Synchronization Between Two Different Chaotic Systems With Uncertainties

Synchronization in chaotic dynamic systems has attracted increasing attention of scientists from various research fields for its advantages in practical applications. Base on Lyapunov stability theory and Gerschgorin disc theorem, the synchronization between two different chaotic systems and its application to secure communications as well as the synchronization between two fly-ball governors are studied here.Firstly, a new method of virtual unknown parameter is proposed to synchronize two different systems with unknown parameters and disturbance in finite time. Virtual unknown parameters are introduced in order to avoid the unknown parameters from appearing in the controllers and parameters update laws when the adaptive control method is applied. A single virtual unknown parameter is used in the design of adaptive controllers and parameters update laws if the Lipschitz constant on the nonlinear function can be found, while multiple virtual unknown parameters are adopted if the Lipschitz constant cannot be determined. Numerical simulations show that the present method does make the two different chaotic systems synchronize in finite time.Secondly, an adaptive method is proposed to synchronize two different systems with unknown parameters and disturbance. Base on Lyapunov stability theory, an adaptive controller and parameters update laws are designed to synchronize two different chaotic systems even though the bounds on disturbance and parameters are unknown. Numerical simulation results verify the effectiveness of the method.Thirdly, Base on Lyapunov stability theory, an adaptive controller and parameters update laws are designed to synchronize hyperchaotic Lüsystem,with unknown parameters and external disturbance,and hyperchaotic R?ssler system. Once synchronization achieved, information signals hidden in the transmitter can be recovered exactly at the receiver. The synchronization strategy only uses the boundedness of the unknown parameters and external disturbance, and does not have to determine those bounds and Lipschitz constants in advance. Numerical simulation results verify the effectiveness of the method. At last,a sinusoidal feedback control is used to synchronize two coupling chaotic fly-ball governors. Some sufficient synchronization criteria in the form of a few algebraic inequalities are obtained by the Lyapunov direct method and Gerschgorin disc theorem. The property that similar matrices have the same eigenvalues is applied to get more flexible criteria. Numerical simulations are given to support our theory.