On the new results of global attractive set and positive invariant set of the Lorenz chaotic system and the applications to chaos control and synchronization

Abstract

Constructing a family of generalized Lyapunov functions, a new method is proposed to obtain new global attractive set and positive invariant set of the Lorenz chaotic system. The method we proposed greatly simplifies the complex proofs of the two famous estimations presented by the Russian scholar Leonov. Our uniform formula can derive a series of the new estimations. Employing the idea of intersection in set theory, we extract a new Leonov formula-like estimation from the family of the estimations. With our method and the new estimation, one can confirm that there are no equilibrium, periodic solutions, almost periodic motions, wandering motions or other chaotic attractors outside the global attractive set. The Lorenz butterfly-like singular attractors are located in the global attractive set only. This result is applied to the chaos control and chaos synchronization. Some feedback control laws are obtained to guarantee that all the trajectories of the Lorenz systems track a periodic solution, or globally stabilize an unstable (or locally stable but not globally asymptotically stable) equilibrium. Further, some new global exponential chaos synchronization results are presented. Our new method and the new results are expected to be applied in real secure communication systems.