The Research On Methods For Chaos Control And Synchronization

Chaos control and synchronization are the frontier problems in nonlinear dynamics research. Sometimes, chaos is useful, as in a mixing process, or in heat transfer, but often it is harmful or unwanted, and must be controlled. It is thus of great importance to control chaos in academic research and practical applications. Different techniques and methods have been proposed to achieve chaos control and synchronization during the last two decades, but not all these approaches are universally applicable for chaos control and other new techniques need be found. Based on this, the paper proposes and develops a few new methods for chaos control and synchronization.The history of dynamic chaos is looked back on simply, and three definitions on chaos are introduced. Two fundamental properties of chaos (sensitive dependence on initial conditions and complex orbit structure) are detailed. Meanwhile, some important conceptions and techniques (Lyapunov exponents, phase portraits, frequency power spectra, the topological entropy, Poincare section, strange attractor) on chaos study are introduced. A few stability theoretics are proposed and prepared for the further research of chaos control and chaos synchronization.The co-existence of the complete synchronization and antisynchronization is found in two linear coupled Lorenz systems, and the inherent reason of the phenomenon is found and the basic property of the system that can display the co-existence phenomenon is investigated. The stability of synchronization in two linear coupled chaotic systems is studied using linear stability theory, the Lyapunov stability theory and the numerical method. Some sufficient conditions of global asymptotic synchronization are attained from rigorously mathematical theory.Generalized synchronization phenomena are studied. Nonlinear feedback control methods are used to achieve generalized synchronization of two arbitrary chaotic systems. The two hyperchaotic Chen system and two non-autonomous unified chaotic systems are treated as numerical examples. the Numerical simulation results show the effectiveness and feasibility of the theoretical analysis. In terms of a dynamical system, the amplitude of its oscillator need to be modified to desired size, meantime the other properties, such as the shape of attractor, can be not changed. The usual control strategy can not work for it. The transformation is presented to obtain a new controller that achieve attractor control. The new controller may modify the size of attractor and make the shape of attractor invariably in spite of the motion of the system is period or chaotic. Based on Lyapunov stability theory, an adaptive control law is derived such that the trajectories of two arbitrary chaotic systems with unknown parameters asymptotically synchronized. Numerical simulations are provided for illustration and verification of the proposed method.The dynamical behavior (example as bifurcation and chaos) of a system consisting of two coupled Duffing equations is investigated. In the situation of weakly nonlinearity, based on the method of multiple scales, approximate solutions of the coupled system are obtained both in the external and internal resonance cases. And in the situation of strongly nonlinearity, bifurcation and chaos behavior in the coupled system are found using numerical methods, for example, phase portraits, frequency power spectra, bifurcation diagrams and Poincare maps etal. Amplitude control and chaos control for uncertain chaotic systems are studied.Chaos behavior of Duffing oscillators and parametrically excited oscillators that derived from nonlinear mechanical models is analyzed. A sliding control strategy is applied to drive the chaotic motion of the chaotic systems to any defined reference signal in spite of modelling errors, parametric variations and perturbing external forces. Attractor control of mechanical systems are investigated. Using the proposed control method in the paper, the size of attractors is modified, meanwhile its shape is not changed. Non-ideal control is presented to achieve chaos control or anticontrol, and a numerical example is used to illustrate the method.