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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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Normally hyperbolic cylinders at double resonance

Normally hyperbolic cylinders at double resonance

Chapter:
(p.106) Chapter Ten Normally hyperbolic cylinders at double resonance
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/9780691202525.003.0010

This chapter proves the geometric picture of double resonance described in Chapter 4. There are two cases. In the simple critical homology case, the chapter shows the homoclinic orbit can be extended to periodic orbits both in positive and negative energy. The union of these periodic orbits forms a normally hyperbolic invariant manifold (which is homotopic to a cylinder with a puncture). In the non-simple homology case, the chapter demonstrates that for positive energy, there exist periodic orbits. The strategy is to prove the existence of these periodic orbits as hyperbolic fixed points of composition of local and global maps. A main technical tool to prove the existence and uniqueness of these fixed points is the Conley-McGehee isolation block.

Keywords:   simple critical homology, homoclinic orbit, periodic orbits, normally hyperbolic invariant manifold, non-simple homology, global maps, Conley-McGehee isolation block

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