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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
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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

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.

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
Appendices
• Appendix A. Hamiltonian vector fields
• Appendix B. Elements of singularity theory
• Bibliography