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

Zhouchao Wei

http://orcid.org/0000-0001-6981-748X

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

E-mail Address: weizc@cug.edu.cn

E-mail Address: weizhouchao@163.com

Corresponding author.

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Yingying Li

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

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Bo Sang

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

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Yongjian Liu

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

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, and

Wei Zhang

http://orcid.org/0000-0003-0210-5642

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

E-mail Address: sandyzhang0@yahoo.com

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https://doi.org/10.1142/S0218127419500950Cited by:39 (Source: Crossref)

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.

Keywords:

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