Forced quasi-periodic oscillations in strongly dissipative systems of any finite dimension

TL;DR

This paper proves the existence of quasi-periodic response solutions in strongly dissipative, finite-dimensional analytic systems under quasi-periodic forcing, without requiring Diophantine conditions on the frequency.

Contribution

It extends the existence results of response solutions to multi-dimensional systems with strong dissipation, relaxing frequency conditions previously needed.

Findings

Response solutions exist for sufficiently large dissipation.

No Diophantine condition on forcing frequency is required.

Results generalize previous one-dimensional cases.

Abstract

We consider a class of singular ordinary differential equations describing analytic systems of arbitrary finite dimension, subject to a quasi-periodic forcing term and in the presence of dissipation. We study the existence of response solutions, i.e. quasi-periodic solutions with the same frequency vector as the forcing term, in the case of large dissipation. We assume the system to be conservative in the absence dissipation, so that the forcing term is --- up to the sign --- the gradient of a potential energy, and both the mass and damping matrices to be symmetric and positive definite. Further, we assume a non-degeneracy condition on the forcing term, essentially that the time-average of the potential energy has a strict local minimum. On the contrary, no condition is assumed on the forcing frequency; in particular we do not require any Diophantine condition. We prove that, under the assumptions above, a response solution always exist provided the dissipation is strong enough. This extends results previously available in literature in the one-dimensional case.