Hopf's last hope:
Spatiotemporal chaos in terms of unstable recurrent patterns
Center for Chaos and Turbulence Studies
Niels Bohr Institute
DK-2100 Copenhagen O
|Hovedfagområde Fysik||Tidsrum Forår 1997|
|Fagområde(r) Teoretisk fysik, dynamiske systemer||Sted NBI Blegdamsvej|
|Hold størelse 1-2|
|Forudsætninger Kaos kursus, differentielle ligninger, programmering|
Bachelor project description
In recent years unstable periodic orbits have been shown to be an effective tool in the description of deterministic dynamical systems of low intrinsic dimension, in diagnosing deterministic chaos in noisy biological systems, and many other applications. It is an open question whether the theory has anything to say about extended systems (hydrodynamics, field theory).
A spatiotemporally periodic solution of the Kuramoto-Sivashinsky equation.
The goal of the project is to show that the periodic orbit theory can be used to describe spatially extended systems by applying it to the Kuramoto-Sivashinsky equation. Hopf's proposal was to think of turbulence as a sequence of near recurrences of a repertoire of unstable spatiotemporal patterns; as we watch a given ``turbulent'' system evolve, every so often we catch a glimpse of a familiar pattern. For any finite spatial resolution, the system follows approximately for a finite time a pattern belonging to a finite alphabet of admissible patterns, and the long term dynamics can be thought of as a walk through the space of such patterns, just as chaotic dynamics with a low dimensional attractor can be thought of as a succession of nearly periodic (but unstable) motions.
The project would be a continuation of the Nonlinearity paper available at http://www.nbi.dk/~predrag/papers/ks.ps.gz . Much of the preparatory program development is already accomplished.