Domains of analyticity and Lindstedt expansions of KAM tori in some dissipative perturbations of Hamiltonian systems

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 µ,ε defined on a 2d-dimensional symplectic manifold M with exact symplectic form Ω; we assume that f µ,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 thatWe 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.

KAM theory for some dissipative systems

“…• The singular limit of zero dissipation. We showed in [19] that, if one fixes the frequency, one can choose the drift parameter as a function of the perturbation in a smooth way: µ = µ(ω, ε) ≡ µ ε (ω). Note however that µ 0 (ω) = 0, but for ε > 0, the function µ ε is invertible so that the function µ ε (ω) is a smooth function with a limit as ε → 0.…”

“…Remark 1. It is useful to make a remark on the non-degeneracy condition (19), when applied to the conservative and dissipative standard maps ((1), ( 3)). For the conservative standard map, the non-degeneracy condition is typically the so-called twist condition, which can be written as…”

“…We remark, however, that ( 20) and ( 21) involve global properties of the system, while ( 19) is a condition involving just the approximate solution, so that ( 19) may be applied in situations where (19), (21) fail.…”

KAM estimates for the dissipative standard map

“…This model has been studied both numerically and theoretically in the literature. For example [53,54,55] consider the breakdown and conjecture universality properties; [10] studies the breakdown even for complex values of the parameters; [11] studies the invariant bundles near the circles and find scaling properties at breakdown; [9,14] study the domains of analyticity in the limit of small dissipation.…”

Existence of whiskered KAM tori of conformally symplectic systems

“…In this Section we investigate the domains of analyticity of whiskered tori in conformally symplectic systems in the limit of small dissipation. The discussion below proceeds along the lines of [CCdlL17]. We develop an asymptotic expansion (Lindstedt series) and use this as a starting point for the application of Theorem 20.…”