Whiskered KAM tori of conformally symplectic systems

Calleja, Renato C. ¹  ; Celletti, Alessandra² ; de la Llave, Rafael³

¹ Department of Mathematics and Mechanics, IIMAS National Autonomous University of Mexico (UNAM), Apdo. Postal 20-126 C.P. 01000, Mexico D.F. (Mexico)
² Department of Mathematics, University of Rome Tor Vergata Via della Ricerca Scientifica 1 00133 Rome (Italy)
³ School of Mathematics, Georgia Institute of Technology 686 Cherry St. Atlanta GA. 30332-1160 (USA)

Mathematics Research Reports, Tome 1 (2020), pp. 15-29.

Résumé

We investigate the existence of whiskered tori in some dissipative systems, called conformally symplectic systems, having the property that they transform the symplectic form into a multiple of itself. We consider a family $f_{\mu }$ of conformally symplectic maps which depends on a drift parameter $\mu$.

We fix a Diophantine frequency of the torus and we assume to have a drift ${\mu }_0$ and an embedding of the torus $K_0$, which satisfy approximately the invariance equation $f_{{\mu }_0}\circ K_0=K_0\circ T_{\omega }$ (where $T_{\omega }$ denotes the shift by $\omega$). We also assume to have a splitting of the tangent space at the range of $K_0$ into three bundles. We assume that the bundles are approximately invariant under $D{f}_{{\mu }_0}$ and the derivative satisfies some rate conditions.

Under suitable nondegeneracy conditions, we prove that there exist ${\mu }_{\infty }$, $K_{\infty }$ invariant under $f_{{\mu }_{\infty }}$, close to the original ones, and a splitting which is invariant under $D{f}_{{\mu }_{\infty }}$. The proof provides an efficient algorithm to construct whiskered tori. Full details of the statements and proofs are given in [10].

Détail

Reçu le : 2019-01-24
Accepté le : 2019-05-22
Publié le : 2020-06-30

DOI : 10.5802/mrr.4

Classification : 70K43, 70K20, 34D30
Keywords: Whiskered tori, Conformally symplectic systems, KAM theory.

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