PhD thesis - IAS

Stability and instability of near-integrable Hamiltonian systems.
Abstract
In this thesis, we study various questions concerning the stability and instability of nearintegrable
Hamiltonian systems. It is divided into four parts and eight chapters.
In a first part, we give an informal introduction to Hamiltonian systems and to the perturbation
theory of integrable Hamiltonian systems in the first chapter, and then, in the second
chapter, we state our results.
A second part is devoted to stability results. In the third chapter, we give a new proof of the
exponential stability theorem of Nekhoroshev in the generic case for an analytic system. Our
method uses only composition of periodic averaging, and therefore it avoids the small divisors
problem. Then, in the fourth chapter, we take advantage of this approach to obtain new
results of exponential and super-exponential stability in the neighbourhood of elliptic fixed
points, invariant Lagrangian quasi-periodic tori and more generally invariant linearly stable
quasi-periodic tori, which are isotropic and reducible. In the fifth chapter, for a quasi-convex
integrable Hamiltonian system, we also prove a result of polynomial stability in the case where
the system is only finitely differentiable.
A third part lies between stability and instability. In the sixth chapter, for a quasi-convex
system which is analytic or Gevrey, we improve the stability exponent by studying the geometry
of simple resonances. Thus we obtain a time of stability which is closer to the known instability
times, and which is certainly optimal.
In the fourth part, we will construct examples of unstable Hamiltonian systems. First,
in the seventh chapter, we give a new example of an a priori unstable system which has a
drifting orbit with an optimal time of instability. Our method uses the symbolic dynamics
created by the transverse intersection between the stable and unstable manifolds of a normally
hyperbolic invariant manifold. In the eighth and last chapter, we also construct an example
of a near-integrable Hamiltonian system, for which the size of the perturbation goes to zero
only when the number of degrees of freedom goes to infinity, and which has an orbit drifting
in a polynomial time. In particular, this gives a new constraint on the threshold of validity for
exponential stability results.
The first part is written in French, and the others in English.
Keywords : Dynamical Systems, Hamiltonian Systems, Perturbation Theory, Nekhoroshev
Theory, Arnold Diffusion.