Global Dynamics of the Chaotic Disk Dynamo System Driven by Noise

The disk dynamo system, which is capable of chaotic behaviours, is obtained experimentally from two disk dynamos connected together. It models the geomagnetic field and is used to explain the reversals in its polarity. Actually, the parameters of the chaotic systems exhibit random fluctuation to a greater or lesser extent, which can carefully describe the disturbance made by environmental noise. The global dynamics of the chaotic disk dynamo system with random fluctuating parameters are concerned, and some new results are presented. Based on the generalized Lyapunov function, the globally attractive and positive invariant set is given, including a two-dimensional parabolic ultimate boundary and a four-dimensional ellipsoidal ultimate boundary. Furthermore, a set of sufficient conditions is derived for all solutions of the stochastic disk dynamo system being global convergent to the equilibrium point. Finally, numerical simulations are presented for verification.

Complex Dynamical Behaviors of Lorenz-Stenflo Equations

A mathematical chaos model for the dynamical behaviors of atmospheric acoustic-gravity waves is considered in this paper. Boundedness and globally attractive sets of this chaos model are studied by means of the generalized Lyapunov function method. The innovation of this paper is that it not only proves this system is globally bounded but also provides a series of global attraction sets of this system. The rate of trajectories entering from the exterior of the trapping domain to its interior is also obtained. Finally, the detailed numerical simulations are carried out to justify theoretical results. The results in this study can be used to study chaos control and chaos synchronization of this chaos system.

New Estimations of Bounds for the Chu Chaotic System

This paper is concerned with positive invariant set for the Chu chaotic system using a technique combing the generalized Lyapunov function theory and maximum principle. For this chaotic system, some new ellipsoid estimations and a cylindrical domain estimation of the globally exponentially attractive set for all the positive parameters values of the system are derived without existence assumptions via employing inequality techniques and the continuous extension principle.

GLOBALLY ATTRACTIVE AND POSITIVE INVARIANT SET OF THE LORENZ SYSTEM

In this paper, based on a generalized Lyapunov function, a simple proof is given to improve the estimation of globally attractive and positive invariant set of the Lorenz system. In particular, a new estimation is derived for the variable x. On the globally attractive set, the Lorenz system satisfies Lipschitz condition, which is very useful in the study of chaos control and chaos synchronization. Applications are presented for globally, exponentially tracking periodic solutions, stabilizing equilibrium points and synchronizing two Lorenz systems.