Local Behavior Near Quasi-Periodic Solutions of Conformally Symplectic Systems

Abstract

Abstract. We study the behavior of conformally symplectic systems near rotational Lagrangian tori. We recall that conformally symplectic systems appear for example in mechanical models including a friction proportional to the velocity.We show that in a neighborhood of these quasi-periodic solutions (either transitive tori of maximal dimension or periodic solutions), one can always find a smooth symplectic change of variables in which the time evolution becomes just a rotation in some direction and a linear contraction in others. In particular quasi-periodic solutions of contractive (expansive) diffeomorphisms are always local attractors (repellors). We present results when the systems are analytic, C r or C ∞ . We emphasize that the results presented here are non-perturbative and apply to systems that are far from integrable; moreover, we do not require any assumption on the frequency and in particular we do not assume any non-resonance condition.We also show that the system of coordinates can be computed rather explicitly and we provide iterative algorithms, which allow to generalize the notion of "isochrones". We conclude by showing that the above results apply to quasi-periodic conformally symplectic flows.

Aubry–Mather Theory for Conformally Symplectic Systems

“…Hence, (γ,γ) is a solution of (10); in particular, proceeding as in [15, Corollary 2.2.12]), it follows that γ is C 2 . Actually, using Proposition 1 and equation (8) one can prove that γ is as smooth as the Lagrangian L (see also Remark 4).…”

“…Remark 8. In [7,8] the authors studied the persistence of KAM tori (i.e., smooth invariant Lagrangian graphs on which the dynamics is conjugated to a rotation) under small perturbations of conformally symplectic vector fields. Observe that whenever a KAM torus exists, then it is unique and it coincides with the Aubry set defined above (this follows from Proposition 8 and Remark 2).…”

“…∂x (x, p) − λp + η Different KAM approaches (e.g., [7,8,24,28]) have been proposed to show the persistence -under suitable assumptions and for small values of ε -of the invariant torus of the integrable case Λ λ,η . This "perturbed" torus does coincide with the Aubry and the Mather sets that we constructed; in particular, it continues to be a local attractor [8].…”

“…. In particular, they describe physical and mechanical systems characterized by a dissipation proportional to the momentum (or to the velocity), plus the action of a drifting term (see (8)).…”

Transient times, resonances and drifts of attractors in dissipative rotational dynamics

“…Indeed the solution (16) provides the natural attractor, which can be found in the original coordinates by integrating equations (13) with ε = 0. The local behavior near quasi-periodic attractors of some dissipative systems, precisely conformally symplectic systems 2 , has been studied in [4]. The main result of [4] is that there exists a transformation of coordinates such that the time evolution becomes a rotation in the angles and a contraction in the actions.…”