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Subcritical instability in shear flows: the shape of the basin boundary

Wednesday 10th September 2008 - 17:10 to 17:30
INI Seminar Room 1
Session Chair: 
B Eckhardt
The boundary of the basin of attraction of the stable, laminar point is investigated for several of the dynamical systems modeling subcritical instability. In the cases thus far considered, this boundary contains a linearly unstable structure (equilibrium point or periodic orbit). The stable manifold of this unstable structure coincides at least locally with the basin boundary. The unstable structure plays a decisive role in mediating the transition in that transition orbits cluster tightly around its (one-dimensional) unstable manifold, illustrating a scenario proposed by Waleffe. The picture that emerges augments the bypass scenario for transition and reconciles it with Waleffe's scenario. We consider a model proposed by Waleffe (W97) for which an unstable equilibrium point U lies on the basin boundary. We find numerically that all orbits starting near U decay to the origin, whereas 'half' of them should remain permanently bounded away from the origin. We offer an interpretation of this tendency toward decay based on the structure of the basin boundary.
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Presentation Material: 
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons