Renato C. Calleja, Alessandra Celletti, Rafael de la Llave
Domains of analyticity of Lindstedt expansions of KAM tori in dissipative perturbations of Hamiltonian systems
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ABSTRACT. Many problems in Physics are described by dynamical systems
that are conformally symplectic (e.g., mechanical systems with a friction proportional to
the velocity,
variational problems with a small discount or thermostated
systems). Conformally symplectic systems
are characterized by the property that they transform a symplectic form
into a multiple of itself. The limit of small dissipation, which is the object of the
present study, is particularly interesting.
We provide all details for maps, but we present also the (somewhat minor) modifications needed
to obtain a direct proof for the case of differential equations.
We consider a family of conformally symplectic maps $f_{\mu, \eps}$ defined
on a $2d$-dimensional symplectic manifold $\M$ with exact symplectic form $\Omega$; we assume that $f_{\mu, \eps}$ satisfies
$f_{\mu, \eps}^* \Omega = \lambda(\eps) \Omega$. We assume that
the family depends on a $d$-dimensional parameter $\mu$ (called \emph{drift}) and also on a small
scalar parameter $\eps$. Furthermore, we assume that the conformal factor $\lambda$
depends on $\eps$, in such a way that for $\eps=0$ we have
$\lambda(0)=1$ (the symplectic case).
We also assume that
$\lambda(\eps) = 1 + lpha \eps^a + O(|\eps|^{a+1})$,
where $a\in\integer_+$, $lpha\in