Superexponential stability

Abstract These lectures are devoted to the main results of classical perturbation theory. We start by recalling the methods of Hamiltonian dynamics, the problem of small divisors, the series of Lindstedt and the method of normal form. Then we discuss the theorem of Kolmogorov with an application to the Sun-Jupiter-Saturn problem in Celestial Mechanics. Finally we discuss the problem of long-time stability, by discussing the concept of exponential stability as introduced by Moser and Littlewood and fully exploited by Nekhoroshev. The phenomenon of superexponential stability is also recalled. 

The rest of this section is devoted to the discussion of the theory of complete stability, exponential stability and superexponential stability. 

Superexponential stability is in a sense the outcome of the combination of perturbation methods. The simplest case to be considered is, again, that of an elliptic equilibrium or of the neighborhood of an invariant KAM torus. For definiteness, let us consider the latter case. 

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