KAM (computer-assisted) results in Celestial Mechanics: the dissipative spin-orbit problem

Abstract: The existence of invariant tori through Kolmogorov-Arnold-Moser (KAM) theory has been proven in several models of Celestial Mechanics through dedicated analytical proofs combined with computer-assisted techniques. After reviewing some of such results, obtained in conservative frameworks, we present a recent result on the existence of invariant attractors for a dissipative model: the spin-orbit problem with tidal torque. This model belongs to the class of conformally symplectic systems, which are characterized by the property that they transform the symplectic form into a multiple of itself. Finding the solution of such systems requires to add a drift parameter.

We describe a KAM theorem for conformally symplectic systems in an a-posteriori format: assuming the existence of an approximate solution, satisfying the invariance equation up to an error term - small enough with respect to explicit condition numbers, - then we can prove the existence of a solution nearby. The theorem, which does not assume that the system is close to integrable, yields an efficient algorithm to construct invariant attractors for the spin-orbit problem and it provides accurate estimates of the breakdown threshold of the invariant attractor.

This talk refers to joint works with R. Calleja, J. Gimeno, and R. de la Llave.

analysis of PDEsclassical analysis and ODEsdynamical systemsfunctional analysisnumerical analysis

Audience: researchers in the discipline

CRM CAMP (Computer-Assisted Mathematical Proofs) in Nonlinear Analysis

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