Boundedness Of Several Chaotic Systems And Its Application

The ultimate bound of a chaotic system is important for the study of the qualitative behavior of a chaotic system, but to estimate the globally exponentially attractive and ultimate bound set for a dynamic system is a quite challenging task in general. A chaotic system is bounded, in the sense that its chaotic attractor is bounded in the phase space, and the estimate of its bound is important in chaos control, chaos synchronization and their applications.In this paper, three chaotic systems are investigated as follows: a general Lorenz chaotic system, a Brushless DC Motors system and a new three-dimensional chaotic system. To the general Lorenz chaotic system, its stability is analyzed, linear feedback control methods are proposed to control the general Lorenz chaotic system to its unstable equilibrium points, linear feedback control and adaptive control methods are used to achieve chaos synchronization. The ultimate bound and positively invariant sets for the Brushless DC Motors system and the new three-dimensional chaotic system were investigated via the generalized Lyapunov function and optimization. Then the result is applied to the study of chaos synchronization of the two systems. Numerical simulations are presented to show the effectiveness of the proposed chaos synchronization scheme.