Construction of response functions in forced strongly dissipative systems

Renato C. Calleja; Alessandra Celletti; Rafael de la Llave

doi:10.3934/dcds.2013.33.4411

Article Contents

2013, Volume 33, Issue 10: 4411-4433. Doi: 10.3934/dcds.2013.33.4411

This issue Previous Article On the large deviation rates of non-entropy-approachable measures Next Article About the unfolding of a Hopf-zero singularity

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
2.
Department of Mathematics, University of Roma Tor Vergata, Via della Ricerca Scientifica 1, 00133 Roma
3.
School of Mathematics, Georgia Institute of Technology, 686 Cherry St., Atlanta, GA 30332-0160

Received: June 30, 2012

Revised: November 30, 2012

Published: September 2013

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Abstract

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.

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Mathematics Subject Classification: 70K43, 70K20, 34D35.

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