Dynamics at Infinity

Mardi 7 juin 2011 15:00-00:00 - Juliette Hell - Freie Universitaet Berlin

Résumé : We interpret phenomena like blow up or grow up as heteroclinic connections between finite invariant sets and infinity - or « transfinite » heteroclinics. We access infinity by applying the Poincaré compactification projecting the phase space on the Poincaré hemisphere. Infinity is thereby projected on its equator, an invariant sphere where infinity unfolds its « celestial » dynamics. The Conley index is a useful tool to analyze the connections structure and wassuccessfully applied on bounded global attractors. In the context of unbounded dynamics, Conley index has to be adapted carefully to be able to detect transfinite heteroclinics : even simple systems like planar quadratic ODEs show behaviors near the sphere at infinity that prevent the isolation required by Conley index theory. In order of overcoming this difficulty, we define the concept of « dynamical complement » of an invariant set. The isolated invariance of the dynamical complement is our requirement for establishing the existence of transfinite heteroclinics via non-classical Conley index techniques.

Lieu : 425 - 121-123

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