Convergent series for quasi-periodically forced strongly dissipative systems

We study the ordinary differential equation εẍ+ ẋ+ ε g(x) = εf(ωt), with f and g analytic and f quasi-periodic in t with frequency vector ω ∈ R. 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 c0 is 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 c0 is positive, then the quasi-periodic solution is locally unique and attractive.