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Whiskered KAM Tori of Conformally Symplectic Systems

Hamiltonian systems, from topology to applications through analysis I October 08, 2018 - October 12, 2018

October 12, 2018 (11:00 AM PDT - 12:00 PM PDT)
Speaker(s): Renato Calleja (Instituto de Matemáticas Aplicadas y Sistemas)
Location: MSRI: Simons Auditorium
Tags/Keywords
  • Whiskered tori

  • Confomrally symplectic systems

  • KAM theory

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Abstract

Many physical problems are described by conformally symplectic systems. We study the existence of whiskered tori in a family $f_\mu$ of conformally symplectic maps depending on parameters $\mu$. Whiskered tori are tori on which the motion is a rotation but having as many contracting/expanding directions as allowed by the preservation of the geometric structure. Our main result is formulated in an a-posteriori format. Given an approximately invariant embedding of the torus for a parameter value $\mu_0$ with an approximately invariant splitting of the tangent space at the range of the embedding into stable/unstable/center bundles, there is an invariant embedding and invariant splittings for new parameters. Using the results of formal expansions as the starting point for the a-posteriori method, we study the domains of analiticity of parameterizations of whiskered tori in perturbations of Hamiltonian Systems with dissipation. The proofs of the results lead to efficient algorithms that are quite practical to implement. Joint work with A. Celletti and R. de la Llave.

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