Study On The Globally Attractive Set And Positively Invariant Set Of The Lorenz System Family And Its Application

The ultimate bound of a chaotic system is important for the study of the qualitative behavior of a chaotic system. If we can show that a system under consideration has a globally attractive set, then we know that the system cannot have equilibrium points, periodic solutions, quasi-periodic solutions, or other chaotic attractors outside the globally attractive set. This greatly simplifies the analysis of the dynamical properties of the system. The ultimate bound also plays an important role in designing scheme for chaos control and chaos synchronization.This paper investigates the ultimate bound and positively invariant set for three chaotic systems via the generalized Lyapunov function and optimization.Some basic dynamical properties are investigated for a new chaotic system, we derive a three-dimensional ellipsoidal ultimate bound and positively invariant set and prove the existence in mathematical theory. And the two-dimensional bound with respect to x ? z and y ? zare established.With the help of the same method, we get the many different ellipsoidal-like formulas for the Qi chaotic system and the Lorenz-Stenflo chaotic system. Then the result is applied to the study of chaos synchronization and the globally synchronization of the two systems is achieved. Numerical simulations are presented to show the effectiveness of the proposed chaos synchronization scheme...