Osinga, Hinke Maria

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Computing invariant manifolds: variations on graph transform

University of Groningen

   University of GroningenComputing invariant manifoldsOsinga, Hinke MariaIMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:1996Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Osinga, H. M. (1996). Computing invariant manifolds: variations on graph transform. Groningen: s.n.CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 12-11-2019AssrRRcr I J JAeSTRACTInvariant manifolds are important in the globai study of dynamical systems. This thesisdescribes the computation of invariant manifolds using the constructive proofs of existencetheorems. Our methods are variations of the graph transform. We restrict to discretedynamical systems, given by maps and focus on normally hyperbolic manifolds and theirstable and unstable manifolds.A normally hyperboiic invariant manifold is compact, for example a circle or a torus.In the simplest case it is a point, called a hyperbolic fixed point. The tangent space at thenormally hyperbolic manifold can be split into stable and unstable subspaces, together withthe subspace tangent to the manifold, such that this splitting is invariant under the derivativemap. This implies that attraction and expansion have a linear nature. The manifold isnorrnally hyperbolic if the attraction and expansion in the stable and unstable subspacesare stronger than the possible attraction and expansion on the manifold, respectively. (Fora hyperbolic fixed point, it simply means that the eigenvalues of the derivative matrix atthe fixcd point are all off the unit circle.)The stable manifold of a normally hyperbolic invariant manifold is the subset of pointsin the state space that converge to an orbit on the normally hyperbolic manifold underforward iteration. If the normally hyperbolic manifold consists of only one (fixed) point, thepoints on the stable manifold necessarily converge to the fixed point. The unstable manifoldis the stable manifold of the inverse map.The Invariant Manifoid Theorem states that a normally hyperbolic invariant manifold,together with its local stable and unstable manifolds, persists under small perturbations.The Local Stabie Manifold Theorem for a hyperbolic fixed point is a special case of theabove theorem. The proofs of both theorems use the graph transform. The idea of thegraph transform is to consider the unknown manifold as the graph of a function from theunperturbed manifold to a space with dimension equal to the codimension of the unperturbedmanifold. The graph transform operates on the space of such continuous functions and canbe proven to be a contraction by using a proper metric. The Banach Fixed Point Theoremnow states that the graph transform has a unique fixed point that can be approximated withany given accuracy.In this thesis the graph transform is adapted such that the algorithm can be implementedon a computer. The implementation is not straightforward, since the domain of the graphtransform, the space of continuous functions, is infinite dimensional. We describe how todiscretize the graph transform and prove the correctness of our algorithm. Moreover, wefind estimates for the speed of convergence.We present two variants of our method: one for the computation of local stable andunstable manifolds of a hyperbolic fixed point, and the other for normally hyperbolic in-variant manifolds. The latter method can also be used for the computation of stable andunstable manifolds of a normally hyperbolic invariant manifold. We have implemented thesemethods giving an algorithm that works for the computation of manifolds with arbitraryfinite dimensions. This is illustrated with several examoles.

Computing invariant manifolds:variations on graph transform

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