Hostname: page-component-8448b6f56d-mp689 Total loading time: 0 Render date: 2024-04-18T07:06:35.948Z Has data issue: false hasContentIssue false
  • English
  • Français

ANALYTIC REDUCIBILITY OF RESONANT COCYCLES TO A NORMAL FORM

Part of: Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems Smooth dynamical systems: general theory Qualitative theory

Published online by Cambridge University Press:  27 November 2014

Claire Chavaudret  and
Laurent Stolovitch
Claire Chavaudret
Affiliation:
Laboratoire J.-A. Dieudonné, U.M.R. 6621, Université de Nice - Sophia Antipolis, Parc Valrose 06108 Nice Cedex 02, France (chavaudret@unice.fr)
Laurent Stolovitch
Affiliation:
CNRS and Laboratoire J.-A. Dieudonné U.M.R. 6621, Université de Nice - Sophia Antipolis, Parc Valrose 06108 Nice Cedex 02, France (stolo@unice.fr)
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider systems of quasi-periodic linear differential equations associated to a ‘resonant’ frequency vector ${\it\omega}$, namely, a vector whose coordinates are not linearly independent over $\mathbb{Z}$. We give sufficient conditions that ensure that a small analytic perturbation of a constant system is analytically conjugate to a ‘resonant cocycle’. We also apply our results to the non-resonant case: we obtain sufficient conditions for reducibility.

Keywords

normal formsquasi-periodic cocyclesreducibilityresonancessmall divisors

MSC classification

Primary: 37C55: Periodic and quasiperiodic flows and diffeomorphisms 37J40: Perturbations, normal forms, small divisors, KAM theory, Arnol'd diffusion 34C20: Transformation and reduction of equations and systems, normal forms
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 15 , Issue 1 , January 2016 , pp. 203 - 223
Copyright
© Cambridge University Press 2014 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Adrianova, L. Ja., The reducibility of systems of n linear differential equations with quasi-periodic coefficients, Vestnik Leningrad. Univ. 17(7) (1962), 14–24.Google Scholar
Avila, A., Fayad, B. and Krikorian, R., A KAM scheme for SL(2, ℝ) cocycles with Liouvillean frequencies, Geom. Funct. Anal. 21(5) (2011), 1001–1019.CrossRefGoogle Scholar
Avila, A. and Krikorian, R., Reducibility or non-uniform hyperbolicity for quasiperiodic Schrödinger cocycles, Ann. of Math. (2) 164(3) (2006), 911–940.CrossRefGoogle Scholar
Arnold, V. I., Chapitres supplémentaires de la théorie des équations différentielles ordinaires. Mir, 1980.Google Scholar
Chavaudret, C., Strong almost reducibility for analytic and Gevrey quasi-periodic cocycles, Bull. Soc. Math. France 141(1) (2013), 47–106.CrossRefGoogle Scholar
Chavaudret, C. and Marmi, S., Reducibility of quasiperiodic cocycles under a Brjuno-Rüssmann arithmetical condition, J. Mod. Dyn. 6(1) (2012), 59–78.Google Scholar
Eliasson, L. H., Floquet solutions for the 1-dimensional quasi-periodic Schrödinger equation, Comm. Math. Phys. 146(3) (1992), 447–482.CrossRefGoogle Scholar
Eliasson, L. H., Almost reducibility of linear quasi-periodic systems, in Smooth Ergodic Theory and its Applications (Seattle, WA, 1999), Proceedings of Symposia in Pure Mathematics, Volume 69, pp. 679–705 (American Mathematical Society, Providence, RI, 2001).Google Scholar
He, H.-L. and You, J., Full measure reducibility for generic one-parameter family of quasi-periodic linear systems, J. Dynam. Differential Equations 20(4) (2008), 831–866.Google Scholar
Johnson, R. and Moser, J., The rotation number for almost periodic potentials, Comm. Math. Phys. 84(3) (1982), 403–438.Google Scholar
Krikorian, R., Réductibilité des systèmes produits-croisés à valeurs dans des groupes compacts, Astérisque (259) 1999.Google Scholar
Krikorian, R., Réductibilité presque partout des flots fibrés quasi-périodiques à valeurs dans des groupes compacts, Ann. Sci. Éc. Norm. Supér. (4) 32(2) (1999), 187–240.Google Scholar
Mitropol’skiĭ, Ju. A. and Samoĭlenko, A. M., On constructing solutions of linear differential equations with quasiperiodic coefficients by the method of improved convergence, Ukraïn Mat. Zh. 17(6) (1965), 42–59.Google Scholar
Stolovitch, L., Forme normale de champs de vecteurs commutants, C. R. Acad. Sci. I 324 (1997), 665–668.CrossRefGoogle Scholar
Zhou, Q. and Wang, J., Reducibility results for quasiperiodic cocycles with Liouvillean frequency, J. Dynam. Differential Equations 24(1) (2012), 61–83.Google Scholar