Matryoshka multistability: Coexistence of an infinite number of exactly self-similar nested attractors in a fractal phase space

Multistability, and its special types such as megastability and extreme multistability, is an important phenomenon in modern nonlinear science that provides several possible practical applications. In this paper, we propose a new special type of multistability when the infinite number of exactly self-similar attractors nested inside each other coexist in a system. We called it matryoshka multistability due to its resemblance to the famous Russian wooden doll. We theoretically explain and experimentally confirm the properties of a new type of multistable behavior using two representative examples based on the Chua and Sprott Case J chaotic systems. In addition, we construct an adaptive controller for synchronizing two Chua-type matryoshka multistable systems when the amplitude of the master system is of arbitrary scale. The proposed type of multistability can find several applications in chaotic communication, cryptography, and data compression.