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Chaos. 2017 Jun;27(6):063102. doi: 10.1063/1.4983468.

On the sighting of unicorns: A variational approach to computing invariant sets in dynamical systems.

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Center for Mathematics, Technical University of Munich, 85747 Garching, Germany.
Department of Chemical and Biological Engineering and PACM, Princeton University, Princeton, New Jersey 08544, USA and Institute of Advanced Studies, TU Munich, and ZIB, FU Berlin, Berlin, Germany.


We propose to compute approximations to invariant sets in dynamical systems by minimizing an appropriate distance between a suitably selected finite set of points and its image under the dynamics. We demonstrate, through computational experiments, that this approach can successfully converge to approximations of (maximal) invariant sets of arbitrary topology, dimension, and stability, such as, e.g., saddle type invariant sets with complicated dynamics. We further propose to extend this approach by adding a Lennard-Jones type potential term to the objective function, which yields more evenly distributed approximating finite point sets, and illustrate the procedure through corresponding numerical experiments.


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