Complex Dynamical Behaviors in a 3D Simple Chaotic Flow with 3D Stable or 3D Unstable Manifolds of a Single Equilibrium

Author:

Wei Zhouchao1ORCID,Li Yingying1,Sang Bo2,Liu Yongjian3,Zhang Wei4ORCID

Affiliation:

1. School of Mathematics and Physics, China University of Geosciences, Wuhan 430074, P. R. China

2. School of Mathematical Sciences, Liaocheng University, Liaocheng 252059, P. R. China

3. Guangxi Colleges and Universities Key Laboratory of Complex System Optimization and Big Data Processing, Yulin Normal University, Yulin, Guangxi 537000, P. R. China

4. College of Mechanical Engineering, Beijing University of Technology, Beijing 100124, P. R. China

Abstract

This paper shows some examples of chaotic systems for the six types of only one hyperbolic equilibrium in changed chameleon-like chaotic system. Two of the six cases have hidden attractors. By adjusting the parameters in the system and controlling the stability of only one equilibrium, we can further obtain chaos with four kinds of conditions: (1) index-0 node; (2) index-3 node; (3) index-0 node foci; (4) index-3 node foci. Based on the method of focus quantities, we study three limit cycles (the outmost and inner cycles are stable, and the intermediate cycle is unstable) bifurcating from an isolated Hopf equilibrium. In addition, one periodic solution can be obtained from a nonisolated zero-Hopf equilibrium. The system may help us in better understanding, revealing an intrinsic relationship of the global dynamical behaviors with the stability of equilibrium point, especially hidden chaotic attractors.

Funder

National Natural Science Foundation of China

Natural Science Foundation of Shandong Province

Guangxi University High Level Innovation Team and Distinguished Scholars Program of China

Publisher

World Scientific Pub Co Pte Lt

Subject

Applied Mathematics,Modelling and Simulation,Engineering (miscellaneous)

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