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

Greene's criterion for twist mappings asserts the existence of smooth invariant circles with preassigned rotation number if and only if the periodic trajectories with frequency approaching that of the quasi-periodic orbit are at the border of linear stability. We formulate an extension of this criterion for conformally symplectic systems in any dimension and prove one direction of the implication, namely, 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 proof of this result is very different from the proof in the area-preserving case, since in the conformally symplectic case the existence of periodic orbits requires adjusting parameters. Also, as shown in [R. Calleja, A. Celletti, and R. de la Llave, J. Dynam. Differential Equations, 55 (2013), pp. 821--841], in the conformally symplectic case there are no Birkhoff invariants giving obstructions to linearization near an invariant torus. As a byproduct of the techniques developed here, we obtain quantitative information on the existence of periodic orbits in the neighborhood of quasi-periodic tori, and we provide upper and lower bounds on the width of the phase-locking regions and of the Arnold tongues in $n$-degrees of freedom conformally symplectic systems.