Journal of Dynamics and Differential Equations

, Volume 25, Issue 3, pp 821–841

# Local Behavior Near Quasi-Periodic Solutions of Conformally Symplectic Systems

• Renato C. Calleja
• Alessandra Celletti
• Rafael de la Llave
Article

## 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 $$n$$ independent frequencies of contractive (expansive) diffeomorphisms are always local attractors (repellors). We present results when the systems are analytic, $$C^r$$ or $$C^\infty$$. 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.

## Keywords

Dissipative systems Conformal mappings Quasi-periodic solutions Attractors

## Mathematics Subject Classification (2000)

70K43 70K20 34D35

## Notes

### Acknowledgments

Part of this work was carried out when R.C. was a Posdoctoral Fellow at the IMA (University of Minnesota) and at the CRM (Montréal). A.C. was partially supported by PRIN-MIUR 2010JJ4$$\hbox {KPA}\_009$$ “Teorie geometriche e analitiche dei sistemi Hamiltoniani in dimensioni finite e infinite” and by the European MC-ITN Astronet-II. R.C. and R.L. were partially supported by NSF Grant DMS-1162544.

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## Authors and Affiliations

• Renato C. Calleja
• 1
Email author
• Alessandra Celletti
• 2
• Rafael de la Llave
• 1
1. 1.School of MathematicsGeorgia Institute of TechnologyAtlantaUSA
2. 2.Department of MathematicsUniversity of Roma Tor VergataRomeItaly
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