Hostname: page-component-7479d7b7d-c9gpj Total loading time: 0 Render date: 2024-07-13T17:37:46.847Z Has data issue: false hasContentIssue false
  • English
  • Français

Chaotic Vibration of a Two-dimensional Non-strictly Hyperbolic Equation

Published online by Cambridge University Press:  20 November 2018

Liangliang Li ,
Jing Tian  and
Goong Chen
Liangliang Li
Affiliation:
IFCEN, Sun Yat-sen University, Zhuhai, 519082, China, e-mail : liliangliang@mail.sysu.edu.cn
Jing Tian
Affiliation:
Department of Mathematics, Towson University, Towson, MD 21252, United States, e-mail : jtian@towson.edu
Goong Chen
Affiliation:
Department of Mathematics and Institute for Quantum Science and Engineering, Texas A&M University, College Station, TX 77843-4242, United States, e-mail : gchen@math.tamu.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Keywords

32H5034C2837K5054H2058J45chaotic vibrationreflection boundary conditionperiod-doubling bifurcationmethod of characteristics
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 61 , Issue 4 , 01 December 2018 , pp. 768 - 786
Copyright
Copyright © Canadian Mathematical Society 2018

References

[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/BFb0084762Google Scholar
[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/S0002-9947-98-02022-4Google Scholar
[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/S0218127498000280Google Scholar
[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/S0218127498000292Google Scholar
[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/10.1063/1.532670Google Scholar
[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/S0218127402004504Google Scholar
[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/S0218127404010540Google Scholar
[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] 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] 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/BF02392730Google Scholar
[11] Courant, R. and Hilbert, D., Methods of mathematical physics, Vol. II: Partial differential equations. Wiley-Interscience, New York-London, 1962.Google Scholar
[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/10.1063/1.3486070Google Scholar
[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.21997Google Scholar