New megastable system with 2-D strip of hidden attractors and analytical solutions

I. M. Burkin ; O. I. Kuznetsova

Čebyševskij sbornik, Tome 22 (2021) no. 4, pp. 361-369.

Résumé

Many real dynamical systems are characterized by the presence of a set coexisting attractors. This property of systems is called multistability. In multistable systems, a sudden transition to unwanted or unknown attractors can occur. Such a transition can lead to catastrophic events. It turned out that multistability is also associated with the emergence of unpredictable attractors, which are called hidden attractors. An attractor is called hidden if its area of attraction does not intersect with small neighborhoods of an unstable fixed point. One of the defining reasons for studying multistable chaotic systems with different characteristics is a wide range of their potential engineering applications - in control theory, computer science, cryptology, artificial neural networks, image encryption, secure communication, and weak signal detection. In recent years, researchers have turned to developing methods for artificially designing systems with desired dynamics. In this case, the main efforts are focused on creating systems with an infinite number of coexisting attractors - extremely multistable and megastable systems. It turned out that such systems open up new possibilities for solving some applied problems, for example, for realizing control of the signal amplitude and polarity in engineering systems or for creating new image encryption systems. In this paper, a new smooth three-dimensional dynamical system is constructed, reversible in time, containing an analytical solution and a strange multifractal hidden attractor. The basin of attraction of the attractor includes almost all three-dimensional space, and its dimension is "almost 3". By replacing one of the variables of the system with a periodic function of this variable, a system is constructed that has a 1-D strip of hidden chaotic attractors of dimension "almost 3" and, at the same time, an infinite number of analytical solutions. A special transformation of the latter system allows us to design a megastable system with a 2-D strip of hidden attractors.

Détail

Keywords: dynamical systems, analytical solutions, chaos, megastability, hidden attractors, Lyapunov exponents, Kaplan-Yorke dimension.

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