On the measure of the KAM-tori in a neighbourhood of a separatrix

A. G. Medvedev

Sbornik. Mathematics, Tome 215 (2024) no. 6, pp. 755-774

Résumé

Consider a Liouville-integrable Hamiltonian system with $n$ degrees of freedom. Assume that the foliation of the phase space by invariant Lagrangian $n$-tori is degenerate on a $(2n-1)$-dimensional singular manifold $\mathbb{W}$ formed by the asymptotic manifolds of hyperbolic $(n-1)$-tori. The system usually ceases to be integrable after a small perturbation of order $\varepsilon$, but in accordance with the KAM-theory most invariant $n$-tori persist. The dynamics on the complement $C$ to this toric set is commonly associated with chaos. The measure of the set of points obtained as the intersection of a neighbourhood of $\mathbb{W}$ with $C$ is considered. Under natural assumptions it has the order of $\sqrt \varepsilon$. This results generalizes and complements the estimates for the measure of $C$ away from $\mathbb{W}$ due to Svanidze, Neishtadt and Pöschel. Bibliography: 14 titles.

Détail

Keywords: KAM-theory, separatrices, systems with small parameter, measure of the invariant tori, perturbation theory.
Mots-clés : chaos

Bibliographie

[1] A. N. Kolmogorov, “On conservation of conditionally periodic motions under small perturbations of the Hamiltonian”, Dokl. Akad. Nauk SSSR, 98 (1954), 527–530 (Russian) | MR | Zbl

[2] V. I. Arnold, “Proof of a theorem of A. N. Kolmogorov on the invariance of quasi-periodic motions under small perturbations of the Hamiltonian”, Russian Math. Surveys, 18:5 (1963), 9–36 | DOI

[3] J. Moser, “On invariant curves of area-preserving mappings of an annulus”, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II, 1962 (1962), 1–20 | MR | Zbl

[4] A. I. Neishtadt, “Estimates in the Kolmogorov theorem on conservation of conditionally periodic motions”, J. Appl. Math. Mech., 45:6 (1981), 766–772 | DOI | MR | Zbl

[5] J. Pöschel, “Integrability of Hamiltonian systems on Cantor sets”, Comm. Pure Appl. Math., 35:5 (1982), 653–695 | DOI | MR | Zbl

[6] N. V. Svanidze, “Small perturbations of an integrable dynamical system with an integral invariant”, Proc. Steklov Inst. Math., 147 (1981), 127–151 | MR | Zbl

[7] L. Biasco and L. Chierchia, “Explicit estimates on the measure of primary KAM tori”, Ann. Mat. Pura Appl. (4), 197:1 (2018), 261–281 | DOI | MR | Zbl

[8] L. Biasco and L. Chierchia, “On the measure of Lagrangian invariant tori in nearly-integrable mechanical systems”, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 26:4 (2015), 423–432 | DOI | MR | Zbl

[9] A. G. Medvedev, A. I. Neishtadt and D. V. Treschev, “Lagrangian tori near resonances of near-integrable Hamiltonian systems”, Nonlinearity, 28:7 (2015), 2105–2130 | DOI | MR | Zbl

[10] V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, “Mathematical aspects of classical and celestial mechanics”, Dynamical systems III, Encyclopaedia Math. Sci., 3, Springer-Verlag, Berlin, 1988, 1–291 | MR | Zbl

[11] L. Biasco and L. Chierchia, Singular KAM theory, 2023, arXiv: 2309.17041v1

[12] J. Moser, “The analytic invariants of an area-preserving mapping near a hyperbolic fixed point”, Comm. Pure Appl. Math., 9:4 (1956), 673–692 | DOI | MR | Zbl

[13] D. Treschev and O. Zubelevich, Introduction to the perturbation theory of Hamiltonian systems, Springer Monogr. Math., Springer-Verlag, Berlin, 2010, x+211 pp. | DOI | MR | Zbl

[14] A. I. Neishtadt, Questions of the perturbation theory of nonlinear resonance systems, D.Sc. dissertation, Moscow State University, Moscow, 1989, 298 pp. (Russian)