived system, to duplicate the identical motion of the original system. In this paper, for several types of nonlinear vibration system, some methods to realize synchronization are presented and proved. For the vibration system with constant parameters and nonlinear restoring force, a derived system and control method are designed to realize synchronization. The simulation is carried out on Φ~6-Duffing system and the result shows that the two systems reach identical synchronization. Then for the vibration system with time-varied parameters and without time-varied elements in restoring force, a principal to design a derived synchronization system is given and proved. For the derived system, its typical characteristic is that the response can only reach synchronization with the original deterministic motion. It can't reach synchronization with the original unstable motion, e.g., chaos. The simulation on Mathieu system proves this point. Finally for the general nonlinear vibration system, namely system with time-varied parameters and forces, it is proved that the linear coupling can keep identical synchronization between the original and derived system. The principal to form the coupling matrix is presented. For parameters-excited Mathieu system and quasi-periodically excited Duffing system, the numerical simulations are carried out and the results prove that the method is effective. In the application of synchronization, a novel method to distinguish the faint difference in exciting force of Duffing system is presented. By simulation, the faint variation in amplitude and frequency of exciting force can lead to the loss of synchronization between the two systems. So the synchronization can be used to detect the faint variation in signal.Finally the further research direction and some thoughts on works are presented for consultation and corrections.