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September  2021, 14(9): 3267-3284. doi: 10.3934/dcdss.2020335

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  September 2021 Early access  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 and Continuous Dynamical Systems - S, 2021, 14 (9) : 3267-3284. doi: 10.3934/dcdss.2020335
References:
[1]

G. ChenS.-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.

[2]

G. ChenS.-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.

[3]

G. ChenS.-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.

[4]

G. ChenS.-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.

[5]

G. ChenT. 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.

[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.

[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.

[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.

[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.

[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.

[11]

Y. HuangJ. 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.

[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.

[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.

[14]

L. L. LiT. 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.

[15] Y. C. Li, Chaos in Partial Differential Equations, Graduate Series in Analysis. International Press, omerville, MA, 2004.  doi: 10.1002/cnm.650.
[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

show all references

References:
[1]

G. ChenS.-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.

[2]

G. ChenS.-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.

[3]

G. ChenS.-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.

[4]

G. ChenS.-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.

[5]

G. ChenT. 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.

[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.

[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.

[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.

[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.

[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.

[11]

Y. HuangJ. 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.

[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.

[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.

[14]

L. L. LiT. 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.

[15] Y. C. Li, Chaos in Partial Differential Equations, Graduate Series in Analysis. International Press, omerville, MA, 2004.  doi: 10.1002/cnm.650.
[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

Figure 1.  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) $.
Figure 2.  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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