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Chaotic Dynamics of a 2D Hyperbolic PDE with the Boundary Conditions of Superlinear Type

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Abstract

The study of the chaotic dynamics of partial differential equation (PDE) system has become the focus of dynamical system. While important progress has been made for the research of chaos theory of one-dimensional wave PDE, the understanding of chaotic vibrations for two-dimensional (2D) hyperbolic PDE is still incomplete. This paper is concerned with a 2D hyperbolic PDE system governed by a linear equation with two suplinear boundary conditions. Based on the chaotic mapping theory and method of characteristics, sufficient conditions for the existence of nonisotropic chaotic vibrations are obtained for such system under three different system parameters. In addition, three numerical examples are provided to illustrate the effectiveness of our theoretical results.

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References

  1. Block, L., Coppel, W.A.: Dynamics in One Dimension. Springer, New York (1992)

    Book  Google Scholar 

  2. Brunovský, P.: Notes on chaos in the cell population partial differential equation. Nonlinear Anal. TMA 7, 167–176 (1983)

    Article  MathSciNet  Google Scholar 

  3. Chen, G., Hsu, S. B., Zhou, J.: Chaotic vibrations of the one-dimensional wave equation due to a self-excitation boundary condition. Part I: controlled hysteresis. Trans. Am. Math. Soc. 350, 4265–4311 (1998)

    Article  Google Scholar 

  4. Chen, G., Hsu, S.B., Zhou, J.: Nonisotropic spatiotemporal chaotic vibration of the wave equation due to mixing energy transport and a van der Pol boundary condition. Int. J. Bifurc. Chaos 12, 535–559 (2002)

    Article  MathSciNet  Google Scholar 

  5. Chen, G., Huang, T.W., Huang, Y.: Chaotic behavior of interval maps and total variations of iterates. Int. J. Bifur. Chaos 14, 2161–2186 (2004)

    Article  MathSciNet  Google Scholar 

  6. Chen, G., Huang, Y.: Chaotic Maps: Dynamics, Fractals and Rapid Fluctuations. Synthesis Lectures on Mathematics and Statistics, ed. Krantz, S.G. Morgan & Claypool Publisher, Williston (2011)

  7. Chen, G., Sun, B., Huang, T.W.: Chaotic oscillations of solutions of the Klein–Gordon equation due to imbalance of distributed and boundary energy flows. Int. J. Bifurc. Chaos 24, 1430021 (2014)

    Article  MathSciNet  Google Scholar 

  8. Dai, X.P., Huang, T.W., Huang, Y., Chen, G.: Chaotic oscillations of solutions of first order hyperbolic systems in 1D with nonlinear boundary conditions. Int. J. Bifurc. Chaos 24, 1450072 (2014)

    Article  MathSciNet  Google Scholar 

  9. Devaney, R.L.: An Introduction to Chaotic Dynamical Systems. Addison-Wesley, New York (1989)

    MATH  Google Scholar 

  10. Hu, C.C.: Chaotic vibrations of the one-dimensional mixed wave system. Int. J. Bifurc. Chaos 19, 579–590 (2009)

    Article  MathSciNet  Google Scholar 

  11. Huang, Y.: Growth rates of total variations of snapshots of the 1D linear wave equation with composite nonlinear boundary reflection relations. Int. J. Bifurc. Chaos 13, 1183–1195 (2003)

    Article  MathSciNet  Google Scholar 

  12. Huang, Y.: A new characterization of nonisotropic chaotic vibrations of the one-dimensional linear wave equation with a van der Pol boundary condition. J. Math. Anal. Appl. 288, 78–96 (2003)

    Article  MathSciNet  Google Scholar 

  13. Huang, Y., Feng, Z. S.: Infinite-dimensional dynamical systems induced by interval maps. Dyn. Contin. Discr. Impuls. Syst. Ser. A 13, 509–524 (2009)

    MathSciNet  MATH  Google Scholar 

  14. Khan, M., Shah, T., Gondal, M.A.: An efficient technique for the construction of substitution box with chaotic partial differential equation. Nonlinear Dyn. 73, 1795–1801 (2013)

    Article  MathSciNet  Google Scholar 

  15. Li, L.L., Chen, Y.L., Huang, Y.: Nonisotropic spatiotemporal chaotic vibrations of the one-dimensional wave equation with a mixing transport term and general nonlinear boundary condition. J. Math. Phys. 51, 102703 (2010)

    Article  MathSciNet  Google Scholar 

  16. Li, L.L., Huang, T.W., Huang, X.Y.: Chaotic oscillations of the 1D wave equation due to extreme imbalance of self-regulations. J. Math. Anal. Appl. 450, 1388–1400 (2017)

    Article  MathSciNet  Google Scholar 

  17. Li, L.L., Tian, J., Chen, G.: Chaotic vibration of a two-dimensional non-strictly hyperbolic equation. Can. Math. Bull. 61, 768–786 (2018)

    Article  MathSciNet  Google Scholar 

  18. Robinson, C.: Dynamical Systems: Stability, Symbolic Dynamics and Chaos. CRC Press, Boca Raton (1995)

  19. Strauss, W.A.: The Energy Method in Nonlinear Partial Differential Equations. Notas de Matemática, Instituto de Matemática Pura e Aplicada, Brazil (1969)

    MATH  Google Scholar 

  20. Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos, 2nd edn. Spinger, New York (2003)

  21. Xiang, Q.M., Yang, Q.G.: Chaotic oscillations of linear hyperbolic PDE with general nonlinear boundary condition. J. Math. Anal. Appl. 472, 94–111 (2019)

    Article  MathSciNet  Google Scholar 

  22. Xiang, Q.M., Yang, Q.G.: Nonisotropic chaotic oscillations of the wave equation due to the interaction of mixing transport term and superlinear boundary condition. J. Math. Anal. Appl. 462, 730–7746 (2018)

    Article  MathSciNet  Google Scholar 

  23. Xiang, Q.M., Yang, Q.G.: Nonisotropic chaotic vibrations of a 2D hyperbolic PDE. Chaos 30, 023127 (2020)

    Article  MathSciNet  Google Scholar 

  24. Xiang, Q.M., Yin, Z.B., Zhu, P.X.: Chaotic dynamics of linear hyperbolic PDEs with nonlinear boundary conditions. Chaos Soliton Fract. 131, 109525 (2020)

    Article  MathSciNet  Google Scholar 

  25. Yang, Q.G., Xiang, Q.M.: Chaotic oscillations of linear hyperbolic PDE with variable coefficients and implicit boundary conditions. Discrete Cont. Dyn. Syst. Ser. S (2020). https://doi.org/10.3934/dcdss.2020335

    Article  Google Scholar 

  26. Yang, X.D., Chen, L.Q.: Bifurcation and chaos of an axially accelerating viscoelastic beam. Chaos Soliton Fract. 23, 249–258 (2005)

    Article  Google Scholar 

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Funding

This work was partly supported by the National Natural Science Foundation of China (No. 11901091, 11871225, 11801096, 12071074), Guangdong Basic and Applied Basic Research Foundation (No. 2020A151501339) and Foundation for Distinguished Young Talents in Higher Education of Guangdong, China (No. 2018KQNCX307).

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Authors and Affiliations

  1. Basic Teaching Department, Guangzhou College of Technology and Business, Guangzhou, 510850, China

    Pengxian Zhu

  2. School of Mathematics and Big Data, Foshan University, Foshan, 528000, China

    Qiaomin Xiang

  3. School of Science, Tianjin University of Commerce, Tianjin, 300134, China

    Kai Lu

Authors
  1. Pengxian Zhu
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  2. Qiaomin Xiang
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  3. Kai Lu
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Correspondence to Qiaomin Xiang.

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Zhu, P., Xiang, Q. & Lu, K. Chaotic Dynamics of a 2D Hyperbolic PDE with the Boundary Conditions of Superlinear Type. Qual. Theory Dyn. Syst. 20, 30 (2021). https://doi.org/10.1007/s12346-021-00470-0

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