Dynamics Day

• 11h-12h Luca Biasco
• 13h30-14h30 Anna Florio
• 14h45-15h45 Donato Scarcella
• 16h15-17h15 Santiago Barbieri

Luca Biasco
Measure and density of KAM tori in generic nearly-integrable Mechanical Systems

We prove that the phase space of a generic nearly-integrable mechanical system is foliated by primary and secondary invariant tori up to the union of a small neighbourhood of double resonances and an exponentially small set.

As a byproduct we prove, for mechanical systems and up to a logarithmic correction, a conjecture by Arnold, Kozlov and Neishtadt stating that the phase space of a generic nearly-integrable Hamiltonian System is foliated by invariant KAM tori up to a set of measure proportional to the size of the perturbation.

Anna Florio
Dynamique sauvage pour 3D flots d'Euler stationnaires

L'évolution d'un fluid idéal en équilibre dans R³ est décrite par l'équation d'Euler stationaire. Parmi ses solutions, les champs de vecteurs Beltrami sont les seuls où des phénomènes dynamiquement intéressants peuvent apparaître. Dans un travail en collaboration avec Pierre Berger et Daniel Peralta-Salas, nous étudions les champs de vecteurs Beltrami Euclidiens et nous montrons que des tangences homoclines et des phénomènes de type Newhouse apparaissent parmi ces champs de vecteurs. De plus, grâce à la théorie de Gonchenko-Shilnikov-Turaev, nous prouvons l'existence de champs de vecteurs Beltrami Euclidiens universels, i.e., qui contiennent une approximation de n'importe quelle dynamique.

Donato Scarcella
Asymptotic quasiperiodic solutions for time dependent Hamiltonians

In 1954 Kolmogorov laid the fondation for the so-called KAM theory. KAM theory shows the persistence of quasiperiodic solutions, in nearly integrable Hamiltonian systems. It is motivated by classical problems in celestial mechanics such as the n-body problem.

In this talk we are interested in perturbations which depend on time non-quasiperiodically. We will analyze some properties of time-dependent Hamiltonians converging asymptotically, when time tends to infinity, to autonomous Hamiltonians having an invariant torus supporting quasiperiodic solutions. We will study the conditions of existence of solutions which converge asymptotically in time to the quasiperiodic solution of the unperturbed autonomous system.

Moreover, we will analyze the exemple in celestial mechanics of a planetary system perturbed by a given comet coming from and going back to infinity, asymptotically along a hyperbolic Keplerian orbit.

Santiago Barbieri
Semi-algebraic geometry and generic long-time stability of intregrable Hamiltonian systems

As it is well known, a Theorem due to Nekhoroshev (1971-1977) shows that the solutions of a sufficiently regular integrable system verifying a transversality property on its gradient - known as "steepness" - are stable over a long time under the effect of any suitably small perturbation. Nekhoroshev also showed (1973) that the steepness property is generic, both in measure and topological sense, in the space of jets (Taylor polynomials) of sufficiently smooth functions. However, the proof of this result kept being poorly understood up to now and, surprisingly, the paper in which it is contained is hardly known, whereas the rest of the theory has been widely studied over the decades. Moreover, the definition of steepness is not constructive and no general rule to establish whether a given function is steep or not existed up to now, thus entailing a major problem in applications.

In this seminar, I will start by explaining the main ideas behind Nekhoroshev's proof of the genericity of steepness by making use of a more modern language. Indeed, the proof strongly relies on arguments of complex analysis and semi-algebraic geometry: the latter was much less developed than nowadays when the article was written, so that many passages appear to be quite obscure in the original reasoning. Especially, Nekhoroshev developed for his specific problem results about analytic reparametrization of semi algebraic sets and a nice estimate of complex analysis (the existence of a Bernstein-Remez inequality for algebraic functions) was buried in the proof. This result was completely generalized and applied in various settings by Roytwarf and Yomdin in 1997 by making use of different argument, but this part of Nekhoroshev's work seems to be unknown to the literature.

Finally, I will show how a deep understanding of the genericity of steepness allows to determine explicit algebraic criteria in the space of jets of any order and any number of variables which ensure that a given function is steep.

References

N. N. Nekhoroshev, "Stable lower estimates for smooth mappings and for gradients of smooth function", Mathematics of the USSR-Sbornik, 1973, vol. 90 (132), no. 3, pp.432-478

N. Roytwarf and Y. Yomdin: Bernstein Classes, Annales de l'Institut Fourier, 47( 3):825–858, 1998

• Sylvain Crovisier (U. Orsay)
• Jacques Fejoz (U. Dauphine, Obs. Paris)
• Laurent Niederman (U. Orsay, Obs. Paris)