Universidad Autónoma de México, México
The flows of mechanical systems with a friction added do not preserve the symplectic structure, but instead they contract the symplectic structure with a factor that does not depend on the phase variables. Geometrically these kinds of systems are conformally symplectic. In this advanced course, I plan to explain the developments in the KAM theory of conformally symplectic dynamical systems, hoping to cover KAM theorems for the existence of Lagrangian and lower-dimensional tori. In this dissipative setting, the KAM tori that one obtains are quasi-periodic attractors for the dynamics given fixed values of the parameters. I will explain the constructive theorems that several mathematicians have developed over the years [CCdlL13, CH14, HCF+16, CCdlL17, CCdlL20b, CCdlL20c], that prove the existence of quasi-periodic attractors in the dissipative setting of conformally symplectic systems. I will also present some of the numerical implementations that can be derived from the constructive theorems. See for example [CC10, CF12, CH17, BC19], for implementations for computing the attractors, their breakdown mechanisms and tori in the limit of small dissipation.
[CCdlL13] Renato C. Calleja, Alessandra Celletti, and Rafael de la Llave. A KAM theory for conformally symplectic systems: efficient algorithms and their validation, J. Differential Equations, 255(5):978-1049, 2013.
[CH14] Marta Canadell and Alex Haro. Parameterization method for computing quasi periodic reducible normally hyperbolic invariant tori, Advances in differential equations and applications, volume 4 of SEMA SIMAI Springer Ser., pages 85–94. Springer, Cham, 2014.
[CCdlL17] Renato C. Calleja, Alessandra Celletti, and Rafael de la Llave. Domains of analyticity and Lindstedt expansions of KAM tori in some dissipative perturbations of Hamiltonian systems, Nonlinearity, 30(8):3151–3202, 2017.
[CCdlL20b] Renato C. Calleja, Alessandra Celletti, and Rafael de la Llave. Existence of whiskered KAM tori of conformally symplectic systems, Nonlinearity, 33(1):538-597, 2020. [CC10] Renato Calleja and Alessandra Celletti. Breakdown of invariant attractors for the dissipative standard map, Chaos, 20(1):013121, 9, 2010.
[CF12] Renato Calleja and Jordi-Lluis Figueras. Collision of invariant bundles of quasi periodic attractors in the dissipative standard map, Chaos, 22(3):033114, 10, 2012. [CCdlL20a] R. Calleja, A. Celletti, and R. de la Llave. KAM estimates for the dissipative standard map, preprint (2020), available at arXiv 2002.10647.
[CCdlL20c] R. Calleja, A. Celletti, R. de la Llave, KAM theory for some dissipative systems, preprint: arXiv 2007.08394