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

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*P. Lochak; J.-P. Marco; D. Sauzin*

In this text we take up the problem of the splitting of invariant manifolds in multidimensional Hamiltonian systems, stressing the canonical features of the problem. We first conduct a geometric study, which for a large part is not restricted to the perturbative situation of near-integrable systems. This point of view allows us to clarify some previously obscure points, in particular the symmetry and variance properties of the splitting matrix (indeed its very definition(s)) and more generally the connection with symplectic geometry. Using symplectic normal forms, we then derive local exponential upper bounds for the splitting matrix in the perturbative analytic case, under fairly general circumstances covering in particular resonances of any multiplicity. The next technical input is the introduction of a canonically invariant scheme for the computation of the splitting matrix. It is based on the familiar Hamilton-Jacobi picture and thus again is symplectically invariant from the outset. It is applied here to a standard Hamiltonian exhibiting many of the important features of the problem and allows us to explore in a unified way the question of finding lower bounds for the splitting matrix, in particular that of justifying a first order computation (the so-called Poincaré-Melnikov approximation). Although we do not specifically address the issue in this paper we mention that the problem of the splitting of the invariant manifold is well-known to be connected with the existence of a global instability in these multidimensional Hamiltonian systems and we hope the present study will ultimately help shed light on this important connection first noted and explored by V. I. Arnold.

#### Table of Contents

# Table of Contents

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

- Contents v6 free
- Chapter 0. Introduction and Some Salient Features of the Model Hamiltonian 110 free
- Chapter 1. Symplectic Geometry and the Splitting of Invariant Manifolds 918 free
- § 1.1. Symplectic geometry: a short reminder 1019
- § 1.2. Hyperbolic invariant manifolds 1322
- § 1.3. Angles of Lagrangian planes: the symplectic viewpoint 1625
- § 1.4. Angles of Lagrangian planes: the Euclidean viewpoint 1928
- § 1.5. Symplectic isomorphisms, angles and splitting forms 2231
- § 1.6. The splitting of Lagrangian submanifolds 2635
- § 1.7. Lagrangian submanifolds in a cotangent bundle 2837
- § 1.8. Hyperbolic tori and normally hyperbolic invariant manifolds 3140
- § 1.9. The perturbative setting 3645

- § 1.10. Lagrangian intersections and homoclinic trajectories 4049
- § 1.11. The splitting of the invariant manifolds of hyperbolic tori 4655

- Chapter 2. Estimating the Splitting Matrix Using Normal Forms 5160
- § 2.1. Resonant normal forms 5362
- § 2.2. Computations in the vicinity of a resonant surface 5968
- § 2.3. Splitting in a perturbative setting, variance and stability 6170
- § 2.4. General exponential estimates for the splitting matrix 6776
- § 2.5. Persistence of tori, invariant manifolds and homoclinic trajectories 7786
- § 2.6. Splitting and stability 8089

- Chapter 3. The Hamilton–Jacobi Method for a Simple Resonance 91100
- § 3.1. Notation and assumptions 92101
- § 3.2. Formal solutions and the Hamilton–Jacobi algorithm 93102
- § 3.3. Convergence and domains of analyticity 99108
- § 3.4. Exponential closeness of the invariant manifolds 106115
- § 3.5. Linear versus nonlinear splitting 115124
- § 3.6. Some variants and possible generalizations 120129
- § 3.7. A short historical tour and some concluding remarks 126135

- Appendix. Invariant Tori With Vanishing or Zero Torsion 133142
- Bibliography 141150