Electronic ISBN:  9781470414283 
Product Code:  MEMO/227/1067.E 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 227; 2014; 115 ppMSC: Primary 37;
In this monograph the authors introduce a new method to study bifurcations of KAM tori with fixed Diophantine frequency in parameterdependent Hamiltonian systems. It is based on Singularity Theory of critical points of a realvalued 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 nontwist invariant tori. Hence, bifurcating points correspond to nontwist 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 closetointegrable case

Appendices

A. Hamiltonian vector fields

B. Elements of singularity theory


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In this monograph the authors introduce a new method to study bifurcations of KAM tori with fixed Diophantine frequency in parameterdependent Hamiltonian systems. It is based on Singularity Theory of critical points of a realvalued 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 nontwist invariant tori. Hence, bifurcating points correspond to nontwist 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 closetointegrable case

Appendices

A. Hamiltonian vector fields

B. Elements of singularity theory