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## Chaotic oscillations of linear hyperbolic PDE with variable coefficients and implicit boundary conditions

 1 Department of Mathematics, South China University of Technology, Guangzhou, 510640, China 2 Department of Mathematics and Big Data, Foshan University, Foshan, 528000, China

* Corresponding author: Qigui Yang

Received  March 2019 Revised  November 2019 Published  April 2020

In this paper, the chaotic oscillations of the initial-boundary value problem of linear hyperbolic partial differential equation (PDE) with variable coefficients are investigated, where both ends of boundary conditions are nonlinear implicit boundary conditions (IBCs). It separately considers that IBCs can be expressed by general nonlinear boundary conditions (NBCs) and cannot be expressed by explicit boundary conditions (EBCs). Finally, numerical examples verify the effectiveness of theoretical prediction.

Citation: Qigui Yang, Qiaomin Xiang. Chaotic oscillations of linear hyperbolic PDE with variable coefficients and implicit boundary conditions. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020335
##### References:
 [1] G. Chen, S.-B. Hsu and J. X. Zhou, Chaotic vibrations of the one-dimensional wave equation due to a self-excitation boundary condition. Ⅰ: Controlled hysteresis, Trans. Amer. Math. Soc., 350 (1998), 4265-4311.  doi: 10.1090/S0002-9947-98-02022-4.  Google Scholar [2] G. Chen, S.-B. Hsu and J. X. Zhou, Chaotic vibrations of the one-dimensional wave equation due to a self-excitation boundary condition. Ⅱ: Energy injection, period doubling and homoclinic orbits, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 8 (1998), 423-445.  doi: 10.1142/S0218127498000280.  Google Scholar [3] G. Chen, S.-B. Hsu and J. X. Zhou, Chaotic vibrations of the one-dimensional wave equation due to a self-excitation boundary condition. Ⅲ: Natural hysteresis memory effects, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 8 (1998), 447-470.  doi: 10.1142/S0218127498000292.  Google Scholar [4] G. Chen, S.-B. Hsu and J. X. 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Huang, Nonisotropic spatiotemporal chaotic vibrations of the one-dimensional wave equation with a mixing transport term and general nonlinear boundary condition, J. Math. Phys., 51 (2010), 102703, 22 pp. doi: 10.1063/1.3486070.  Google Scholar [13] L. L. Li, Y. Huang, G. Chen and T. W. Huang, Chaotic oscillations of second order linear hyperbolic equations with nonlinear boundary conditions: A factorizable but noncommutative case, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 25 (2015), 1530032, 20 pp. doi: 10.1142/S0218127415300323.  Google Scholar [14] L. L. Li, T. W. Huang and X. Y. Huang, Chaotic oscillations of the 1D wave equation due to extreme imbalance of self-regulations, J. Math. Anal. Appl., 450 (2017), 1388-1400.  doi: 10.1016/j.jmaa.2017.01.095.  Google Scholar [15] Y. C. Li, Chaos in Partial Differential Equations, Graduate Series in Analysis. International Press, omerville, MA, 2004.  doi: 10.1002/cnm.650.  Google Scholar [16] S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, 2$^nd$ edition, Texts in Applied Mathematics, 2. Springer-Verlag, New York, 2003. doi: 10.1063/1.4822950.  Google Scholar [17] Q. M. Xiang and Q. G. Yang, Nonisotropic chaotic oscillations of the wave equation due to the interaction of mixing transport term and superlinear boundary condition, J. Math. Anal. Appl., 462 (2018), 730-746.  doi: 10.1016/j.jmaa.2018.02.031.  Google Scholar [18] Q. M. Xiang and Q. G. Yang, Chaotic oscillations of a linear hyperbolic PDE with a general nonlinear boundary condition, J. Math. Anal. Appl., 472 (2019), 94-111.  doi: 10.1016/j.jmaa.2018.10.083.  Google Scholar [19] Q. G. Yang, G. R. Jiang and T. S. Zhou, Chaotification of linear impulsive differential systems with applications, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 22 (2012), 1250297, 12 pp. doi: 10.1142/S0218127412502975.  Google Scholar [20] Q. G. Yang and Q. M. Xiang, Existence of chaotic oscillations in second-order linear hyperbolic PDEs with implicit boundary conditions, J. Math. Anal. Appl., 457 (2018), 751-775.  doi: 10.1016/j.jmaa.2017.08.018.  Google Scholar [21] Z. B. Yin and Q. G. Yang, Distributionally scrambled set for an annihilation operator, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 25 (2015), 1550178, 13 pp. doi: 10.1142/S0218127415501783.  Google Scholar [22] Z. B. Yin and Q. G. Yang, Generic distributional chaos and principal measure in linear dynamics, Ann. Pol. Math., 118 (2016), 71-94.  doi: 10.4064/ap3908-9-2016.  Google Scholar [23] Z. B. Yin and Q. G. Yang, Distributionally n-chaotic dynamics for linear operators, Rev. Mat. Complut., 31 (2018), 111-129.  doi: 10.1007/s13163-017-0226-5.  Google Scholar [24] Z. B. Yin and Q. G. Yang, Distributionally n-scrambled set for weighted shift operators, J. Dyn. Control Syst., 23 (2017), 693-708.  doi: 10.1007/s10883-017-9359-6.  Google Scholar

show all references

##### References:
 [1] G. Chen, S.-B. Hsu and J. X. Zhou, Chaotic vibrations of the one-dimensional wave equation due to a self-excitation boundary condition. Ⅰ: Controlled hysteresis, Trans. Amer. Math. Soc., 350 (1998), 4265-4311.  doi: 10.1090/S0002-9947-98-02022-4.  Google Scholar [2] G. Chen, S.-B. Hsu and J. X. Zhou, Chaotic vibrations of the one-dimensional wave equation due to a self-excitation boundary condition. Ⅱ: Energy injection, period doubling and homoclinic orbits, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 8 (1998), 423-445.  doi: 10.1142/S0218127498000280.  Google Scholar [3] G. Chen, S.-B. Hsu and J. X. Zhou, Chaotic vibrations of the one-dimensional wave equation due to a self-excitation boundary condition. Ⅲ: Natural hysteresis memory effects, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 8 (1998), 447-470.  doi: 10.1142/S0218127498000292.  Google Scholar [4] G. Chen, S.-B. Hsu and J. X. Zhou, Nonisotropic spatiotemporal chaotic vibration of the wave equation due to mixing energy transport and a van der Pol boundary condition, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 12 (2002), 535-559.  doi: 10.1142/S0218127402004504.  Google Scholar [5] G. Chen, T. W. Huang and Y. Huang, Chaotic behavior of interval maps and total variations of iterates, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 14 (2004), 2161-2186.  doi: 10.1142/S0218127404010540.  Google Scholar [6] G. Chen, B. Sun and T. W. Huang, Chaotic oscillations of solutions of the Klein-Gordon equation due to inbalance of distributed and boundary energy flows, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 24 (2014), 1430021, 19 pp. doi: 10.1142/S0218127414300213.  Google Scholar [7] X. P. Dai, Chaotic dynamics of continuous-time topological semiflow on Polish spaces, J. Differential Equations, 258 (2015), 2794-2805.  doi: 10.1016/j.jde.2014.12.027.  Google Scholar [8] M. W. Hirsch, S. Smale and R. L. Devaney, Differential Equations, Dynamical Systems, and an Introduction to Chaos, Second edition, Pure and Applied Mathematics (Amsterdam), 60. Elsevier/Academic Press, Amsterdam, 2004. doi: 10.1016/C2009-0-61160-0.  Google Scholar [9] C.-C. Hu, Chaotic vibrations of the one-dimensional mixed wave system, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 19 (2009), 579-590.  doi: 10.1142/S0218127409023202.  Google Scholar [10] Y. Huang, Growth rates of total variations of snapshots of the 1D linear wave equation with composite nonlinear boundary reflection relations, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), 1183-1195.  doi: 10.1142/S0218127403007138.  Google Scholar [11] Y. Huang, J. Luo and Z. L. Zhou, Rapid fluctuations of snapshots of one-dimensional linear wave equations with a van der Pol nonlinear boundary condition, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15 (2005), 567-580.  doi: 10.1142/S0218127405012223.  Google Scholar [12] L. L. Li, Y. L. Chen and Y. Huang, Nonisotropic spatiotemporal chaotic vibrations of the one-dimensional wave equation with a mixing transport term and general nonlinear boundary condition, J. Math. Phys., 51 (2010), 102703, 22 pp. doi: 10.1063/1.3486070.  Google Scholar [13] L. L. Li, Y. Huang, G. Chen and T. W. Huang, Chaotic oscillations of second order linear hyperbolic equations with nonlinear boundary conditions: A factorizable but noncommutative case, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 25 (2015), 1530032, 20 pp. doi: 10.1142/S0218127415300323.  Google Scholar [14] L. L. Li, T. W. Huang and X. Y. Huang, Chaotic oscillations of the 1D wave equation due to extreme imbalance of self-regulations, J. Math. Anal. Appl., 450 (2017), 1388-1400.  doi: 10.1016/j.jmaa.2017.01.095.  Google Scholar [15] Y. C. Li, Chaos in Partial Differential Equations, Graduate Series in Analysis. International Press, omerville, MA, 2004.  doi: 10.1002/cnm.650.  Google Scholar [16] S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, 2$^nd$ edition, Texts in Applied Mathematics, 2. Springer-Verlag, New York, 2003. doi: 10.1063/1.4822950.  Google Scholar [17] Q. M. Xiang and Q. G. Yang, Nonisotropic chaotic oscillations of the wave equation due to the interaction of mixing transport term and superlinear boundary condition, J. Math. Anal. Appl., 462 (2018), 730-746.  doi: 10.1016/j.jmaa.2018.02.031.  Google Scholar [18] Q. M. Xiang and Q. G. Yang, Chaotic oscillations of a linear hyperbolic PDE with a general nonlinear boundary condition, J. Math. Anal. Appl., 472 (2019), 94-111.  doi: 10.1016/j.jmaa.2018.10.083.  Google Scholar [19] Q. G. Yang, G. R. Jiang and T. S. Zhou, Chaotification of linear impulsive differential systems with applications, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 22 (2012), 1250297, 12 pp. doi: 10.1142/S0218127412502975.  Google Scholar [20] Q. G. Yang and Q. M. Xiang, Existence of chaotic oscillations in second-order linear hyperbolic PDEs with implicit boundary conditions, J. Math. Anal. Appl., 457 (2018), 751-775.  doi: 10.1016/j.jmaa.2017.08.018.  Google Scholar [21] Z. B. Yin and Q. G. Yang, Distributionally scrambled set for an annihilation operator, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 25 (2015), 1550178, 13 pp. doi: 10.1142/S0218127415501783.  Google Scholar [22] Z. B. Yin and Q. G. Yang, Generic distributional chaos and principal measure in linear dynamics, Ann. Pol. Math., 118 (2016), 71-94.  doi: 10.4064/ap3908-9-2016.  Google Scholar [23] Z. B. Yin and Q. G. Yang, Distributionally n-chaotic dynamics for linear operators, Rev. Mat. Complut., 31 (2018), 111-129.  doi: 10.1007/s13163-017-0226-5.  Google Scholar [24] Z. B. Yin and Q. G. Yang, Distributionally n-scrambled set for weighted shift operators, J. Dyn. Control Syst., 23 (2017), 693-708.  doi: 10.1007/s10883-017-9359-6.  Google Scholar
The spatiotemporal profiles of system (23) with $(\alpha_1,\beta_1) = (0.1,1)$, $(\alpha_2,\beta_2) = (0.5,1)$, $x\in [0,1]$ and $t\in [60,64]$: (a) $w_{x}(x,t)$; (b) $w_{t}(x,t)$.
The spatiotemporal profiles of system (23) with $\gamma_1 = 1.1\pi$ and $\gamma_2 = 0.4\pi$, $x\in [0,1]$ and $t\in [60,64]$: (a) $w_{x}(x,t)$; (b) $w_{t}(x,t)$.
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