Self-excited vibrations for damped and delayed 1-dimensional wave equations

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.

Link to Journal

N. Kosovalic and B. Pigott, "Self-excited vibrations for damped and delayed 1-dimensional wave equations." J. Dynam. Differential Equations. 31 (2019).