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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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Double Resonance: Forcing Equivalence

Double Resonance: Forcing Equivalence

Chapter:
(p.39) Chapter Five Double Resonance: Forcing Equivalence
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/9780691202525.003.0005

This chapter assesses the choice of cohomology and Aubry-Mather type at the double resonance. It begins by choosing cohomology classes for the (unperturbed) slow mechanical system. As in the case of single-resonance, the strategy is to choose a continuous curve in the cohomology space and prove forcing equivalence up to a residual perturbation. To do this, one needs to use the duality between homology and cohomology. The chapter then proves Aubry-Mather type for the perturbed slow mechanical system and reverts to the original coordinates. As the system has been perturbed, one needs to modify the choice of cohomology classes to connect the single and double resonances. Finally, the chapter proves Theorem 2.2, proving the main theorem.

Keywords:   cohomology, Aubry-Mather type, double resonance, slow mechanical system, forcing equivalence, cohomology classes, homology

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