Designing megastable systems with multidimensional lattice of chaotic attractors

I. M. Burkin ; O. I. Kuznetsova

Čebyševskij sbornik, Tome 22 (2021) no. 1, pp. 105-117

Résumé

Multistable systems and their dynamic properties are interesting topics in nonlinear dynamics. Small differences in the initial conditions (for example, due to rounding errors in numerical calculations) lead to completely different results for such dynamic systems, which makes a long-term prediction of their behavior practically impossible. This happens even if such systems are deterministic, that is, their future behavior is completely determined by the choice of initial conditions without the participation of random elements. In other words, the deterministic nature of these systems does not make them predictable. The behavior of the solutions of a dynamic system depends both on the choice of their initial conditions and on the values of the system parameters. The coexistence of several attractors, or multistability, corresponds to the simultaneous existence of more than one nontrivial attractor for the same set of system parameters. This phenomenon was discovered in almost all natural sciences, including electronics, optics, biology. In recent years, the efforts of many researchers have been aimed at creating so-called megastable systems, that is, systems that, at constant values of their parameters, have a countable number of coexisting attractors. Interest in such systems is due to a wide range of applications, for example, for hiding information in communication systems and audio encryption schemes, biomedical engineering and fuzzy control. The article proposes methods for the synthesis of megastable systems using systems in Lurie form. Megastable systems containing a $1$-D lattice of chaotic attractors can be obtained by replacing the nonlinearity in the original system with a periodic function. By replacing some variables with periodic functions in the original system of order $n$, one can construct a megastable system containing an $n$-D lattice of chaotic attractors. As one example, a fourth-order system with a $4$-D lattice of chaotic attractors is constructed for the first time. The Lyapunov exponents and Kaplan – Yorke dimension are calculated for attractors belonging to lattices.

Détail

Keywords: dynamical systems, chaos, countable number of coexisting attractors, Lyapunov exponents, Kaplan-Yorke dimension.