Renato C. Calleja, Alessandra Celletti, Rafael de la Llave
KAM estimates for the dissipative standard map
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ABSTRACT. From the beginning of KAM theory, it was realized that its
applicability to realistic problems depended on developing
quantitative estimates on the sizes of the perturbations allowed.
In this paper we present results on the existence of
quasi-periodic solutions for
conformally symplectic systems in non-perturbative
regimes. We recall that, for conformally symplectic
systems, finding the solution requires also to find
a mph{drift parameter}. We present a proof on the existence of
solutions for
values of the parameters which agree with more than three
figures with the numerically conjectured optimal values.
The first step of the strategy is to establish a very
explicit quantitative theorem in an a-posteriori format.
We recall that in numerical analysis, an a-posteriori theorem
assumes the existence of
an approximate solution, which satisfies an invariance
equation up to an error which is small enough with respect to
explicit condition numbers, and then concludes the existence of
a solution. In the case of conformally symplectic systems,
an a-posteriori theorem was proved in