Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self-excited Attractors

Abstract

"In the last few years, entropy has been a basic and important concept in information theory. It is also often used as a measure of the degree of chaos in systems, e.g., Lyapunov exponents, fractal dimension, and entropy are usually used to describe the complexity of chaotic systems. Thus, it would be very interesting to study entropy in nonlinear systems. Additionally, there has been an increasing interest in a new classification of nonlinear dynamical systems including two kinds of attractors: Self-excited attractors and hidden attractors. Self-excited attractors can be localized straight forwardly by applying a standard computational procedure. Some interesting examples of systems with self-excited attractors are chaotic systems with different kinds of symmetry, with multi-scroll attractors, with multiple attractors, and with extreme multistability.

In systems with hidden attractors we have to develop a specific computational procedure to identify the hidden attractors due to the fact that the equilibrium points do not help in their localization. Some examples of this kind of systems are chaotic dynamical systems with no equilibrium points, with only stable equilibria, with curves of equilibria, with surfaces of equilibria, and with non-hyperbolic equilibria. There is evidence that hidden attractors play an important role in the various fields ranging from phase-locked loops, oscillators, describing convective fluid motion, model of drilling system, information theory, cryptography to multilevel DC/DC converter. Furthermore, hidden attractors may lead to unexpected and disastrous responses."