KAM theory for conformally symplectic systems
Calleja, Renato C.
Celletti, A. (Alessandra)
Llave, Rafael de la
Centre de Recerca Matemàtica

Publicació: Centre de Recerca Matemàtica 2011
Descripció: 78 p.
Resum: We present a KAM theory for some dissipative systems (geometrically, these are conformally symplectic systems, i. e. systems that transform a symplectic form into a multiple of itself). For systems with n degrees of freedom depending on n parameters we show that it is possible to find solutions with n-dimensional (Diophantine) frequencies by adjusting the parameters. We do not assume that the system is close to integrable, but we use an a-posteriori format. Our unknowns are a parameterization of the solution and a parameter. We show that if there is a sufficiently approximate solution of the invariance equation, which also satisfies some explicit non-degeneracy conditions, then there is a true solution nearby. We present results both in Sobolev norms and in analytic norms. The a-posteriori format has several consequences: A) smooth dependence on the parameters, including the singular limit of zero dissipation; B) estimates on the measure of parameters covered by quasi-periodic solutions; C) convergence of perturbative expansions in analytic systems; D) bootstrap of regularity (i. e. , that all tori which are smooth enough are analytic if the map is analytic); E) a numerically efficient criterion for the break-down of the quasi-periodic solutions. The proof is based on an iterative quadratically convergent method and on suitable estimates on the (analytical and Sobolev) norms of the approximate solution. The iterative step takes advantage of some geometric identities, which give a very useful coordinate system in the neighborhood of invariant (or approximately invariant) tori. This system of coordinates has several other uses: A) it shows that for dissipative conformally symplectic systems the quasi-periodic solutions are attractors, B) it leads to efficient algorithms, which have been implemented elsewhere. Details of the proof are given mainly for maps, but we also explain the slight modifications needed for flows and we devote the appendix to present explicit algorithms for flows.
Drets: Aquest document està subjecte a una llicència d'ús Creative Commons. Es permet la reproducció total o parcial, la distribució, i la comunicació pública de l'obra, sempre que no sigui amb finalitats comercials, i sempre que es reconegui l'autoria de l'obra original. No es permet la creació d'obres derivades. Creative Commons
Llengua: Anglès
Col·lecció: Centre de Recerca Matemàtica. Prepublicacions
Col·lecció: Prepublicacions del Centre de Recerca Matemàtica ; 1076
Document: Article ; Prepublicació ; Versió de l'autor
Matèria: Varietats simplèctiques ; Aplicacions conformes

78 p, 604.1 KB

El registre apareix a les col·leccions:
Documents de recerca > Prepublicacions

 Registre creat el 2018-10-23, darrera modificació el 2024-05-18

   Favorit i Compartir