An Extension of Greene's Criterion for Conformally Symplectic Systems and a Partial Justification

Periodic and quasi-periodic orbits are important objects that explain much of the dynamics in several Hamiltonian models in Celestial Mechanics. Adding a friction proportional to the velocity of the particles , an increasingly common asumption in Celestial Mechanics, gives rise to conformally symplectic models. Greene's criterion for twist mappings asserts the existence of a KAM torus by examining the lin-earization of periodic orbits of rotation numbers close to it. I will formulate an extension of Greene's criterion for the existence of smooth invariant tori with preassigned rotation number for conformally symplectic systems in any dimension. I will prove that if there is a smooth invariant attractor, we can predict the eigenvalues of the periodic orbits whose frequencies approximate that of the torus for values of the parameters close to that of the attractor. The result proves one direction of the implication in our extension of Greene's criterion. As a byproduct of the techniques developed here, we obtain quantitative information on the existence of periodic orbits, upper and lower bounds on the width of the phase-locking regions and of Arnold tongues, and a numerically implementable criterion that agrees with other breakdown criteria in example models.