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].

Received:

Accepted:

Published online:

DOI:
10.5802/mrr.4

Accepted:

Published online:

Classification:
70K43, 70K20, 34D30

Keywords: Whiskered tori, Conformally symplectic systems, KAM theory.

Keywords: Whiskered tori, Conformally symplectic systems, KAM theory.

Author's affiliations:

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

@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}, pages = {15--29}, publisher = {MathOA foundation}, volume = {1}, year = {2020}, doi = {10.5802/mrr.4}, language = {en}, url = {https://mrr.centre-mersenne.org/articles/10.5802/mrr.4/} }

TY - JOUR AU - Renato C. Calleja AU - Alessandra Celletti AU - Rafael de la Llave TI - Whiskered KAM tori of conformally symplectic systems JO - Mathematics Research Reports PY - 2020 SP - 15 EP - 29 VL - 1 PB - MathOA foundation UR - https://mrr.centre-mersenne.org/articles/10.5802/mrr.4/ DO - 10.5802/mrr.4 LA - en ID - MRR_2020__1__15_0 ER -

%0 Journal Article %A Renato C. Calleja %A Alessandra Celletti %A Rafael de la Llave %T Whiskered KAM tori of conformally symplectic systems %J Mathematics Research Reports %D 2020 %P 15-29 %V 1 %I MathOA foundation %U https://mrr.centre-mersenne.org/articles/10.5802/mrr.4/ %R 10.5802/mrr.4 %G en %F 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/articles/10.5802/mrr.4/

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