New Chaotic Systems with Two Closed Curve Equilibrium Passing the Same Point: Chaotic Behavior, Bifurcations, and Synchronization

Xinhe Zhu,Wei-Shih Du

doi:10.3390/sym11080951

Abstract

In this work, we introduce a chaotic system with infinitely many equilibrium points laying on two closed curves passing the same point. The proposed system belongs to a class of systems with hidden attractors. The dynamical properties of the new system were investigated by means of phase portraits, equilibrium points, Poincaré section, bifurcation diagram, Kaplan–Yorke dimension, and Maximal Lyapunov exponents. The anti-synchronization of systems was obtained using the active control. This study broadens the current knowledge of systems with infinite equilibria.

Highlights

IntroductionChaotic systems have been widely studied and used in various practical fields by mathematicians, physicists, scientists, and engineers in the past four decades; see [1,2,3,4] and the references therein
Chaotic systems have been widely studied and used in various practical fields by mathematicians, physicists, scientists, and engineers in the past four decades; see [1,2,3,4] and the references therein.Many chaotic systems with different shapes of attractors have been reported, such as chaotic systems with butterfly attractors and systems with multiscroll chaotic attractors.Recent developments include some different types of chaotic systems with no equilibrium points, with a single stable equilibrium, with a line of equilibrium points, with a circular equilibrium, with circle and square equilibrium, with rounded square loop equilibrium, and with different closed curve equilibrium
A system with infinitely many equilibrium points can be classified as one system with hidden attractors, for the reason that we do not know which part of the equilibria may be used to localize the hidden attractors, which should be treated in detail

Summary

Introduction

Chaotic systems have been widely studied and used in various practical fields by mathematicians, physicists, scientists, and engineers in the past four decades; see [1,2,3,4] and the references therein. Recent developments include some different types of chaotic systems with no equilibrium points (see, e.g., [7]), with a single stable equilibrium (see, e.g., [8]), with a line of equilibrium points (see, e.g., [9]), with a circular equilibrium (see, e.g., [10]), with circle and square equilibrium (see, e.g., [11]), with rounded square loop equilibrium (see, e.g., [12]), and with different closed curve equilibrium (see, e.g., [13]).

A New Family with Two Closed Curve Equilibrium

Anti-Synchronization of New Systems

Conclusions