Domains of Analyticity of Lindstedt Expansions of Kam Tori in Dissipative Perturbations of Hamiltonian Systems

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 fμ,ε defined on a 2d-dimensional symplectic manifold M with exact symplectic form Ω; we assume that fμ,ε satisfies f ∗ μ,ε Ω = λ(ε)Ω. We assume that the family depends on a d-dimensional parameter μ (called drift) and also on a small scalar parameter ε. Furthermore, we assume that the conformal factor λ depends on ε, in such a way that for ε = 0 we have λ(0) = 1 (the symplectic case). We also assume that λ(ε) = 1 + αε +O(|ε|), where a ∈ Z+, α ∈ C\{0}. We study the domains of analyticity in ε near ε = 0 of perturbative expansions (Lindstedt series) 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 Lindstedt series are analytic in a domain in the complex ε plane, which is 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 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; the zeroth order Lindstedt series is provided by the solution of the invariance equation of the symplectic case. Starting from such approximate solution, we implement an a-posteriori KAM theorem to get the true solution of the invariance equation, and we show that the procedure converges. This allows also the study of monogenic and Withney extensions. 2010 Mathematics Subject Classification. 70K43, 70K20, 37J40.