# American Institute of Mathematical Sciences

October  2013, 33(10): 4411-4433. doi: 10.3934/dcds.2013.33.4411

## Construction of response functions in forced strongly dissipative systems

 1 Institute for Mathematics and its Applications, University of Minnesota, 307 Church St SE, Minneapolis, MN 55455, United States 2 Department of Mathematics, University of Roma Tor Vergata, Via della Ricerca Scientifica 1, 00133 Roma, Italy 3 School of Mathematics, Georgia Institute of Technology, 686 Cherry St., Atlanta, GA 30332-0160, United States

Received  July 2012 Revised  December 2012 Published  April 2013

We study the existence of quasi--periodic solutions of the equation $ε \ddot x + \dot x + ε g(x) = ε f(\omega t)\ ,$ where $x: \mathbb{R} \rightarrow \mathbb{R}$ is the unknown and we are given $g:\mathbb{R} \rightarrow \mathbb{R}$, $f: \mathbb{T}^d \rightarrow \mathbb{R}$, $\omega \in \mathbb{R}^d$ (without loss of generality we can assume that $\omega\cdot k\not=0$ for any $k \in \mathbb{Z}^d\backslash\{0\}$). We assume that there is a $c_0\in \mathbb{R}$ such that $g(c_0) = \hat f_0$ (where $\hat f_0$ denotes the average of $f$) and $g'(c_0) \ne 0$. 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 $\omega$ satisfies some very mild irrationality conditions, there are families of quasi--periodic solutions with frequency $\omega$. These families depend analytically on $ε$, when $ε$ 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 $|\omega \cdot k|^{-1}$ grow exponentially with $k$. In the case that $f$ is a trigonometric polynomial, we do not need any condition on $|\omega \cdot k|$. 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.
Citation: Renato C. Calleja, Alessandra Celletti, Rafael de la Llave. Construction of response functions in forced strongly dissipative systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4411-4433. doi: 10.3934/dcds.2013.33.4411
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##### References:
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