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Habilitation à diriger des recherches

STABILITE GENERIQUE DES SYSTEMES HAMILTONIENS QUASI-INTEGRABLES

Abstract : The study of stability and unstability of nearly-integrable Hamiltonian systems is an old and difficult problem in dynamical systems.

There are two kind of theorems :

i) Results of stability over infinite times provided by K.A.M. theory which are valid over a Cantor set of big measure but we have almost no information on the other trajectories and even a strong unstability can occur.

ii) Results of stability for open sets of initial values but only valid over an exponentially long time with respect to the size of the perturbation.

This second kind of results was introduced by N.N. Nekhorochev who proved in 1977 a global stability theorem over exponentially long times provided that the unperturbed (integrable) Hamiltonian meets some generic transversality condition known as steepness. Especially, the convex functions are steep.

The study of this notion of steepness and its consequences has almost not been resumed since Nekhorochev original work despite the genericity of this class of functions among differentiable functions and various physical examples where the integrable Hamiltonian is steep but not convex.

Here, we give a proof of Nekhorochev's theorem which is significantly simplified with respect to the original one. This allows to derive accurate estimates on the times of stability which are essentially optimal in the convex case.

Then, using theorems of real subanalytic geometry, we derive a geometric criterion for steepness : a real valued function which is real analytic around a compact set is steep if and only if its restriction to any affine subspace admits only isolated critical points. This is an extension of a previous result of Y. Ilyashenko in the holomorphic case. Our study is based on theorems of real subanalytic geometry (curve selection lemma and Lojaciewicz's exponents). This also allows to find a necessary condition for exponential stability which is close to steepness. We also give methods to compute explicitly the constants involved in this kind of theorems.

Finally, we prove a theorem of exponential stability for nearly integrable Hamiltonian systems with a non degeneracy condition on the unperturbed hamiltonian which is strictly weaker than steepness. The point in this refinement lies in the fact that it allows to exhibit a generic class of real analytic integrable Hamiltonians which are exponentially stable with fixed exponents.

Genericity is proved in the sense of measure since we exhibit a prevalent set of integrable Hamiltonian which satisfy the latter property. This is obtained by an application of a quantitative Sard theorem given by Yomdin.
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Habilitation à diriger des recherches
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Submitted on : Monday, January 15, 2007 - 12:32:35 PM
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Laurent Niederman. STABILITE GENERIQUE DES SYSTEMES HAMILTONIENS QUASI-INTEGRABLES. Mathématiques [math]. Université Paris Sud - Paris XI, 2006. ⟨tel-00124486⟩

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