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Arnold Diffusion for Smooth Systems of Two and a Half Degrees of Freedom(AMS-208)$
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Vadim Kaloshin and Ke Zhang

Print publication date: 2020

Print ISBN-13: 9780691202525

Published to Princeton Scholarship Online: May 2021

DOI: 10.23943/princeton/9780691202525.001.0001

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Normal Forms and Cohomology Classes at Single Resonances

Normal Forms and Cohomology Classes at Single Resonances

Chapter:
(p.24) Chapter Three Normal Forms and Cohomology Classes at Single Resonances
Source:
Arnold Diffusion for Smooth Systems of Two and a Half Degrees of Freedom
Author(s):

Kaloshin Vadim

Zhang Ke

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691202525.003.0003

This chapter discusses the single-resonance non-degeneracy conditions and normal forms. It then formulates Theorem 3.3, which covers the forcing equivalence in the single-resonance regime. The classical partial averaging theory indicates that after a coordinate change, the system has the normal form away from punctures. In order to state the normal form, one needs an anisotropic norm adapted to the perturbative nature of the system. The chapter also uses the idea of Lochak to cover the action space with double resonances. A double resonance corresponds to a periodic orbit of the unperturbed system. Finally, the chapter looks at a lemma which is an easy consequence of the Dirichlet theorem.

Keywords:   single-resonance non-degeneracy conditions, normal forms, forcing equivalence, single-resonance regime, partial averaging theory, double resonances, Dirichlet theorem

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