О НЕКОТОРЫХ ДИНАМИЧЕСКИХ СИСТЕМАХ С КОНТИНУУМОМ ХАОТИЧЕСКИХ АТТРАКТОРОВ

Игорь Михайлович Буркин

doi:10.17308/sait/1995-5499/2025/3/5-14

Authors

Igor M. Burkin Tula State University https://orcid.org/0000-0002-7362-3917 (unauthenticated)

DOI:

https://doi.org/10.17308/sait/1995-5499/2025/3/5-14

Keywords:

dynamic system, extreme multistability, chaos, continuum of coexisting attractors, Lyapunov exponents, Kaplan — Yorke dimension

Abstract

The phenomenon of multistability of dynamic systems has attracted the attention of many researchers in recent years. A multistable system contains several coexisting attractors, each with its own basin of attraction, and which can be visualized with an appropriate choice of the initial condition. An extremely multistable system contains an infinite number of coexisting nontrivial attractors. The presence of an infinite number of attractors causes extreme uncertainty and unpredictability in the behavior of the system, which, in turn, opens up the possibility of using such a system, for example, in cryptography and encryption of transmitted video and audio information in communication systems. Of particular interest is the understanding of the fundamental law of formation of extreme multistability. Understanding this law allows generating systems with the desired behavior. Extreme multistability of many currently known systems can be explained by the presence of the phenomenon of offset boosting, which assumes the presence of a displacement parameter in the system. As it turned out, the cancellation of a displacement parameter can lead to the presence in the system of a continuum of coexisting attractors that are continuously located in the phase space and extend to infinity in a certain direction. One of the methods for generating an extremely multistable system with an uncountable number of attractors based on a known system containing a chaotic attractor is a special expansion of the dimensionality of this system. In this paper, the method of expanding the dimensionality is used to construct two fourth-order systems containing a continuum of coexisting chaotic attractors. The f irst system is based on the known three-dimensional Sprott system, and the second is based on a third-order system proposed by the author, which has a pair of hidden chaotic twin attractors.

Author Biography

Igor M. Burkin, Tula State University

Doctor of Physical and Mathematical Sciences, Associate Professor, Professor of the Department of Computational Mechanics and Mathematics

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