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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 Not yet published.Expected publication date is January 6, 2014. 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 Appendices Appendix A. Hamiltonian vector fields Appendix B. Elements of singularity theory Bibliography