Fractal Synchronization of Chaotic Modes

Abstract

The work develops methods and techniques for monitoring the synchronization of chaotic generators and measuring of nonlinear dynamical quantities. Chaotic modes are selected as the key objects for synchronization control. A universal definition of a chaotic mode is proposed. It is a type of chaotic oscillation that is characteristic of the considered nonlinear oscillatory system, globally stable, distinguishable, and reproducible. A chaotic mode is treated as a state vector of the system in phase space. All possible values of the state vector form the phase portrait of the system. The phase portrait of a chaotic system exhibits a fractal structure. The work puts forward a hypothesis on the possibility of using fractal dimension to evaluate and measure the degree of synchronization of chaotic modes. As a result, the conditions of applicability, as well as the advantages and limitations of the proposed fractal approach, are established.