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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
Front Cover for Singularity Theory for Non-Twist KAM Tori
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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
Front Cover for Singularity Theory for Non-Twist KAM Tori
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  • Front Cover for Singularity Theory for Non-Twist KAM Tori
  • Back Cover for Singularity Theory for Non-Twist KAM Tori
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.

  • Table of Contents
     
     
    • 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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