Existence of whiskered KAM tori of conformally symplectic systems. (4th December 2019)

• Record Type: Journal Article
• Authors: Calleja, Renato C; Celletti, Alessandra; de la Llave, Rafael
• Is Part Of: Nonlinearity. Volume 33:Number 1(2020)
• Page Start: 538
• Page End: 597
• Publication Date: 2019-12-04
• Subjects: whiskered tori -- conformally symplectic systems -- KAM theory; 70K43 -- 70K20 -- 34D35; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/1361-6544/ab4c80
• Languages: English
• ISSNs: 0951-7715

Abstract

Abstract: Many physical problems are described by conformally symplectic systems (i.e. systems whose evolution in time transforms a symplectic form into a multiple of itself). We study the existence of whiskered tori in a family of conformally symplectic maps depending on parameters (often called drifts ). We recall that whiskered tori are tori on which the motion is a rotation, but they have as many expanding/contracting directions as allowed by the preservation of the geometric structure. Our main result is formulated in an a posteriori format. We fix satisfying Diophantine conditions. We assume that we are given (1) a value of the parameter, (2) an embedding of the torus K 0 into the phase space, approximately invariant under in the sense that (where is the shift by ) is small (in some norm), (3) a splitting of the tangent space at the range of K 0, into three bundles which are approximately invariant under and such that the derivative satisfies 'rate conditions' on each of the components. Then, if some non-degeneracy conditions (verifiable by a finite calculation on the approximate solution and which do not require any global property of the map) are satisfied, we show that there is another parameter, an embedding and splittings close to the original ones which are invariant under . We also bound, and the distance of the initial and final splittings in terms of the initial error. We allow that the stable/unstable bundles are nontrivial (i.e. not homeomorphic to a product bundle). On the other hand, we show that the geometric set up has the global consequence that the center bundle is necessarily trivial (i.e. homeomorphic to a product bundle). The proof of the main theorem consists in describing an iterative process that takes advantage of cancellations coming from the geometry. Then, we show that the process converges to a true solution when started from an approximate enough solution. The iterative process leads to an efficient algorithm that is quite practical to implement. The a posteriori format of the theorem implies the usual formulation of persistence under perturbations (the solutions for the original systems are approximate solutions for the perturbation), but it also allows to justify approximate solutions produced by any method (for example numerical solutions or asymptotic formal expansions). As an application, we study the singular problem of effects of small dissipation on whiskered tori. We develop formal (presumably not convergent) expansions in the perturbative parameter (which generates dissipation) and use them as input for the a posteriori theorem. This allows to obtain lower bounds for the domain of analyticity of the tori as function of the perturbative parameter. Even if we state only the theory for maps, our results apply also to flows.