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Automated computation of robust normal forms of planar analytic vector fields
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
Matematisk Institutt, Universitetet i Bergen.
2009 (English)In: Discrete and continuous dynamical systems. Series B, ISSN 1531-3492, Vol. 12, no 4, 769-782 p.Article in journal (Refereed) Published
Abstract [en]

We construct an auto-validated algorithm that calculates a close to identity change of variables which brings a general saddle point into a normal form. The transformation is robust in the underlying vector field, and is analytic on a computable neighbourhood of the saddle point. The normal form is suitable for computations aimed at enclosing the flow close to the saddle, and the time it takes a trajectory to pass it. Several examples illustrate the usefulness of this method.

Place, publisher, year, edition, pages
2009. Vol. 12, no 4, 769-782 p.
Keyword [en]
Normal Forms, Hyperbolic fixed points, Numerical integration
National Category
Mathematics
Identifiers
URN: urn:nbn:se:uu:diva-103152DOI: 10.3934/dcdsb.2009.12.769ISI: 000271091400005OAI: oai:DiVA.org:uu-103152DiVA: diva2:217593
Available from: 2009-05-14 Created: 2009-05-14 Last updated: 2010-07-01Bibliographically approved
In thesis
1. Computer-aided Computation of Abelian integrals and Robust Normal Forms
Open this publication in new window or tab >>Computer-aided Computation of Abelian integrals and Robust Normal Forms
2009 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This PhD thesis consists of a summary and seven papers, where various applications of auto-validated computations are studied.

In the first paper we describe a rigorous method to determine unknown parameters in a system of ordinary differential equations from measured data with known bounds on the noise of the measurements.

Papers II, III, IV, and V are concerned with Abelian integrals. In Paper II, we construct an auto-validated algorithm to compute Abelian integrals. In Paper III we investigate, via an example, how one can use this algorithm to determine the possible configurations of limit cycles that can bifurcate from a given Hamiltonian vector field. In Paper IV we construct an example of a perturbation of degree five of a Hamiltonian vector field of degree five, with 27 limit cycles, and in Paper V we construct an example of a perturbation of degree seven of a Hamiltonian vector field of degree seven, with 53 limit cycles. These are new lower bounds for the maximum number of limit cycles that can bifurcate from a Hamiltonian vector field for those degrees.

In Papers VI, and VII, we study a certain kind of normal form for real hyperbolic saddles, which is numerically robust. In Paper VI we describe an algorithm how to automatically compute these normal forms in the planar case. In Paper VII we use the properties of the normal form to compute local invariant manifolds in a neighbourhood of the saddle.

Place, publisher, year, edition, pages
Uppsala: Acta Universitatis Upsaliensis, 2009. vi+24 p.
Series
Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 66
Keyword
Ordinary differential equations, parameter estimation, planar Hamiltonian systems, bifurcation theory, Abelian integrals, limit cycles, normal forms, hyperbolic fixed points, numerical integration, invariant manifolds, interval analysis. 2000 Mathematics Subject Classification. 34A60, 34C07, 34C20, 37D10, 37G15, 37M20, 37M99, 65G20, 65L09, 65L70.
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:uu:diva-107519 (URN)978-91-506-2093-1 (ISBN)
Public defence
2009-10-09, Polhemsalen, Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, 09:00 (English)
Opponent
Supervisors
Available from: 2009-09-17 Created: 2009-08-14 Last updated: 2009-09-21

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Johnson, Tomas

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