Speaker: Stanislaus Maier-Paape, RWTH, Aachen, Germany
Title: Computer assisted proofs of heteroclinic connections

Abstract: As the title suggests, we propose a method which allows to prove with computer aid connecting orbits of dynamical systems generated by an ODE. We want to prove the existence of connecting orbits between two hyperbolic equilibria $x^R$ (repeller) and $x^A$ (attractor) of this ODE, whose Morse indices are $k$ and $k-1$, respectively. The main idea is to analytically investigate neighborhoods of the equilibria (to obtain for instance good estimates on the location of their invariant manifolds in form of box enclosures), and combine this with a fast rigorous transport algorithm for the ODE (like e.g. the CAPD software, http://capd.wsb-nlu.edu.pl). We transport small neighborhoods of a part of the unstable manifold of $x^R$ by means of an enclosing algorithm from CAPD into a neighborhood of $x^A$. Comparing this transported enclosure with a box enclosure of the stable manifold of $x^A$, we can state conditions whose verification with the computer results in a proof that both manifolds intersect, giving the desired connecting orbit. Our method applies for the cases $k=1$, $k=2$ and by time reversal for the cases $k = n$ and $k = n-1$ ($n=$ space dimension). Although we clearly yet are not even close, we think this method does have the potential to be applicable for all $k \in \{1, \ldots, n\}$, if not for evolution equations in infinite dimensions.

Time: Friday, Sept. 19, 2008, 1:30-2:30 p.m.

Place: Science and Tech I, Room 242

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