A KAM theory for conformally symplectic systems: efficient algorithms and their validation

We present a KAM theory for some dissipative systems (geometrically, these are conformally symplectic systems, i.e. systems that transform a symplectic form into a multiple of itself). For systems with n degrees of freedom depending on n parameters we show that it is possible to find solutions with a fixed n-dimensional (Diophantine) frequency by adjusting the parameters. We do not assume that the system is close to integrable, but we present the results in an a-posteriori format. Our unknowns are a parameterization of the quasi-periodic solution and some parameters in the system. We formulate an invariance equation that expresses that the system with the parameters leaves invariant the solution given by the embedding. We show that if there is a sufficiently approximate solution of the invariance equation, which also satisfies some non-degeneracy conditions, then there is a true solution nearby. The smallness assumptions above can be understood either in Sobolev or in analytic norms. The a-posteriori format has several consequences: A) smooth dependence on the parameters, including the singular limit of zero dissipation; B) estimates on the measure of parameters covered by quasi-periodic solutions; C) convergence of perturbative expansions in dissipative analytic systems; D) bootstrap of regularity (i.e. that all tori which are smooth enough are analytic if the map is analytic); E) a numerically efficient criterion for the breakdown of the quasi-periodic solutions. The proof is based on an iterative quadratically convergent method. The iterative step takes advantage of some geometric identities; these identities also lead to an efficient algorithm. If we discretize the parameterization with N terms, a modified Newton step requires O(N) storage and O(N log(N)) operations. The a-posteriori theorems allow one to be confident on the numerical results even very close to breakdown. The algorithm does not require that the system is close to integrable, so that a continuation algorithm is guaranteed to continue the tori till the breakdown (in practice, only limitation being the total memory and the precision). The algorithms have been implemented and run in other papers