On the connection between the Nekhoroshev theorem and Arnold Diffusion

The analytical techniques of the Nekhoroshev theorem are used to provide estimates on the coefficient of Arnold diffusion along a particular resonance in the Hamiltonian model of Froeschl\'{e} et al. (2000). A resonant normal form is constructed by a computer program and the size of its remainder $||R_{opt}||$ at the optimal order of normalization is calculated as a function of the small parameter $\epsilon$. We find that the diffusion coefficient scales as $D\propto||R_{opt}||^3$, while the size of the optimal remainder scales as $||R_{opt}|| \propto\exp(1/\epsilon^{0.21})$ in the range $10^{-4}\leq\epsilon \leq 10^{-2}$. A comparison is made with the numerical results of Lega et al. (2003) in the same model.Comment: Accepted in Celestial Mechanics and Dynamical Astronom

The connection between computability of a nonlinear problem and its linearization: the Hartman-Grobman theorem revisited

As one of the seven open problems in the addendum to their 1989 book Computability in Analysis and Physics Pour-El and Richards (1989)[17], Pour-El and Richards asked, "What is the connection between the computability of the original nonlinear operator and the linear operator which results from it?" Yet at present, systematic studies of the issues raised by this question seem to be missing from the literature. In this paper, we study one problem in this direction: the Hartman-Grobman linearization theorem for ordinary differential equations (ODEs). We prove, roughly speaking, that near a hyperbolic equilibrium point x(0) of a nonlinear ODE (x) over dot = f(x), there is a computable homeomorphism H such that H circle phi = L circle H, where phi is the solution to the ODE and L is the solution to its linearization (x) over dot = Df (x(0)) x. (C) 2012 Elsevier B.V. All rights reserved.Fundacao para a Ciencia e a Tecnologia; EU FEDER POCTI/POCI via SQIG - Instituto de Telecomunicacoes through the FCT [PEst-OE/EEI/LA0008/2011

Magnetic Domain Studies of Cobalt Nanostructures

International audienceThe pulsed laser deposition technique associated with a low energy cluster beam is used to deposit cobalt thin films with a thickness 100-200 nm and cobalt dots of a diameter 100-200 nm on silicon substrates. The deposited thin films of Co are composed of clusters of a size 10- 50 nm, with very few large grains as revealed by atomic force microscopy. The observations performed by magnetic force microscopy on as-grown thin films reveal randomly distributed out-of-plane magnetic domain structures. These magnetic domains are aligned linearly by applying an external magnetic field either perpendicular or parallel to the substrate during the deposition. In addition, the effect of film thickness and roughness on multidomains is reported. The increase of roughness resulted in the decrease of magnetic domain width from 200 to 100 nm. This decrease is accompanied by the appearance of instability in the stripe domain pattern. Well separated cobalt dots of diameter in the range of 100-200 nm are also deposited on silicon substrates, which show arc-like multidomains. The domains seem to be oriented along the long axis of the dots. The domain structure of Co nanodots is similar to that of Co thin films indicating strong magnetic coupling of clusters

Cobalt cluster-assembled thin films deposited by low energy cluster beam deposition : structural and magnetic investigations of deposited layers

International audienceCobalt cluster-assembled thin films were deposited on amorphous carbon coated copper grids and on silicon substrates at room temperature by Low Energy Cluster Beam Deposition (LECBD). Characterisations using High-Resolution Transmission Electronic Microscopy (HRTEM) and Atomic Force Microscopy (AFM) reveal randomly stacked agglomerates of 9 to 11 nm diameter, which are themselves composed of small 3.6 nm diameter fcc cobalt clusters. The magnetic response of these films was analysed using Vibrating Sample Magnetometer (VSM) measurements, performed between 15 K to 300 K, and Magnetic Force Microscopy (MFM) investigations, realized at room temperature. The films are ferromagnetic up to room temperature and above, which implies that the clusters are exchange coupled. The approach to saturation is analysed within the Random Anisotropy Model (RAM). The values of the exchange coefficient A and the anisotropy constant K then derived are discussed. The temperature dependence of the coercivity below 100 K is discussed in terms of thermal activation effects. All results indicate that the fundamental entity governing the magnetic behaviours is constituted by the 9-11 nm diameter agglomerates rather than by the clusters themselves