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