Extremely multistable matryoshka systems with hidden attractors

Multistability in chaotic systems is typically classified based on the number of coexisting attractors, their shapes, and emergence mechanisms. However, combinations of multiple types of multistability are rarely explored. This study proposes a technique for constructing systems in the Lurie form with three types of multistability simultaneously. The proposed systems have infinite 1D or 2D lattices of infinitely nested self-similar pairs of twin attractors, which refer to megastability, matryoshka multistability, and coexistence of symmetric twin attractors, respectively. All attractors in the proposed systems are hidden. The mechanisms to achieve these types of multistability rely on manipulating the nonlinear function of a system in the Lurie form. When adding periodic behavior results in a 1D lattice of identical attractors, log-periodic behavior results in matryoshka-type nested self-similar attractors, and symmetry about zero causes the emergence of twin attractors. Offset boosting allows the formation of a 2D lattice of attractors. Theoretical and numerical analyses of the proposed systems are given, including one- and two-dimensional bifurcation diagrams, the largest Lyapunov exponent calculation, and phase space representation. The findings offer deep insights into the nature of the studied types of multistability as well as a framework for further developing systems with combined multistability in chaotic systems.