Response solutions for forced systems with large dissipation and arbitrary frequency vectors

TL;DR

This paper extends the existence of response solutions in strongly dissipative quasi-periodic systems to arbitrary frequency vectors, broadening previous results that required Diophantine conditions.

Contribution

It generalizes prior findings by removing the Diophantine condition, proving response solutions exist for any frequency vector in strongly dissipative systems.

Findings

Response solutions exist for arbitrary frequency vectors.

Previous Diophantine restrictions are no longer necessary.

Results apply to a broader class of dissipative systems.

Abstract

We study the behaviour of one-dimensional strongly dissipative systems subject to a quasi-periodic force. In particular we are interested in the existence of response solutions, that is quasi-periodic solutions having the same frequency vector as the forcing term. Earlier results available in the literature show that, when the dissipation is large enough and a suitable function involving the forcing has a simple zero, response solutions can be proved to exist and to be attractive provided some Diophantine condition is assumed on the frequency vector. In this paper we show that the results extend to the case of arbitrary frequency vectors.