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A Versatile Algorithm for Computing Invariant Manifolds

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Model Reduction and Coarse-Graining Approaches for Multiscale Phenomena

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Summary

This paper deals with the numerical computation of invariant manifolds using a method of discretizing global manifolds. It provides a geometrically natural algorithm that converges regardless of the restricted dynamics. Common examples of such manifolds include limit sets, co-dimension 1 manifolds separating basins of attraction (separatrices), stable/unstable/center manifolds, nested hierarchies of attracting manifolds in dissipative systems and manifolds appearing in bifurcations. The approach is based on the general principle of normal hyperbolicity, where the graph transform leads to the numerical algorithms. This gives a highly multiple purpose method. The algorithm fits into a continuation context, where the graph transform computes the perturbed manifold. Similarly, the linear graph transform computes the perturbed hyperbolic splitting. To discretize the graph transform, a discrete tubular neighborhood and discrete sections of the associated vector bundle are constructed. To discretize the linear graph transform, a discrete (un)stable bundle is constructed. Convergence and contractivity of these discrete graph transforms are discussed, along with numerical issues. A specific numerical implementation is proposed. An application to the computation of the ‘slow-transient’ surface of an enzyme reaction is demonstrated.

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Author information

Authors and Affiliations

  1. Department of Mathematics and Computing Science, University of Groningen, P.O. Box 800, 9700 AV, Groningen, The Netherlands

    H. W. Broer & G. Vegter

  2. Department of Mathematics, University of Texas at Arlington, 411 S. Nedderman Dr., Arlington, TX, 76019, USA

    A. Hagen

Authors
  1. H. W. Broer
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  2. A. Hagen
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  3. G. Vegter
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Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Leicester, University Road, LE1 7RH, Leicester, UK

    Alexander N. Gorban

  2. Institute of Computational Modeling, Russian Academy of Sciences, Russia

    Alexander N. Gorban

  3. Department of Chemical Engineering, Princeton University, Engineering Quadrangle A-217, Princeton, NJ, 08544-5263, USA

    Ioannis G. Kevrekidis

  4. School of Chemical Engineering and Analytical Science, University of Manchester, PO Box 88, Sackville St., Manchester, M60 1QD, UK

    Constantinos Theodoropoulos

  5. Department of Chemical Engineering, Worcester Polytechnic Institute, Institute Road 100, Worcester, MA, 01609-2280, USA

    Nikolaos K. Kazantzis

  6. Institut für Polymere, ETH Zürich, Wolfgang Pauli-Straße 10, CH-8093, Zürich, Switzerland

    Hans Christian Öttinger

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© 2006 Springer-Verlag Berlin Heidelberg

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Broer, H.W., Hagen, A., Vegter, G. (2006). A Versatile Algorithm for Computing Invariant Manifolds. In: Gorban, A.N., Kevrekidis, I.G., Theodoropoulos, C., Kazantzis, N.K., Öttinger, H.C. (eds) Model Reduction and Coarse-Graining Approaches for Multiscale Phenomena. Springer, Berlin, Heidelberg . https://doi.org/10.1007/3-540-35888-9_2

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