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Singularity Theory for Non-Twist KAM Tori

A. González-Enríquez Universitat de Barcelona, Barcelona, Spain
A. Haro Universitat de Barcelona, Barcelona, Spain
R. de la Llave Georgia Institute of Technology, Atlanta, GA
Available Formats:
Electronic ISBN: 978-1-4704-1428-3
Product Code: MEMO/227/1067.E
List Price: $76.00 MAA Member Price:$68.40
AMS Member Price: $45.60 Click above image for expanded view Singularity Theory for Non-Twist KAM Tori A. González-Enríquez Universitat de Barcelona, Barcelona, Spain A. Haro Universitat de Barcelona, Barcelona, Spain R. de la Llave Georgia Institute of Technology, Atlanta, GA Available Formats:  Electronic ISBN: 978-1-4704-1428-3 Product Code: MEMO/227/1067.E  List Price:$76.00 MAA Member Price: $68.40 AMS Member Price:$45.60
• Book Details

Memoirs of the American Mathematical Society
Volume: 2272014; 115 pp
MSC: Primary 37;

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.

• 1. Introduction and preliminaries
• 1. Introduction
• 2. Preliminaries
• 2. Geometrical properties of KAM invariant tori
• 3. Geometric properties of an invariant torus
• 4. Geometric properties of fibered Lagrangian deformations
• 3. KAM results
• 5. Nondegeneracy on a KAM procedure with fixed frequency
• 6. A KAM theorem for symplectic deformations
• 7. A Transformed Tori Theorem
• 4. Singularity theory for KAM tori
• 8. Bifurcation theory for KAM tori
• 9. The close-to-integrable case
• Appendices
• A. Hamiltonian vector fields
• B. Elements of singularity theory
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Volume: 2272014; 115 pp
MSC: Primary 37;

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.

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