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Chaos of the coupled Klein-Gordon like equations with superlinear boundary conditions

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Abstract

The complicated dynamics is considered for a system of coupled Klein-Gordon like equations with two symmetric superlinear boundary conditions. The coupled Klein-Gordon like equations become coupled damped Klein-Gordon equations or telegraph equations or wave equations at a certain parameter value. The system is reduced to a discrete iteration of a 2D map which is different from the unary map induced by the non-coupled hyperbolic PDEs. In this article, the approach of snapback repeller is applied to establish the sufficient condition on the Li-Yorke chaos of the system. Moreover, by characterizing the asymptotic behavior of the 2D map, the global stability of the system is obtained under a certain range of parameter. The obtained dynamics not only extends the chaotic results on coupled wave equations with van der Pol type boundary conditions, but also gives the criterion of global stability.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 12201136), National Natural Science Foundation of China (Grant No. 12071151), Youth Foundation of Department of Education of Guangdong Province (Grant No. 2021KQNCX019), Natural Science Foundation of Guangdong Province (Grant No. 2021A1515010052), Basic and Applied Basic Research Foundation of Guangzhou (Grant No. 202201010278).

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Authors and Affiliations

  1. School of Mathematics, South China University of Technology, Guangzhou, 510640, China

    Pengxian Zhu

  2. School of Mathematics and Statistics, Guangdong University of Foreign Studies, Guangzhou, 510006, China

    Jiaopeng Yang

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  1. Pengxian Zhu
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Correspondence to Jiaopeng Yang.

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Zhu, P., Yang, J. Chaos of the coupled Klein-Gordon like equations with superlinear boundary conditions. Nonlinear Dyn 111, 10425–10439 (2023). https://doi.org/10.1007/s11071-023-08371-4

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