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

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.