Journal of Dynamics and Differential Equations

, Volume 25, Issue 3, pp 821–841 | Cite as

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 

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Renato C. Calleja
    • 1
  • 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
  3. 3.Departamento de Matemáticas y MecánicaInstituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de MéxicoMexico, D.F.Mexico

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