Livia Corsi, Roberto Feola, Guido Gentile
Domains of analyticity for response solutions
in strongly dissipative forced systems
(211K, pdf)
ABSTRACT. We study the analyticity properties on $arepsilon$ of the ordinary
differential equation $arepsilon\ddot x + \dot x + arepsilon g(x) = \e
f(\omega t)$, where $g$ and $f$ are real-analytic functions, with $f$
quasi-periodic in $t$ with frequency vector $\omega$. A response solution is a
quasi-periodic solution to the equation with the same frequency vector as the
forcing. If $c_{0} \in \mathbb{R}$ is such that $g(c_0)$ equals the average of
$f$ and $g'(c_0)
eq0$, under very mild assumptions on $\omega$ there exists a
response solution close to $c_0$. In particular no assumption at all is
required on $\omega$ if $f$ is a trigonometric polynomial. We show that such a
response solution depends analytically on $arepsilon$ in a domain of the
complex plane tangent more than quadratically to the imaginary axis at the
origin.