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

Abstract

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For a positive integer n and R>0, we set B R n = x ∈ ℝ n | x ∞ < R . Given R>1 and n≥4 we construct a sequence of analytic perturbations (H j) of the completely integrable Hamiltonian h r = 1 2 r 1 2 + . . . 1 2 r n - 1 2 + r n on 𝕋 n × 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 𝕋 n × B R n , and setting ε j : = h - H j 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.

How to cite

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Pierre 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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