Whiskered KAM tori of conformally symplectic systems

Renato C.  Calleja; Alessandra Celletti; Rafael de la Llave

doi:10.5802/mrr.4

Renato C. Calleja ¹ ; Alessandra Celletti ²; Rafael de la Llave ³

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, Volume 1 (2020), pp. 15-29.

Abstract

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 $f_{μ}$ of conformally symplectic maps which depends on a drift parameter $μ$ .

We fix a Diophantine frequency of the torus and we assume to have a drift $μ_{0}$ and an embedding of the torus $K_{0}$ , which satisfy approximately the invariance equation $f_{μ_{0}} \circ K_{0} = K_{0} \circ T_{ω}$ (where $T_{ω}$ denotes the shift by $ω$ ). 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 $D f_{μ_{0}}$ and the derivative satisfies some rate conditions.

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

Received: 2019-01-24
Accepted: 2019-05-22
Published online: 2020-06-30

DOI: 10.5802/mrr.4

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

Author's affiliations:

Renato C. Calleja ¹; Alessandra Celletti ²; Rafael de la Llave ³

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Renato C.  Calleja; Alessandra Celletti; Rafael de la Llave. Whiskered KAM tori of conformally symplectic systems. Mathematics Research Reports, Volume 1 (2020), pp. 15-29. doi : 10.5802/mrr.4. https://mrr.centre-mersenne.org/articles/10.5802/mrr.4/

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