Whiskered KAM tori of conformally symplectic systems
Mathematics Research Reports, Volume 1 (2020) , pp. 15-29.

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].

Accepted: 2019-05-21
Published online: 2020-06-30
DOI: https://doi.org/10.5802/mrr.4
Classification: 70K43,  70K20,  34D30
Keywords: Whiskered tori, Conformally symplectic systems, KAM theory.
@article{MRR_2020__1__15_0,
author = {Renato C.  Calleja and Alessandra Celletti and Rafael de la Llave},
title = {Whiskered KAM tori of conformally symplectic systems},
journal = {Mathematics Research Reports},
publisher = {MathOA foundation},
volume = {1},
year = {2020},
pages = {15-29},
doi = {10.5802/mrr.4},
language = {en},
url = {mrr.centre-mersenne.org/item/MRR_2020__1__15_0/}
}
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/item/MRR_2020__1__15_0/

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