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ANALYTIC REDUCIBILITY OF RESONANT COCYCLES TO A NORMAL FORM

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

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 ([email protected])
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 ([email protected])
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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 

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