# American Institute of Mathematical Sciences

May 2019, 39(5): 2413-2435. doi: 10.3934/dcds.2019102

## Self-excited vibrations for damped and delayed higher dimensional wave equations

 1 Department of Mathematics and Statistics, University of South Alabama, 411 University Blvd North, Mobile AL 36688, USA 2 Department of Mathematics, Wofford College, 429 North Church Street, Spartanburg, SC 29303, USA

* Corresponding author: Nemanja Kosovalić

Received  October 2017 Revised  October 2018 Published  January 2019

In the article [12] it is shown that time delay induces self-excited vibrations in a one dimensional damped wave equation. Here we generalize this result for higher spatial dimensions. We prove the existence of branches of nontrivial time periodic solutions for spatial dimensions $d\ge 2$. For $d> 2$, the bifurcating periodic solutions have a fixed spatial frequency vector, which is the solution of a certain Diophantine equation. The case $d = 2$ must be treated separately from the others. In particular, it is shown that an arbitrary number of symmetry breaking orbitally distinct time periodic solutions exist, provided $d$ is big enough, with respect to the symmetric group action. The direction of bifurcation is also obtained.

Citation: Nemanja Kosovalić, Brian Pigott. Self-excited vibrations for damped and delayed higher dimensional wave equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2413-2435. doi: 10.3934/dcds.2019102
##### References:
 [1] J. Bourgain, Construction of periodic solutions of nonlinear wave equations in higher dimension, Geometric and Functional Analysis, 5 (1995), 629-639. doi: 10.1007/BF01902055. [2] S. A. Campbell, J. Belair, T. Ohira and J. Milton, Complex dynamics and multistability in a damped harmonic oscillator with delayed negative feedback, Chaos: An Interdisciplinary Journal of Nonlinear Science, 5 (1995), 640-645. doi: 10.1063/1.166134. [3] W. Craig and E. C. Wayne, Newton's method and periodic solutions of nonlinear wave equations, Communications on Pure and Applied Mathematics, 46 (1993), 1409-1498. doi: 10.1002/cpa.3160461102. [4] M. Golubitsky and I. Stewart, Hopf bifurcation in the presence of symmetry, Archive for Rational Mechanics and Analysis, 87 (1985), 107-165. doi: 10.1007/BF00280698. [5] M. Golubitsky and I. Stewart, The Symmetry Perspective, Birkhäuser Verlag, Basel, 2002. doi: 10.1007/978-3-0348-8167-8. [6] C. Gugg, T.J. Healey, H. Kielhófer and S. Maier-Paape, Nonlinear standing and rotating waves on the sphere, Journal of Differential Equations, 166 (2000), 402-442. doi: 10.1006/jdeq.2000.3791. [7] T. J. Healey and H. Kielhöfer, Free nonlinear vibrations for a class of two-dimensional plate equations: Standing and discrete-rotating waves, Applications, 29 (1997), 501-531. doi: 10.1016/S0362-546X(96)00062-4. [8] A. Jenkins, Self-oscillation, Physics Reports, 525 (2013), 167-222. doi: 10.1016/j.physrep.2012.10.007. [9] H. Kielhöfer, Bifurcation Theory: An Introduction with Applications to PDEs, Springer Verlag, New York, 2004. doi: 10.1007/b97365. [10] Yu. S. Kolesov and N. Kh. Rozov, The parametric buffer phenomenon in a singularly perturbed telegraph equation with pendulum nonlinearity, Mathematical Notes, 69 (2001), 790-798. doi: 10.1023/A:1010230431593. [11] N. Kosovalić, Quasi-periodic self-excited travelling waves for damped beam equations, Journal of Differential Equations, 265 (2018), 2171-2190. doi: 10.1016/j.jde.2018.04.022. [12] N. Kosovalić and B. Pigott, Self-excited vibrations for damped and delayed 1-dimensional wave equations, Journal of Dynamics and Differential Equations, (2018), 1–24. doi: 10.1007/s10884-018-9654-2.

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##### References:
 [1] J. Bourgain, Construction of periodic solutions of nonlinear wave equations in higher dimension, Geometric and Functional Analysis, 5 (1995), 629-639. doi: 10.1007/BF01902055. [2] S. A. Campbell, J. Belair, T. Ohira and J. Milton, Complex dynamics and multistability in a damped harmonic oscillator with delayed negative feedback, Chaos: An Interdisciplinary Journal of Nonlinear Science, 5 (1995), 640-645. doi: 10.1063/1.166134. [3] W. Craig and E. C. Wayne, Newton's method and periodic solutions of nonlinear wave equations, Communications on Pure and Applied Mathematics, 46 (1993), 1409-1498. doi: 10.1002/cpa.3160461102. [4] M. Golubitsky and I. Stewart, Hopf bifurcation in the presence of symmetry, Archive for Rational Mechanics and Analysis, 87 (1985), 107-165. doi: 10.1007/BF00280698. [5] M. Golubitsky and I. Stewart, The Symmetry Perspective, Birkhäuser Verlag, Basel, 2002. doi: 10.1007/978-3-0348-8167-8. [6] C. Gugg, T.J. Healey, H. Kielhófer and S. Maier-Paape, Nonlinear standing and rotating waves on the sphere, Journal of Differential Equations, 166 (2000), 402-442. doi: 10.1006/jdeq.2000.3791. [7] T. J. Healey and H. Kielhöfer, Free nonlinear vibrations for a class of two-dimensional plate equations: Standing and discrete-rotating waves, Applications, 29 (1997), 501-531. doi: 10.1016/S0362-546X(96)00062-4. [8] A. Jenkins, Self-oscillation, Physics Reports, 525 (2013), 167-222. doi: 10.1016/j.physrep.2012.10.007. [9] H. Kielhöfer, Bifurcation Theory: An Introduction with Applications to PDEs, Springer Verlag, New York, 2004. doi: 10.1007/b97365. [10] Yu. S. Kolesov and N. Kh. Rozov, The parametric buffer phenomenon in a singularly perturbed telegraph equation with pendulum nonlinearity, Mathematical Notes, 69 (2001), 790-798. doi: 10.1023/A:1010230431593. [11] N. Kosovalić, Quasi-periodic self-excited travelling waves for damped beam equations, Journal of Differential Equations, 265 (2018), 2171-2190. doi: 10.1016/j.jde.2018.04.022. [12] N. Kosovalić and B. Pigott, Self-excited vibrations for damped and delayed 1-dimensional wave equations, Journal of Dynamics and Differential Equations, (2018), 1–24. doi: 10.1007/s10884-018-9654-2.
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