15-44 Renato C. Calleja, Alessandra Celletti, Rafael de la Llave

Domains of analyticity of Lindstedt expansions of KAM tori in dissipative perturbations of Hamiltonian systems (763K, PDF) May 31, 15

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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

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