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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: Geometric Description

Double Resonance: Geometric Description

Chapter:
(p.31) Chapter Four Double Resonance: Geometric Description
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/9780691202525.003.0004

This chapter examines the geometrical structure for the system at double resonance. After describing the normal form near a double resonance, it reduces the system to the slow mechanical system with perturbation. The system is conjugate to a perturbation of a two degrees of freedom mechanical system after a coordinate change and an energy reduction. The chapter then formulates the non-degeneracy conditions and theorems about their genericity. It also considers the normally hyperbolic invariant cylinders, and sketches the proof using local and global maps. The periodic orbits obtained in Theorem 4.4 correspond to the fixed points of compositions of local and global maps, when restricted to the suitable energy surfaces.

Keywords:   geometrical structure, double resonance, normal form, slow mechanical system, perturbation, non-degeneracy conditions, normally hyperbolic invariant cylinders, global maps

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