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:
https://doi.org/10.5802/mrr.4

Classification: 70K43, 70K20, 34D30

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

Accepted:

Published online:

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}, 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/} }

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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