×

Chaotic oscillations of a linear hyperbolic PDE with a general nonlinear boundary condition. (English) Zbl 1437.35448

Summary: This article establishes a theorem that guarantees the occurrence of chaotic oscillations in a system governed by a linear hyperbolic partial differential equation (PDE) with a nonlinear boundary condition (NBC). Compared with the NBCs in all previous related literature, such an NBC is more general. Both the left end and the right end of the system parameter interval for the occurrence of chaotic oscillations are precisely characterized. The chaotic results obtained are further applied to two specific NBCs and the telegraph equation. Finally, numerical examples verify the effectiveness of theoretical prediction.

MSC:

35L20 Initial-boundary value problems for second-order hyperbolic equations
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Banasiak, J.; Mika, J. R., Singularly perturbed telegraph equations with applications in the random walk theory, J. Appl. Math. Stoch. Anal., 11, 1, 9-28 (1998) · Zbl 0909.35011
[2] Barrow, J., Chaotic behaviour in general relativity, Phys. Rep., 85, 1-49 (1982)
[3] Casati, G.; Chirikov, B. V.; Shepelyansky, D. L.; Guarneri, I., Relevance of classical chaos in quantum mechanics: the hydrogen atom in a monochromatic field, Phys. Rep., 154, 2, 77-123 (1987)
[4] 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. Amer. Math. Soc., 350, 4265-4311 (1998) · Zbl 0916.35065
[5] Chen, G.; Hsu, S. B.; Zhou, J., Chaotic vibrations of the one-dimensional wave equation due to a self-excitation boundary condition. Part II: energy injection, period doubling and homoclinic orbits, Internat. J. Bifur. Chaos, 8, 423-445 (1998) · Zbl 0938.35088
[6] Chen, G.; Hsu, S. B.; Zhou, J., Chaotic vibrations of the one-dimensional wave equation due to a self-excitation boundary condition. Part III: natural hysteresis memory effects, Internat. J. Bifur. Chaos, 8, 447-470 (1998) · Zbl 0938.35089
[7] 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, Internat. J. Bifur. Chaos, 12, 535-559 (2002) · Zbl 1044.37019
[8] Chen, G.; Huang, T. W.; Huang, Y., Chaotic behavior of interval maps and total variations of iterates, Internat. J. Bifur. Chaos, 14, 2161-2186 (2004) · Zbl 1077.37510
[9] Chen, G.; Sun, B.; Huang, Y., Chaotic oscillations of solutions of the Klein-Gordon equation due to inbalance of distributed and boundary energy flows, Internat. J. Bifur. Chaos, 24, 1-19 (2014)
[10] Dai, X., Chaotic dynamics of continuous-time topological semiflow on Polish spaces, J. Differential Equations, 258, 2794-2805 (2015) · Zbl 1320.34069
[11] Hirsch, M. W.; Smale, S.; Devaney, R., Differential Equations, Dynamical Systems, and an Introduction to Chaos (2004), Elsevier: Elsevier Amsterdam · Zbl 1135.37002
[12] Hu, C. C., Chaotic vibrations of the one-dimensional mixed wave system, Internat. J. Bifur. Chaos, 19, 2, 579-590 (2009) · Zbl 1170.35469
[13] Huang, Y., Growth rates of total variations of snapshots of the 1D linear wave equation with composite nonlinear boundary reflection relations, Internat. J. Bifur. Chaos, 13, 1183-1195 (2003) · Zbl 1062.35035
[14] Huang, Y.; Luo, J.; Zhou, Z. L., Rapid fluctuations of snapshots of one-dimensional linear wave equations with a van der Pol nonlinear boundary condition, Internat. J. Bifur. Chaos, 15, 567-580 (2005) · Zbl 1076.35066
[15] Jordan, P. M.; Puri, A., Digital signal propagation in dispersive media, J. Appl. Phys., 85, 3, 1273-1282 (1999)
[16] Li, Y. G., Chaos in Partial Differential Equations (2004), International Press: International Press Somerville, MA
[17] 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, Article 102703 pp. (2010) · Zbl 1314.35070
[18] Li, L. L.; Huang, Y., Growth rates of total variations of snapshots of 1D linear wave equations with nonlinear right-end boundary conditions, J. Math. Anal. Appl., 361, 69-85 (2010) · Zbl 1183.35189
[19] Li, L. L.; Huang, Y.; Chen, G.; Huang, T. W., Chaotic oscillations of second order linear hyperbolic equations with nonlinear boundary conditions: a factorizable but noncommutative case, Internat. J. Bifur. Chaos, 25, 11, 2161-2186 (2015)
[20] Li, L. L.; Huang, T. W.; Huang, Y., Chaotic oscillations of the 1D wave equation due to extreme imbalance of self-regulations, J. Math. Anal. Appl., 450, 1388-1400 (2017) · Zbl 1379.35174
[21] Pozar, D., Microwave Engineering (2004), Wiley
[22] Ulaby, F., Fundamentals of Applied Electromagnetics (2007), Prentice Hall
[23] Weston, V. H.; He, S., Wave splitting of the telegraph equation in \(R^3\) and its application to inverse scattering, Inverse Probl., 9, 789-812 (1993) · Zbl 0797.35168
[24] Wiggins, S., Introduction to Applied Nonlinear Dynamical Systems and Chaos (2003), Springer: Springer New York · Zbl 1027.37002
[25] 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, 730-746 (2018) · Zbl 1447.35210
[26] Yang, Q. G.; Jiang, G. R.; Zhou, T. S., Chaotification of linear impulsive differential systems with applications, Internat. J. Bifur. Chaos, 22, 12, Article 1250297 pp. (2012) · Zbl 1258.34029
[27] Yang, Q. G.; Xiang, Q. M., Existence of chaotic oscillations in second-order linear hyperbolic PDEs with implicit boundary conditions, J. Math. Anal. Appl., 457, 751-775 (2018) · Zbl 1434.35021
[28] Yin, Z. B.; Yang, Q. G., Distributionally scrambled set for an annihilation operator, Internat. J. Bifur. Chaos, 25, 13, Article 1550178 pp. (2015) · Zbl 1330.81103
[29] Yin, Z. B.; Yang, Q. G., Generic distributional chaos and principal measure in linear dynamics, Ann. Polon. Math., 118, 1, 71-94 (2016) · Zbl 1448.47019
[30] Yin, Z. B.; Yang, Q. G., Distributionally n-scrambled set for weighted shift operators, J. Dyn. Control Syst., 23, 4, 693-708 (2017) · Zbl 06793137
[31] Yin, Z. B.; Yang, Q. G., Distributionally n-chaotic dynamics for linear operators, Rev. Mat. Complut., 31, 1, 111-129 (2018) · Zbl 1390.37066
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.