Extremely multistable dynamical systems with a continuum of hidden chaotic attractors

Abstract

In recent years, many researchers have focused on studying the phenomenon of extreme multistability of dynamic systems. An extremely multistable system contains an infinite number of coexisting attractors determined by different initial conditions. The latter circumstance introduces extreme uncertainty into its behavior and opens up the possibility of using such a system, for example, in cryptography and the organization of secure communication in
information transmission systems. Therefore, understanding the fundamental principle of the formation of extreme multistability is of particular interest. Only by understanding this principle we can generate 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 suggests the presence of an offset parameter in the system. As it turned out, the cancellation
of the offset parameter can lead to the presence of a continuum of coexisting attractors in the system, which are continuously located in the phase space and extend to infinity in a certain direction. This discovery can become, for example, an explanation for the occurrence
and propagation of tornadoes and turbulence. In this paper, using the dimension expansion technique, two fourth-order systems without equilibrium states containing a continuum of coexisting hidden chaotic attractors are constructed. The first system is based on the wellknown
three-dimensional Sprott system, and the second is based on the three-dimensional system proposed earlier by the authors, which has a single hidden chaotic attractor of dimension “almost 3”. The second system contains a 2D lattice, which is a union of a countable number of
strips, each of which contains a continuum of attractors.