Ultimate boundary estimation and topological horseshoe analysis on a parallel 4D hyperchaotic system with any number of attractors and its multi-scroll

  • Enzeng DongEmail author
  • Zhijun Zhang
  • Mingfeng Yuan
  • Yuehui Ji
  • Xuesong Zhou
  • Zenghui Wang
Original Paper


This paper constructs a new four-dimensional autonomous hyperchaotic system with complex dynamic behaviors, and its boundary is estimated based on the proposed method and the optimization idea. Inspired by the “parallel universe theory,” a trigonometric function is used to do coordinate transformation of the original new system to generate any number of attractors. Simulation results show that there are infinite equilibriums in the transformed system. Compared with the original new system, the transformed system is more sensitive to the initial values. Based on the estimated boundary of the original system, the boundary of transformed system could be obtained. To verify the existence of chaos of the transformed system, the topology horseshoe of the system is investigated. The positive topological entropy of the transformed system verifies the existence of hyperchaos. Furthermore, selecting proper parameter values, the transformed system shows a multi-scroll attractor. Applying multi-variable trigonometric transformation can also induce any number of attractors and multi-scroll phenomenon in multi-dimension, which is an interesting phenomenon.


Hyperchaotic system Initial sensitivity Estimated boundary Any number of attractors Topological horseshoe Multi-scroll 



This work was partially supported by the Natural Science Foundation of China under Grant (Nos. 61603274 and 61502340), the Foundation of the Application Base and Frontier Technology Research Project of Tianjin (No. 15JCYBJC51800), South African National Research Foundation Grants (Nos. 112108 & 112142), South African National Research Foundation Incentive Grant (No. 114911) and Tertiary Education Support Programme (TESP) of South African ESKOM.

Compliance with ethical standards

Conflicts of interest

The authors declare no conflict of interest.


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Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Tianjin Key Laboratory For Control Theory and Applications in Complicated SystemsTianjin University of TechnologyTianjinChina
  2. 2.Department of Electrical and Mining EngineeringUniversity of South AfricaFloridaSouth Africa

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