Design of multi-wing 3D chaotic systems with only stable equilibria or no equilibrium point using rotation symmetry

Abstract

Some dissipative dynamical systems only have stable equilibria or have no equilibrium point but are able to generate chaos of non-Sil’nikov type. This category of chaotic attractors belong to hidden attractor, which is difficult to be located in the phase space. In this work, we aim at designing multi-wing 3D chaotic systems with hidden attractors. We first proposes a quadratic system with only two stable node-foci, which can generate double-wing chaotic hidden attractor. Using rotation symmetry, we construct multi-fold cover of the quadratic system and obtain multiple symmetric stable equilibria. Two examples of 2-fold and 3-fold covers of the chaotic system are presented. Furthermore, the rotation symmetry is applied to no-equilibrium system to generate multi-wing chaotic hidden attractors. By means of bifurcation diagram, finite-time Lyapunov exponents, phase portraits, and attraction basins, it is shown that the multi-wing chaotic attractors always coexist with stable point attractors. The local initial condition areas and complex boundaries for different coexisting attractors are addressed. Finally, the designed systems are implemented by analog circuit and microcontroller to verify the multi-wing hidden attractors.