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.

Families of whiskered tori for a-priori stable/unstable Hamiltonian systems and construction of unstable orbits / E. Valdinoci. - In: MPEJ. - ISSN 1086-6655. - 6:(2000), pp. 1-31.

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

E. Valdinoci
2000

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.
Settore MAT/05 - Analisi Matematica
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/248777
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