International Solvay Institutes
8-9 October 2004
Université Libre de Bruxelles, Campus
Plaine, Boulevard du Triomphe, B-1050 Brussels
Abstracts
Leonid Shilnikov (Nizhnii Novgorod State Univ., Russia):
Poincaré Homoclinic Orbits: The State of the Art
After a brief overview of the history of dynamical systems with homoclinic
orbits, the main part of the talk will be focused on results related to
homoclinic tangencies. The importance of this problem is due to the fact
that, in this framework, one can answer many fundamental questions both
in the theory of general smooth dynamical systems and of Hamiltonian ones.
In this respect, we enumerate several topics that will be discussed in this
talk:
1. Classification of the principal types of homoclinic tangencies.
2. Description of the principal continuous invariants - Omega-modulae.
3. Existence of Newhouse
regions, that is, regions of everywhere dense structural instability.
4. Examples of systems
with wild attractors: spiral attractors and attractors of Lorenz-type systems
under an external forcing.
5. Coexistence of countable sets of hyperbolic periodic orbits of different
topological types .
6. Everywhere density in
Newhouse regions of 2D diffeomorphisms, including symplectic ones, having
countable sets of periodic orbits of arbitrary order of degeneracy in the
space of diffeomorphisms endowed with any finite C_r- topology.
7. Existence of a countable
set of periodic orbits of the general elliptic type for 2D symplectic diffeomorphisms
with homoclinic tangencies.
8. Existence of a countable set of periodic orbits being KAM stable for
4D symplectic diffeomorphisms with a tangent homoclinic orbit to a saddle-focus
fixed point.
Pierre Gaspard (ULB, Belgium):
From Dynamical Systems Theory to Nonequilibrium
Thermodynamics
An overview is given of recent work on the relationships between dynamical
systems theory and nonequilibrium thermodynamics. In these recent works,
the hydrodynamic modes of diffusion are constructed in translationally invariant
chaotic Hamiltonian systems. These modes describe the relaxation toward
the thermodynamic equilibrium and are given as the eigenstates associated
with Pollicott-Ruelle resonances. The hydrodynamic modes have a fractal
structure characterized by a Hausdorff dimension which is given in terms
of the diffusion coefficient and the Lyapunov exponent characterizing the
dynamical instability. The singular character of the hydrodynamic modes
turns out to be an essential feature in order to understand the production
of entropy in Hamiltonian systems.
References:
P. Gaspard, I. Claus, T. Gilbert, and J. R. Dorfman, The Fractality of
the Hydrodynamic Modes of Diffusion, Physical Review Letters 86 (2001) 1506-1509.
J. R. Dorfman, P. Gaspard, and T. Gilbert, Entropy production of diffusion
in spatially periodic deterministic systems, Physical Review E 66 (2002)
026110 (9 pages);
preprint nlin.CD/0203046.