We study the behaviour 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 Lagrangian, invariant, rotational tori, 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 with independent frequencies of contractive (expansive) diffeomorphisms are always local attractors (repellors). We present results when the systems are analytic, or . 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.

Calleja, R., Celletti, A., de la Llave, R. (2013). Local behavior near quasi-periodic solutions of conformally symplectic systems. JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS, 25(3), 821-841 [10.1007/s10884-013-9319-0].

Local behavior near quasi-periodic solutions of conformally symplectic systems

CELLETTI, ALESSANDRA;
2013-01-01

Abstract

We study the behaviour 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 Lagrangian, invariant, rotational tori, 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 with independent frequencies of contractive (expansive) diffeomorphisms are always local attractors (repellors). We present results when the systems are analytic, or . 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.
2013
Pubblicato
Rilevanza internazionale
Articolo
Esperti anonimi
Settore MAT/07 - FISICA MATEMATICA
English
Con Impact Factor ISI
dissipative systems; conformal mappings; quasi-periodic solutions; attractors
Calleja, R., Celletti, A., de la Llave, R. (2013). Local behavior near quasi-periodic solutions of conformally symplectic systems. JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS, 25(3), 821-841 [10.1007/s10884-013-9319-0].
Calleja, R; Celletti, A; de la Llave, R
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2108/90247
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