A powerful tool to investigate the stability of the orbits of natural and artificial bodies is represented by perturbation theory, which allows one to provide normal form estimates for nearly-integrable problems in Celestial Mechanics. In particular, we consider the orbital stability of point-mass satellites moving around the Earth. On the basis of the J 2 model, we investigate the stability of the semimajor axis. Using a secular Hamiltonian model including also lunisolar perturbations, the so-called geolunisolar model, we study the stability of the other orbital elements, namely the eccentricity and the inclination. We finally discuss the applicability of Nekhoroshev’s theorem on the exponential stability of the action variables. To this end, we investigate the non-degeneracy properties of the J 2 and geolunisolar models. We obtain that the J 2 model satisfies a “three-jet” non-degeneracy condition, while the geolunisolar model is quasi-convex non-degenerate.

Let $f$ be a holomorphic endomorphism of $\mathbb{P}^{2}$ of degree $d\geq 2$. We estimate the local directional dimensions of closed positive currents $S$ with respect to ergodic dilating measures $\unicode[STIX]{x1D708}$. We infer several applications. The first one is an upper bound for the lower pointwise dimension of the equilibrium measure, towards a Binder–DeMarco’s formula for this dimension. The second one shows that every current $S$ containing a measure of entropy $h_{\unicode[STIX]{x1D708}}>\log d$ has a directional dimension ${>}2$, which answers a question of de Thélin–Vigny in a directional way. The last one estimates the dimensions of the Green current of Dujardin’s semi-extremal endomorphisms.

We consider systems of quasi-periodic linear differential equations associated to a ‘resonant’ frequency vector ${\it\omega}$, namely, a vector whose coordinates are not linearly independent over $\mathbb{Z}$. We give sufficient conditions that ensure that a small analytic perturbation of a constant system is analytically conjugate to a ‘resonant cocycle’. We also apply our results to the non-resonant case: we obtain sufficient conditions for reducibility.

We study the secular dynamics of extrasolar planetary systems by extending the Lagrange-Laplace theory to high order and by including the relativistic effects. We investigate the long-term evolution of the planetary eccentricities via normal form and we find an excellent agreement with direct numerical integrations. Finally we set up a simple analytic criterion that allows to evaluate the impact of the relativistic effects in the long-time evolution.

A meadow is a commutative ring with an inverse operator satisfying 0−1 = 0. We determine the initial algebra of the meadows of characteristic 0 and prove a normal form theorem for it. As an immediate consequence we obtain the decidability of the closed term problem for meadows and the computability of their initial object.

In this paper, we studythe linear Schrödinger equation over the d-dimensional torus,with small values of the perturbing potential.We consider numerical approximations of the associated solutions obtainedby a symplectic splitting method (to discretize the time variable) in combination with the Fast Fourier Transform algorithm (to discretize the space variable).In this fully discrete setting, we prove that the regularity of the initialdatum is preserved over long times, i.e. times that are exponentially longwith the time discretization parameter. We here refer to Gevrey regularity, and our estimatesturn out to be uniform in the space discretization parameter.This paper extends [G. Dujardin and E. Faou, Numer. Math.97 (2004) 493–535], where a similar result has been obtained inthe semi-discrete situation, i.e. when the mere time variable is discretized and spaceis kept a continuous variable.

We give a local normal form for Dirac structures. As a consequence, we show that the dimensions of the pre-symplectic leaves of a Dirac manifold have the same parity. We also show that, given a point m of a Dirac manifold M, there is a well-defined transverse Poisson structure to the pre-symplectic leaf P through m. Finally, we describe the neighborhood of a pre-symplectic leaf in terms of geometric data. This description agrees with that given by Vorobjev for the Poisson case.

Normal forms for logic programs under stable/answer set semantics are introduced. We argue that these forms can simplify the study of program properties, mainly consistency. The first normal form, called the kernel of the program, is useful for studying existence and number of answer sets. A kernel program is composed of the atoms which are undefined in the Well-founded semantics, which are those that directly affect the existence of answer sets. The body of rules is composed of negative literals only. Thus, the kernel form tends to be significantly more compact than other formulations. Also, it is possible to check consistency of kernel programs in terms of colorings of the Extended Dependency Graph program representation which we previously developed. The second normal form is called 3-kernel. A 3-kernel program is composed of the atoms which are undefined in the Well-founded semantics. Rules in 3-kernel programs have at most two conditions, and each rule either belongs to a cycle, or defines a connection between cycles. 3-kernel programs may have positive conditions. The 3-kernel normal form is very useful for the static analysis of program consistency, i.e. the syntactic characterization of existence of answer sets. This result can be obtained thanks to a novel graph-like representation of programs, called Cycle Graph which presented in the companion article Costantini (2004b).

In this paper, we study the motion planning problem for generic sub-Riemannian metrics of co-rank one. We give explicit expressions for the metric complexity (in the sense of Jean [CITE]), in terms of the elementary invariants of the problem. We construct asymptotic optimal syntheses. It turns out that among the results we show, the most complicated case is the 3-dimensional. Besides the generic C ∞ case, we study some non-generic generalizations in the analytic case.

Soit ${{H}_{0}}\,=\,\frac{{{x}^{2}}+{{y}^{2}}}{2}$ un hamiltonien isochrone du plan ${{\mathbb{R}}^{2}}$. On met en évidence une classe d’hamiltoniens isochrones qui sont des perturbations polynomiales de ${{H}_{0}}$. On obtient alors une condition nécessaire d’isochronisme, et un critère de choix pour les hamiltoniens isochrones. On voit ce résultat comme étant une généralisation du caractère isochrone des perturbations hamiltoniennes homogènes considérées dans $\left[ \text{L} \right],\,\left[ \text{P} \right],\,\left[ \text{S} \right]$.