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Self-Excited Vibrations for Damped and Delayed 1-Dimensional Wave Equations

  • Nemanja KosovalićEmail author
  • Brian Pigott
Article

Abstract

It is shown that time delay induces self-excited vibrations in one dimensional damped wave equations, via Hopf bifurcation. In order to obtain classical solutions and \(C^{\infty }\) smoothness with respect to the amplitude parameter, we avoid the use of any abstract Hopf bifurcation theorem, and instead work directly in Sobolev spaces on the 2-torus. As a result, a “derivative loss” problem arises, which is overcome using the regularizing effects of the time delay and damping. Only the classical implicit function theorem is used. The direction of bifurcation is also obtained.

Keywords

Wave equation Hopf bifurcation Time delay Periodic solution 

Mathematics Subject Classification

35L05 37K50 

Notes

Acknowledgements

We would like to thank Chris Lin for many helpful discussions during the preliminary stages of this work, as well as the anonymous reviewer for suggestions which have significantly improved this work.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.University of South AlabamaMobileUSA
  2. 2.Wofford CollegeSpartanburgUSA

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