Families of whiskered tori for a-priori stable/unstable Hamiltonian systems and construction of unstable orbits

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Abstract

We give a detailed statement of a KAM theorem about the conservation of partially hyperbolic tori on a fixed energy level for an analytic Hamiltonian H(I, φ, p, q) = h(I, pq; μ) + μf, (I, φ, p, q; μ), where φ is a (d - 1)-dimensional angle, I is in a domain of ℝd-1, p and q are real in a neighborhood 0, and μ is a small parameter. We show that invariant whiskered tori covering a large measure exist for sufficiently small perturbations. The associated stable and unstable manifolds also cover a large measure. Moreover, we show that there is a geometric organization to these tori. Roughly, the whiskered tori we construct are organized in smooth families, indexed by a Cantor parameter. The whole set of tori as well as their stable and unstable manifolds is smoothly interpolated. In particular, we emphasize the following items: sharp estimates on the relative measure of the surviving tori on the energy level, analyticity properties, including dependence upon parameters, geometric structures. We apply these results to both "a-priori unstable" and "a-priori stable" systems. We also show how to use the information obtained in the KAM Theorem we prove to construct unstable orbits.

Original languageEnglish
Pages (from-to)1-30
Number of pages30
JournalMathematical Physics Electronic Journal
Volume6
Publication statusPublished - 1 Dec 2000
Externally publishedYes

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Hamiltonian Systems
Torus
Orbit
Unstable
orbits
KAM Theorem
theorems
energy levels
Stable and Unstable Manifolds
Energy Levels
conservation
coverings
Invariant Tori
Cantor
Geometric Structure
Analyticity
perturbation
Small Perturbations
Small Parameter
estimates

Cite this

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title = "Families of whiskered tori for a-priori stable/unstable Hamiltonian systems and construction of unstable orbits",
abstract = "We give a detailed statement of a KAM theorem about the conservation of partially hyperbolic tori on a fixed energy level for an analytic Hamiltonian H(I, φ, p, q) = h(I, pq; μ) + μf, (I, φ, p, q; μ), where φ is a (d - 1)-dimensional angle, I is in a domain of ℝd-1, p and q are real in a neighborhood 0, and μ is a small parameter. We show that invariant whiskered tori covering a large measure exist for sufficiently small perturbations. The associated stable and unstable manifolds also cover a large measure. Moreover, we show that there is a geometric organization to these tori. Roughly, the whiskered tori we construct are organized in smooth families, indexed by a Cantor parameter. The whole set of tori as well as their stable and unstable manifolds is smoothly interpolated. In particular, we emphasize the following items: sharp estimates on the relative measure of the surviving tori on the energy level, analyticity properties, including dependence upon parameters, geometric structures. We apply these results to both {"}a-priori unstable{"} and {"}a-priori stable{"} systems. We also show how to use the information obtained in the KAM Theorem we prove to construct unstable orbits.",
author = "Enrico Valdinoci",
year = "2000",
month = "12",
day = "1",
language = "English",
volume = "6",
pages = "1--30",
journal = "Mathematical Physics Electronic Journal",
issn = "1086-6655",
publisher = "Angel Jorba & Jaume Timoneda, Eds. & Pubs.",

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TY - JOUR

T1 - Families of whiskered tori for a-priori stable/unstable Hamiltonian systems and construction of unstable orbits

AU - Valdinoci, Enrico

PY - 2000/12/1

Y1 - 2000/12/1

N2 - We give a detailed statement of a KAM theorem about the conservation of partially hyperbolic tori on a fixed energy level for an analytic Hamiltonian H(I, φ, p, q) = h(I, pq; μ) + μf, (I, φ, p, q; μ), where φ is a (d - 1)-dimensional angle, I is in a domain of ℝd-1, p and q are real in a neighborhood 0, and μ is a small parameter. We show that invariant whiskered tori covering a large measure exist for sufficiently small perturbations. The associated stable and unstable manifolds also cover a large measure. Moreover, we show that there is a geometric organization to these tori. Roughly, the whiskered tori we construct are organized in smooth families, indexed by a Cantor parameter. The whole set of tori as well as their stable and unstable manifolds is smoothly interpolated. In particular, we emphasize the following items: sharp estimates on the relative measure of the surviving tori on the energy level, analyticity properties, including dependence upon parameters, geometric structures. We apply these results to both "a-priori unstable" and "a-priori stable" systems. We also show how to use the information obtained in the KAM Theorem we prove to construct unstable orbits.

AB - We give a detailed statement of a KAM theorem about the conservation of partially hyperbolic tori on a fixed energy level for an analytic Hamiltonian H(I, φ, p, q) = h(I, pq; μ) + μf, (I, φ, p, q; μ), where φ is a (d - 1)-dimensional angle, I is in a domain of ℝd-1, p and q are real in a neighborhood 0, and μ is a small parameter. We show that invariant whiskered tori covering a large measure exist for sufficiently small perturbations. The associated stable and unstable manifolds also cover a large measure. Moreover, we show that there is a geometric organization to these tori. Roughly, the whiskered tori we construct are organized in smooth families, indexed by a Cantor parameter. The whole set of tori as well as their stable and unstable manifolds is smoothly interpolated. In particular, we emphasize the following items: sharp estimates on the relative measure of the surviving tori on the energy level, analyticity properties, including dependence upon parameters, geometric structures. We apply these results to both "a-priori unstable" and "a-priori stable" systems. We also show how to use the information obtained in the KAM Theorem we prove to construct unstable orbits.

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M3 - Article

VL - 6

SP - 1

EP - 30

JO - Mathematical Physics Electronic Journal

JF - Mathematical Physics Electronic Journal

SN - 1086-6655

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