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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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

M. Golubitsky and I. Stewart, The Symmetry Perspective, Birkhäuser Verlag, Basel, 2002. doi: 10.1007/978-3-0348-8167-8.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

A. Jenkins, Self-oscillation, Physics Reports, 525 (2013), 167-222.  doi: 10.1016/j.physrep.2012.10.007.  Google Scholar

[9]

H. Kielhöfer, Bifurcation Theory: An Introduction with Applications to PDEs, Springer Verlag, New York, 2004. doi: 10.1007/b97365.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

M. Golubitsky and I. Stewart, The Symmetry Perspective, Birkhäuser Verlag, Basel, 2002. doi: 10.1007/978-3-0348-8167-8.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

A. Jenkins, Self-oscillation, Physics Reports, 525 (2013), 167-222.  doi: 10.1016/j.physrep.2012.10.007.  Google Scholar

[9]

H. Kielhöfer, Bifurcation Theory: An Introduction with Applications to PDEs, Springer Verlag, New York, 2004. doi: 10.1007/b97365.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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