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Hidden and self-excited coexisting attractors in a Lorenz-like system with two equilibrium points

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Abstract

This paper reports the finding of unusual hidden and self-excited coexisting dynamical behaviors in an existing Lorenz-like system. For different parameters, the system has different types of equilibrium points, such as saddle-nodes, stable focus-nodes, saddle-foci and nonhyperbolic equilibrium points, which can be used to find different types of hidden and self-excited attractors. The different types of attractors have been vividly demonstrated by several numerical techniques including phase portraits, bifurcation diagrams and basins of attraction. Very interestingly, we find the rare coexistence of chaotic attractor and periodic orbits in the Lorenz-like system with two saddle-foci.

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Acknowledgements

We would like to thank the anonymous referees for their valuable suggestions and questions. This work is partly supported by the National Natural Science Foundation of China (Grant Nos. 61873186 and 61773282), the Application Base and Frontier Technology Research Project of Tianjin of China (Grant No. 13JCQNJC03600) and South African National Research Foundation (Grant Nos. 112142 and 112108), South African National Research Foundation Incentive Grant (No. 114911) and Tertiary Education Support Programme (TESP) of South African ESKOM.

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Authors and Affiliations

  1. Department of Product Design, Tianjin University of Science and Technology, Tianjin, 300457, People’s Republic of China

    Shijian Cang

  2. School of Electronic Information and Automation, Tianjin University of Science and Technology, Tianjin, 300222, People’s Republic of China

    Shijian Cang & Yue Li

  3. School of Electrical and Information Engineering, Tianjin University, Tianjin, 300072, People’s Republic of China

    Shijian Cang & Ruiye Zhang

  4. Department of Electrical and Mining Engineering, University of South Africa, Florida, 1710, South Africa

    Zenghui Wang

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  1. Shijian Cang
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Correspondence to Shijian Cang.

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Cang, S., Li, Y., Zhang, R. et al. Hidden and self-excited coexisting attractors in a Lorenz-like system with two equilibrium points. Nonlinear Dyn 95, 381–390 (2019). https://doi.org/10.1007/s11071-018-4570-x

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