KAM estimates for the dissipative standard map

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 quasiperiodic solutions for conformally symplectic systems in non-perturbative regimes. We recall that, for conformally symplectic systems, finding the solution requires also to find a 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 [12]. Our first task is to make all the constants fully explicit.We emphasize that our result allows to conclude the existence of the true solution by verifying mainly that the approximate solution satisfies the equation up to a small error and that some condition numbers are finite. The method used to produce the approximate solution does not need to be examined.The second step in the strategy is to produce numerically very accurate solutions in a concrete problem. We have implemented the algorithm indicated in [12] in a model problem, widely considered in the literature; we constructed numerically very accurate solutions of the invariance equations (discretizations with 2 18 Fourier coefficients, each one computed with 100 digits of precision). From the point of view of rigorous mathematics, we note that the first step is a fully rigorous theorem, the second step is a high precision calculation which produces an impact for the theorem in the first part.The third and final step is to present a numerical verification of the hypotheses of the theorem stated in the first part on the numerical solutions presented in the second part. Using these estimates we would conclude the existence of tori for certain values of the drift parameter. The perturbation parameters we can consider coincide with more than 3 significant figures with the values conjectured as optimal by numerical experiments.The verification of the estimates presented here is not completely rigorous since we do not control the round-off error. Nevertheless, running with different precision shows very little difference in the results. Given the high precision of the calculation and the simplicity of the estimates, this does not seem to affect the results. A full verification should be done implementing interval arithmetic.We make available the approximate solutions, the highly efficient algorithms to generate them (incorporating high precision based on the MPFR library) and the routines used to...