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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 |
In the article [
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. |
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. |
[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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