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Singularity Theory for Non-Twist KAM Tori
A. González-Enríquez and A. Haro, Universitat de Barcelona, Spain, and R. de la Llave, Georgia Institute of Technology, Atlanta, GA

Memoirs of the American Mathematical Society
2014; 115 pp; softcover
Volume: 227
ISBN-10: 0-8218-9018-2
ISBN-13: 978-0-8218-9018-9
List Price: US$76
Individual Members: US$45.60
Institutional Members: US$60.80
Order Code: MEMO/227/1067
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In this monograph the authors introduce a new method to study bifurcations of KAM tori with fixed Diophantine frequency in parameter-dependent Hamiltonian systems. It is based on Singularity Theory of critical points of a real-valued function which the authors call the potential. The potential is constructed in such a way that: nondegenerate critical points of the potential correspond to twist invariant tori (i.e. with nondegenerate torsion) and degenerate critical points of the potential correspond to non-twist invariant tori. Hence, bifurcating points correspond to non-twist tori.

Table of Contents

Part 1: Introduction and preliminaries
  • Introduction
  • Preliminaries
Part 2: Geometrical properties of KAM invariant tori
  • Geometric properties of an invariant torus
  • Geometric properties of fibered Lagrangian deformations
Part 3: KAM results
  • Nondegeneracy on a KAM procedure with fixed frequency
  • A KAM theorem for symplectic deformations
  • A Transformed Tori Theorem
Part 4: Singularity theory for KAM tori
  • Bifurcation theory for KAM tori
  • The close-to-integrable case
  • Appendix A. Hamiltonian vector fields
  • Appendix B. Elements of singularity theory
  • Bibliography
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