Forced Frequency Locking for Semilinear Dissipative Hyperbolic PDEs

TL;DR

This paper investigates how small periodic forcing causes frequency locking in 1D dissipative semilinear hyperbolic PDEs, extending known phenomena from ODEs and parabolic PDEs with new analytical challenges.

Contribution

It establishes the occurrence of forced frequency locking in hyperbolic PDEs and derives a scalar equation characterizing the surviving time shifts under forcing.

Findings

Frequency locking occurs similarly to ODEs and parabolic PDEs.

Non-resonance conditions are essential for the results.

A scalar equation describes the surviving time shifts.

Abstract

This paper concerns the behavior of time-periodic solutions to 1D dissipative autonomous semilinear hyperbolic PDEs under the influence of small time-periodic forcing. We show that the phenomenon of forced frequency locking happens similarly to the analogous phenomena known for ODEs or parabolic PDEs. However, the proofs are essentially more difficult than for ODEs or parabolic PDEs. In particular, non-resonance conditions are needed, which do not have counterparts in the cases of ODEs or parabolic PDEs. We derive a scalar equation which answers the main question of forced frequency locking: Which time shifts of the solution to the unforced equation do survive under which forcing?