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

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.

Link to Journal

N. Kosovalic and B. Pigott, "Self-excited vibrations for damped and delayed higher dimensional wave equations." Discrete Contin. Dyn. Syst.. 39 (5) (2019).