Geodesic
Chaotic Vibration of a Two-dimensional Non-strictly Hyperbolic Equation
Canadian mathematical bulletin, Tome 61 (2018) no. 4, pp. 768-786

Voir la notice de l'article provenant de la source Cambridge University Press

The study of chaotic vibration for multidimensional PDEs due to nonlinear boundary conditions is challenging. In this paper, we mainly investigate the chaotic oscillation of a two-dimensional non-strictly hyperbolic equation due to an energy-injecting boundary condition and a distributed self-regulating boundary condition. By using the method of characteristics, we give a rigorous proof of the onset of the chaotic vibration phenomenon of the zD non-strictly hyperbolic equation. We have also found a regime of the parameters when the chaotic vibration phenomenon occurs. Numerical simulations are also provided.
DOI : 10.4153/CMB-2018-012-1
Mots-clés : 32H50, 34C28, 37K50, 54H20, 58J45, chaotic vibration, reflection boundary condition, period-doubling bifurcation, method of characteristics 
Li, Liangliang; Tian, Jing; Chen, Goong. Chaotic Vibration of a Two-dimensional Non-strictly Hyperbolic Equation. Canadian mathematical bulletin, Tome 61 (2018) no. 4, pp. 768-786. doi: 10.4153/CMB-2018-012-1
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[1] [1] Block, L. S. and Coppel, W. A., Dynamics in one dimension. Lecture Notes in Mathematics, 1513, Springer-Verlag, Berlin, 1992. http://dx.doi.Org/10.1007/BFb0084762 Google Scholar

[2] [2] 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. Appendix C. by G. Chen amd G. Crosta. Trans. Amer. Math. Soc. 350 (1998), no. 11, 4265-4311. http://dx.doi.org/10.1090/S0002-9947-98-02022-4 Google Scholar

[3] [3] Chen, G., Hsu, S. B., Zhou, J., Chaotic vibration the one-dimensional wave equation due to a self-excitation boundary condition Partll: Energy Pumping, Period doubling and homoclinic orbits. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 8 (1998), no. 3, 423-445. http://dx.doi.Org/10.1142/S0218127498000280 Google Scholar

[4] [4] Chen, G., Hsu, S. B., Zhou, J., 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), no. 3, 447-470. http://dx.doi.Org/10.1142/S0218127498000292 Google Scholar

[5] [5] Chen, G., Hsu, S. B., Zhou, J., Snapback repellers as a cause of chaotic vibration of the wave equation with a van derPol boundary condition and energy injection at the middle of the span. J. Math. Phys. 39 (1998), 6459–6489. http://dx.doi.Org/10.1063/1.532670 Google Scholar

[6] [6] 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 Appl. Sci. Engrg. 12 (2002), no. 3, 535-559. http://dx.doi.Org/10.1142/S0218127402004504 Google Scholar

[7] [7] Chen, G., Huang, T., and Huang, Y., Chaotic behavior of interval maps and total variations of iterates. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 14 (2004), no. 7, 2161-2186. http://dx.doi.Org/10.1142/S0218127404010540 Google Scholar

[8] [8] Columbini, F., De Giorgi, E., and Spagnolo, S., Sur les équations hyperboliques avec des coefficients qui ne dépendent que du temps. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 6 (1979), no. 3, 511-559. Google Scholar

[9] [9] Columbini, F., Jannelli, E., and Spagnolo, S., Well-posedness in the Gevrey classes of the Cauchy problem for a non-strictly hyperbolic equation with coefficients depending on time. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 10 (1983), no. 2, 291-312. Google Scholar

[10] [10] Columbini, F. and Spagnolo, S., An example of a weakly hyperbolic Cauchy problem not well posed in C°°. Acta Math. 148 (1982), no. 1, 243-253. http://dx.doi.org/10.1007/BF02392730 Google Scholar

[11] [11] Courant, R. and Hilbert, D., Methods of mathematical physics, Vol. II: Partial differential equations. Wiley-Interscience, New York-London, 1962. Google Scholar

[12] [12] Li, L., Chen, Y., and 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 (2010), no. 10,102703. http://dx.doi.Org/10.1063/1.3486070 Google Scholar

[13] [13] Liu, J., Huang, Y., Sun, H., and Xiao, M., Numerical methods for weak solution of wave equation with van der Pol type nonlinear boundary conditions. Numer. Methods Partial Differential Equations 32 (2016), no. 2, 373-398. http://dx.doi.org/10.1002/num.21997 Google Scholar

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