On the Splitting of Invariant Manifolds in Multidimensional Near-Integrable Hamiltonian Systems.

Intro -- Contents -- Chapter 0. Introduction and Some Salient Features of the Model Hamiltonian -- Chapter 1. Symplectic Geometry and the Splitting of Invariant Manifolds -- 1.1. Symplectic geometry: a short reminder -- 1.2. Hyperbolic invariant manifolds -- 1.3. Angles of Lagrangian planes: the symplectic viewpoint -- 1.4. Angles of Lagrangian planes: the Euclidean viewpoint -- 1.5. Symplectic isomorphisms, angles and splitting forms -- 1.6. The splitting of Lagrangian submanifolds -- 1.7. Lagrangian submanifolds in a cotangent bundle -- 1.8. Hyperbolic tori and normally hyperbolic invariant manifolds -- 1.9. The perturbative setting -- 1.10. Lagrangian intersections and homoclinic trajectories -- 1.11. The splitting of the invariant manifolds of hyperbolic tori -- Chapter 2. Estimating the Splitting Matrix Using Normal Forms -- 2.1. Resonant normal forms -- 2.2. Computations in the vicinity of a resonant surface -- 2.3. Splitting in a perturbative setting, variance and stability -- 2.4. General exponential estimates for the splitting matrix -- 2.5. Persistence of tori, invariant manifolds and homoclinic trajectories -- 2.6. Splitting and stability -- Chapter 3. The Hamilton-Jacobi Method for a Simple Resonance -- 3.1. Notation and assumptions -- 3.2. Formal solutions and the Hamilton-Jacobi algorithm -- 3.3. Convergence and domains of analyticity -- 3.4. Exponential closeness of the invariant manifolds -- 3.5. Linear versus nonlinear splitting -- 3.6. Some variants and possible generalizations -- 3.7. A short historical tour and some concluding remarks -- Appendix. Invariant Tori With Vanishing or Zero Torsion -- Bibliography.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.