Among the spectacular results of modern celestial mechanics, the discovery of chaotic diffusion of eccentricities and inclination is probably the most interesting aspect for the broader community interested in general chaos theory. Indeed it reveals a mechanism that should be generic in moderate order resonances among the fast degrees of freedom of any degenerate Hamiltonian system. This paper discusses the resonant structure that is at the base of this chaotic diffusion process and illustrates an approximated method to estimate the diffusion timescale. Examples on the relevance of chaotic diffusion for the dynamics of small bodies of the solar system are also provided.

A planet can be described by an homogeneous rigid ellipsoid with flatness $\eta$, moving on a Keplerian orbit around a star and subject only to Newtonian forces. It was proposed in 1994 in [2] that, for suitable initial data, the precession cone can change $O(1)$ in a finite time, no matter how small $\eta$ is, as a consequence of Arnold diffusion mechanism. One can start introducing some simplifications in the original model, neglecting a term in its Hamiltonian so that the problem is reduced to a priori unstable three time scale system; for such systems a general theory of Arnold diffusion can indeed be developed (mainly in [2], [8], [10], [11]). In this paper we will review the main results about Arnold diffusion in three time scale a priori unstable systems and we discuss their relevance for a complete understanding of the precession problem.

We illustrate a completely analytic approach to Mel'nikov theory, which is based on a suitable extension of a classical method, and which is parallel and — at least in part — complementary to the standard procedure. This approach can be also applied to some "degenerate" situations, as to the case of nonhyperbolic unstable points, or of critical points located at the infinity (thus giving rise to unbounded orbits, e.g. the Keplerian parabolic orbits), and it is naturally "compatible" with the presence of general symmetry properties of the problem. These peculiarities may clearly make this approach of great interest in celestial mechanics, as shown by some classical examples.

The Lindstedt series were introduced in the XIX^th century in Astronomy to study perturbatively quasi-periodic motions in Celestial Mechanics. In Mathematical Physics, after getting the attention of Poincaré, who studied them widely by pursuing to all orders the analysis of Lindstedt and Newcomb, their use was somehow superseded by other methods usually referred to as KAM theory. Only recently, after Eliasson's work, they have been reconsidered as a tool to prove KAM-type results, in a spirit close to that of the Renormalization Group in quantum field theory. Following this new approach we discuss here the use of the Lindstedt series in the context of some model problems, like the standard map and natural generalizations, with particular attention to the properties of analyticity in the perturbative parameter.

The linearized averaged system is a classical integrable approximation for a system of n planets. It presents a strange kind of degeneracy that we call "Herman's resonance". The unexpected identity in the expansion of the perturbating function that causes this degeneracy is shown to be part of a family of similar identities. We give a simple Lemma putting together these identities. We also give a precise construction of the averaged system, that allows us to discuss its intrinsic character.

In this article we discuss our solutions for two of the questions asked in the history of the $n$-body problem: the construction of the global power series solution of the $n$-body problem and the bifurcation of the integral manifold of the three-body problem.

We study the degree growth of the iterates of the initial conditions for a class of third-order integrable mappings which result from the coupling of a discrete Painlevé equation to an homographic mapping. We show that the degree grows like $n^3$. In the special cases where the mapping satisfies the singularity confinement requirement we find a slower, quadratic growth. Finally we present a method for the construction of integrable $N$th-order mappings with degree growth $n^N$.

We study the differential equations of fourth order, in the polynomial class. We give the list of equations of this type, whose the indices of Fuchs are integers, then equations who can be with fixed critical points.