# Diffusion times and stability exponents for nearly integrable analytic systems

Pierre Lochak; Jean-Pierre Marco

Open Mathematics (2005)

- Volume: 3, Issue: 3, page 342-397
- ISSN: 2391-5455

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topPierre Lochak, and Jean-Pierre Marco. "Diffusion times and stability exponents for nearly integrable analytic systems." Open Mathematics 3.3 (2005): 342-397. <http://eudml.org/doc/268862>.

@article{PierreLochak2005,

abstract = {For a positive integer n and R>0, we set \[B\_R^n = \left\lbrace \{x \in \mathbb \{R\}^n |\left\Vert x \right\Vert \_\infty < R\} \right\rbrace \]
. Given R>1 and n≥4 we construct a sequence of analytic perturbations (H j) of the completely integrable Hamiltonian \[h\left( r \right) = \tfrac\{1\}\{2\}r\_1^2 + ...\tfrac\{1\}\{2\}r\_\{n - 1\}^2 + r\_n \]
on \[\mathbb \{T\}^n \times B\_R^n \]
, with unstable orbits for which we can estimate the time of drift in the action space. These functions H j are analytic on a fixed complex neighborhood V of \[\mathbb \{T\}^n \times B\_R^n \]
, and setting \[\varepsilon \_j : = \left\Vert \{h - H\_j \} \right\Vert \_\{C^0 (V)\} \]
the time of drift of these orbits is smaller than (C(1/ɛ j)1/2(n-3)) for a fixed constant c>0. Our unstable orbits stay close to a doubly resonant surface, the result is therefore almost optimal since the stability exponent for such orbits is 1/2(n−2). An analogous result for Hamiltonian diffeomorphisms is also proved. Two main ingredients are used in order to deal with the analytic setting: a version of Sternberg’s conjugacy theorem in a neighborhood of a normally hyperbolic manifold in a symplectic system, for which we give a complete (and seemingly new) proof; and Easton windowing method that allow us to approximately localize the wandering orbits and estimate their speed of drift.},

author = {Pierre Lochak, Jean-Pierre Marco},

journal = {Open Mathematics},

keywords = {37J40; 37B; 37D},

language = {eng},

number = {3},

pages = {342-397},

title = {Diffusion times and stability exponents for nearly integrable analytic systems},

url = {http://eudml.org/doc/268862},

volume = {3},

year = {2005},

}

TY - JOUR

AU - Pierre Lochak

AU - Jean-Pierre Marco

TI - Diffusion times and stability exponents for nearly integrable analytic systems

JO - Open Mathematics

PY - 2005

VL - 3

IS - 3

SP - 342

EP - 397

AB - For a positive integer n and R>0, we set \[B_R^n = \left\lbrace {x \in \mathbb {R}^n |\left\Vert x \right\Vert _\infty < R} \right\rbrace \]
. Given R>1 and n≥4 we construct a sequence of analytic perturbations (H j) of the completely integrable Hamiltonian \[h\left( r \right) = \tfrac{1}{2}r_1^2 + ...\tfrac{1}{2}r_{n - 1}^2 + r_n \]
on \[\mathbb {T}^n \times B_R^n \]
, with unstable orbits for which we can estimate the time of drift in the action space. These functions H j are analytic on a fixed complex neighborhood V of \[\mathbb {T}^n \times B_R^n \]
, and setting \[\varepsilon _j : = \left\Vert {h - H_j } \right\Vert _{C^0 (V)} \]
the time of drift of these orbits is smaller than (C(1/ɛ j)1/2(n-3)) for a fixed constant c>0. Our unstable orbits stay close to a doubly resonant surface, the result is therefore almost optimal since the stability exponent for such orbits is 1/2(n−2). An analogous result for Hamiltonian diffeomorphisms is also proved. Two main ingredients are used in order to deal with the analytic setting: a version of Sternberg’s conjugacy theorem in a neighborhood of a normally hyperbolic manifold in a symplectic system, for which we give a complete (and seemingly new) proof; and Easton windowing method that allow us to approximately localize the wandering orbits and estimate their speed of drift.

LA - eng

KW - 37J40; 37B; 37D

UR - http://eudml.org/doc/268862

ER -

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