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Local bifurcation analysis and topological horseshoe of a 4D hyper-chaotic system

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Abstract

In this paper, a 4D autonomous system with complex hyper-chaotic dynamics is introduced. The Lyapunov exponent spectrum, bifurcation diagram and phase portrait are provided. Basic dynamical properties are also analyzed. In order to clarify the evolution of the complex dynamic behaviors of the system, the local bifurcation is studied and a Hopf bifurcation is proved to occur when the appropriate bifurcation parameter passes the critical value. All the conditions of Hopf bifurcation are derived by applying center manifold theorem and Poincaré–Andronov–Hopf bifurcation theorem. Numerical simulation results are consistent with the theoretical analysis. Besides, we present a rigorous study on the hyper-chaotic system by combining the topological horseshoe theory with a computer-assisted approach of Poincaré maps. Utilizing the algorithm for finding horseshoes in 3D hyper-chaotic maps, a horseshoe with two-directional expansion in the 4D hyper-chaotic system has been found, which rigorously proves the existence of hyper-chaos in theory.

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Acknowledgments

This work is partially supported by Natural Science Foundation of China Grants No.61174094 , No.61573199 and No.61533011. Tianjin Nature Science Foundation Grant No. 14JCYBJC18700, and Shandong Provincial Natural Science Foundation Grant No.ZR2014FQ019.

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Authors and Affiliations

  1. College of Control Science and Engineering, Shandong University, Jinan, 250061, China

    Zhonglin Wang

  2. Department of Physics and Electronics, Binzhou University, Binzhou, 256600, China

    Zhonglin Wang

  3. College of Computer and Control Engineering, Nankai University, Tianjin, 300071, China

    Leilei Zhou & Zengqiang Chen

  4. College of Science, Civil Aviation University of China, Tianjin, 300300, China

    Zengqiang Chen & Jiezhi Wang

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  1. Zhonglin Wang
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  2. Leilei Zhou
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  4. Jiezhi Wang
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Correspondence to Leilei Zhou.

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Wang, Z., Zhou, L., Chen, Z. et al. Local bifurcation analysis and topological horseshoe of a 4D hyper-chaotic system. Nonlinear Dyn 83, 2055–2066 (2016). https://doi.org/10.1007/s11071-015-2464-8

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