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Résultats exploitables
The aim of the proposed research is to understand the structure of the phase space of generic Hamiltonian systems (mainly analytic) and of nearby systems. Two types of Hamiltonian systems have been investigated intensively in recent decades: integrable ones, the motion in which is completely regular, and Anosov systems (e.g. geodesic flows on manifolds with negative curvature) where the motion is completely chaotic. However, most systems encountered in applications are neither integrable nor Anosov and they display both kinds of motion in their phase space. It follows from this that the study of this general situation is of great practical and theoretical importance. The aim is to give a description of the co-existence of regular and chaotic motions in a Hamiltonian system with two or more degrees of freedom (in the case of one degree of freedom only a regular component exists) by use of both theoretical and numerical approaches, and to describe typical bifurcations which may occur in such systems or nearby ones. Among the many topics covered by the general objective, those selected have importance and correspond to the expertise of the teams. Selected items for the actual project are: integrability and conservation laws, stability and diffusion in nearby integrable Hamiltonian systems, slow-fast systems, averaging methods, variational methods for homoclinic orbits, splitting of separatrices, positive entropy problem, fractal structures, breakdown of invariant tori and renormalisation, semi-analytic, numerical and graphical tools. Another objective of the project is to strengthen relationships between the different teams. Researchers of each team have already visited several others and, in some cases, they have produced joint papers. It is important to continue and to formalize these relations in a systematic way. Joint work will produce more and better results than the isolated efforts of each team.
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