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

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

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

Nemanja Kosovalić ^1,, and Brian Pigott ^2,

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.

Keywords: Wave equation, time delay, standing waves, Hopf bifurcation, self-excited vibration, symmetry breaking, symmetric group action.

Mathematics Subject Classification: Primary: 35L05; Secondary: 37K50.

Citation: Nemanja Kosovalić, Brian Pigott. Self-excited vibrations for damped and delayed higher dimensional wave equations. Discrete and Continuous Dynamical Systems, 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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