Construction of response functions in forced strongly dissipative systems

We study the existence of quasi-periodic solutions of the equation epsilon x + x + epsilon g(x)= epsilon f(omega t), where x : R -> R is the unknown and we are given g : R -> R, f : T-d -> R, w is an element of R-d (without loss of generality we can assume that omega center dot kappa not equal 0 for any k is an element of Z(d)\{0}). We assume that there is a c(0) is an element of R such that g(c(0)) = f(0) (where f(0) denotes the average of f) and g(c(0)) not equal 0. Special cases of this equation, for example when g(x) - x(2), are called the "varactor problem'' in the literature. We show that if f, g are analytic, and w satisfies some very mild irrationality conditions, there are families of quasi-periodic solutions with frequency omega. These families depend analytically on epsilon, when epsilon ranges over a complex domain that includes cones or parabolic domains based at the origin. The irrationality conditions required in this paper are very weak. They allow that the small denominators vertical bar omega center dot kappa vertical bar(-1) grow exponentially with kappa. In the case that f is a trigonometric polynomial, we do not need any condition on vertical bar omega center dot kappa vertical bar. This answers a delicate question raised in [8]. We also consider the periodic case, when omega is just a number (d=1). We obtain that there are solutions that depend analytically in a domain which is a disk removing countably many disjoint disks. This shows that in this case there is no Stokes phenomenon (different resummations on different sectors) for the asymptotic series. The approach we use is to reduce the problem to a fixed point theorem. This approach also yields results in the case that g is a finitely differentiable function; it provides also very effective numerical algorithms and we discuss how they can be implemented.