Domains of analyticity and Lindstedt expansions of KAM tori in some dissipative perturbations of Hamiltonian 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 consider a family of conformally symplectic maps $\newcommand{\ep}{\varepsilon} \newcommand{\eps}{\varepsilon} f_{\mu, \eps}$ (very simple modifications, which we present, give results for differential equations) defined on a 2d-dimensional symplectic manifold $\newcommand{\M}{{\mathcal M}} \M$ with exact symplectic form $\Omega$; we assume that $\newcommand{\ep}{\varepsilon} \newcommand{\eps}{\varepsilon} f_{\mu, \eps}$ satisfies $\newcommand{\ep}{\varepsilon} \newcommand{\eps}{\varepsilon} f_{\mu, \eps}^* \Omega = \lambda(\eps) \Omega$. The d-dimensional parameter μ is called drift. We assume that $\newcommand{\ep}{\varepsilon} \newcommand{\eps}{\varepsilon} \lambda(\eps) = 1 + \alpha \eps^a + O(\vert \eps\vert ^{a+1})$, where $\newcommand{\integer}{{\mathbb Z}} a\in\integer_+$, $\newcommand{\complex}{{\mathbb C}} \alpha\in\complex\backslash\{0\}$.

We study the perturbative expansions and the domains of analyticity in $\newcommand{\ep}{\varepsilon} \newcommand{\eps}{\varepsilon} \eps$ near $\newcommand{\ep}{\varepsilon} \newcommand{\eps}{\varepsilon} \eps = 0$ of the parameterization of the quasi-periodic orbits of frequency ω (assumed to be Diophantine) and of the parameter μ. Notice that this is a singular perturbation, since any friction (no matter how small) reduces the set of quasi-periodic solutions in the system. We prove that the tori are analytic in a domain in the complex $\newcommand{\ep}{\varepsilon} \newcommand{\eps}{\varepsilon} \eps$ plane, obtained by taking from a ball centered at zero a sequence of smaller balls with center along smooth lines going through the origin. The radii of the excluded balls decrease faster than any power of the distance of the center to the origin. We state also a conjecture on the optimality of our results. The boundary of the domain is very thin, so that we can perform unique analytic continuation of the invariant tori along closed circles enclosing the points in the complement of the analyticity. We show that there is no monodromy of these continuations, either for the tori, for the invariant manifolds or for the drift.

The proof is based on the following procedure. To find a quasi-periodic solution, one solves an invariance equation for the embedding of the torus, depending on the parameters of the family. Assuming that the frequency of the torus satisfies a Diophantine condition, under mild non-degeneracy assumptions, using a Lindstedt procedure we construct an approximate solution to all orders of the invariance equation describing the KAM torus. Starting from such approximate solution, we use an a posteriori KAM theorem to get the true solution of the invariance equation. This allows also the study of monogenic and Whitney differentiability properties of the extensions as well as the monodromy.