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Revealing the true and pseudo-singularly degenerate heteroclinic cycles

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Abstract

Through revisiting the four-dimensional chaotic system discussed in Wang et al. (Dyn Syst Control 8:129, 2019), its hidden dynamical behaviors that were not reported previously can be uncovered, such as the distribution and local stability of equilibrium points, generic and degenerate pitchfork bifurcation, Hopf bifurcation, the coexistence of true and pseudo-singularly degenerate heteroclinic cycles, and the existence of globally exponentially attractive sets as well as heteroclinic orbits. The innovation of the paper lies in the following results: (1) Coexisting pseudo-singularly degenerate heteroclinic cycles (the solution approaching infinity with a short-duration transient of singularly degenerate heteroclinic cycles) and true cycles consisting of normally hyperbolic saddle foci (or saddle nodes) and stable node foci are numerically illustrated with nearby hyperchaotic attractors, verifying a kind of mechanism for formulating hyperchaos. (2) Globally exponentially attractive sets with different exponential rates are located with the aid of the Lyapunov function. (3) The existence of a pair of symmetrically heteroclinic orbits is rigorously proven by the Lyapunov function; and the definitions of both the \(\alpha \)-limit set and \(\omega \)-limit set. Since these findings improve and complement the known results, we expect to provide a reference for real-world applications.

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Acknowledgements

This work is supported in part by National Natural Science Foundation of China under Grant 12001489, in part by Natural Science Foundation of Zhejiang Guangsha Vocational and Technical University of construction under Grant 2022KYQD-KGY, in part by Zhejiang public welfare Technology Application Research Project of China under Grant LGN21F020003, in part by Natural Science Foundation of Taizhou University under Grant T20210906033, in part by NSF of Zhejiang Province under Grant LQ18A010001.

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Authors and Affiliations

  1. School of Electronic and Information Engineering (School of Big Data Science), Taizhou University, Taizhou, 318000, People’s Republic of China

    H. Wang & Q. Su

  2. School of Information, Zhejiang Guangsha Vocational and Technical University of Construction, Dongyang, 322100, Zhejiang, People’s Republic of China

    G. Ke

  3. School of Information Engineering, GongQing Institute of Science and Technology, Gongqingcheng, 332020, People’s Republic of China

    G. Ke

  4. Department of Big Data Science, School of Science, Zhejiang University of Science and Technology, Hangzhou, 310023, People’s Republic of China

    J. Pan & H. Fan

  5. School of Innovation and Entrepreneurship/Center of Engineering Training, Zhejiang University of Science and Technology, Hangzhou, 310023, People’s Republic of China

    G. Dong

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  1. H. Wang
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  2. G. Ke
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Correspondence to G. Ke.

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Wang, H., Ke, G., Pan, J. et al. Revealing the true and pseudo-singularly degenerate heteroclinic cycles. Indian J Phys 97, 3601–3615 (2023). https://doi.org/10.1007/s12648-023-02689-w

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  • DOI: https://doi.org/10.1007/s12648-023-02689-w

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