Convergent series for quasi-periodically forced strongly dissipative systems

Abstract

We study the ordinary differential equation εẍ + ẋ + εg(x) = εf(ωt), with f and g analytic and f quasi-periodic in t with frequency vector ω ∈ ℝd. We show that if there exists c0∈ ℝ such that g(c0) equals the average of f and the first non-zero derivative of g at c0is of odd order 𝔫, then, for ε small enough and under very mild Diophantine conditions on ω, there exists a quasi-periodic solution close to c0, with the same frequency vector as f. In particular if f is a trigonometric polynomial the Diophantine condition on ω can be completely removed. This extends results previously available in the literature for 𝔫 = 1. We also point out that, if 𝔫 = 1 and the first derivative of g at c0is positive, then the quasi-periodic solution is locally unique and attractive.