Singularity Theory for Non-Twist KAM Tori.
González-Enríquez, A.
Singularity Theory for Non-Twist KAM Tori. - 1st ed. - 1 online resource (128 pages) - Memoirs of the American Mathematical Society ; v.227 . - Memoirs of the American Mathematical Society .
Intro -- Contents -- Part 1 . Introduction and preliminaries -- Chapter 1. Introduction -- 1.1. Towards a singularity theory for KAM tori -- 1.2. Methodology (a brief description) -- 1.3. Outline of this monograph -- Chapter 2. Preliminaries -- 2.1. Elementary notations -- 2.2. Geometric preliminaries -- 2.3. Symplectic deformations and moment maps -- 2.4. Analytic preliminaries -- 2.5. Cohomology equations -- Part 2 . Geometrical properties of KAM invariant tori -- Chapter 3. Geometric properties of an invariant torus -- 3.1. Automatic reducibility -- 3.2. Geometric definition of non-twist tori -- 3.3. Intrinsic character of the reducibility and of the torsion -- Chapter 4. Geometric properties of fibered Lagrangian deformations -- 4.1. The potential of a fibered Lagrangian deformation -- 4.2. A parametric version of the potential -- Part 3 . KAM results -- Chapter 5. Nondegeneracy on a KAM procedure with fixed frequency -- 5.1. Approximate reducibility of approximately invariant tori -- 5.2. Dummy and modifying parameters -- Chapter 6. A KAM theorem for symplectic deformations -- 6.1. Functional equations and nondegeneracy condition -- 6.2. Statement of the KAM theorem -- 6.3. Proof of the KAM Theorem -- Chapter 7. A Transformed Tori Theorem -- 7.1. Nondegeneracy condition -- 7.2. Statement of the Transformed Tori Theorem -- 7.3. Proof of the Transformed Tori Theorem -- Part 4 . Singularity theory for KAM tori -- Chapter 8. Bifurcation theory for KAM tori -- 8.1. Classification of KAM invariant tori -- 8.2. Local equivalence of Bifurcations diagrams -- Chapter 9. The close-to-integrable case -- 9.1. The integrable case -- 9.2. Persistence of invariant tori in quasi-integrable systems -- 9.3. Unfolding non-twist tori -- 9.4. The Birkhoff potential and the potential of an invariant torus -- Appendices. Appendix A. Hamiltonian vector fields -- A.1. Cohomology equations -- A.2. Automatic reducibility of invariant tori -- A.3. Families of Hamiltonians and moment maps -- A.4. Potential and moment of an invariant FLD -- A.5. Transformed Tori Theorem -- A.6. A KAM Theorem for families of Hamiltonians -- Appendix B. Elements of singularity theory -- Bibliography.
9781470414283
Bifurcation theory.
Perturbation (Mathematics).
Ergodic theory.
Electronic books.
QA380 -- .G66 2013eb
515.39
Singularity Theory for Non-Twist KAM Tori. - 1st ed. - 1 online resource (128 pages) - Memoirs of the American Mathematical Society ; v.227 . - Memoirs of the American Mathematical Society .
Intro -- Contents -- Part 1 . Introduction and preliminaries -- Chapter 1. Introduction -- 1.1. Towards a singularity theory for KAM tori -- 1.2. Methodology (a brief description) -- 1.3. Outline of this monograph -- Chapter 2. Preliminaries -- 2.1. Elementary notations -- 2.2. Geometric preliminaries -- 2.3. Symplectic deformations and moment maps -- 2.4. Analytic preliminaries -- 2.5. Cohomology equations -- Part 2 . Geometrical properties of KAM invariant tori -- Chapter 3. Geometric properties of an invariant torus -- 3.1. Automatic reducibility -- 3.2. Geometric definition of non-twist tori -- 3.3. Intrinsic character of the reducibility and of the torsion -- Chapter 4. Geometric properties of fibered Lagrangian deformations -- 4.1. The potential of a fibered Lagrangian deformation -- 4.2. A parametric version of the potential -- Part 3 . KAM results -- Chapter 5. Nondegeneracy on a KAM procedure with fixed frequency -- 5.1. Approximate reducibility of approximately invariant tori -- 5.2. Dummy and modifying parameters -- Chapter 6. A KAM theorem for symplectic deformations -- 6.1. Functional equations and nondegeneracy condition -- 6.2. Statement of the KAM theorem -- 6.3. Proof of the KAM Theorem -- Chapter 7. A Transformed Tori Theorem -- 7.1. Nondegeneracy condition -- 7.2. Statement of the Transformed Tori Theorem -- 7.3. Proof of the Transformed Tori Theorem -- Part 4 . Singularity theory for KAM tori -- Chapter 8. Bifurcation theory for KAM tori -- 8.1. Classification of KAM invariant tori -- 8.2. Local equivalence of Bifurcations diagrams -- Chapter 9. The close-to-integrable case -- 9.1. The integrable case -- 9.2. Persistence of invariant tori in quasi-integrable systems -- 9.3. Unfolding non-twist tori -- 9.4. The Birkhoff potential and the potential of an invariant torus -- Appendices. Appendix A. Hamiltonian vector fields -- A.1. Cohomology equations -- A.2. Automatic reducibility of invariant tori -- A.3. Families of Hamiltonians and moment maps -- A.4. Potential and moment of an invariant FLD -- A.5. Transformed Tori Theorem -- A.6. A KAM Theorem for families of Hamiltonians -- Appendix B. Elements of singularity theory -- Bibliography.
9781470414283
Bifurcation theory.
Perturbation (Mathematics).
Ergodic theory.
Electronic books.
QA380 -- .G66 2013eb
515.39