Coexisting Infinite Equilibria and Chaos

Abstract

Equilibria are a class of attractors that host inherent stability in a dynamic system. Infinite number of equilibria and chaos sometimes coexist in a system with some connections. Hidden chaotic attractors exist independent of any equilibria rather than being excited by them. However, the equilibria can modify, distort, eliminate, or even instead coexist with the chaotic attractor depending on the distance between the equilibria and chaotic attractor. In this paper, chaotic systems with infinitely many equilibria are considered and explored. Extra surfaces of equilibria are introduced into the chaotic flows, showing that a chaotic system can maintain its basic dynamics if the newly added equilibria do not intersect the original attractor. The offset-boostable plane of equilibria rescales the frequency of the chaotic oscillation with an almost linearly modified largest Lyapunov exponent or conversely drives the system into periodic oscillation, even ending in a divergent state. Furthermore, additional infinite number of equilibria or even a solid space of equilibria are safely nested into the chaotic system without destroying the original dynamics, which provides an alternate permanent location for a dynamical system. A circuit simulation agrees with the numerical calculation.