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An improved lower bound on the number of limit cycles bifurcating from a quintic Hamiltonian planar vector field under quintic perturbation
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
Matematisk Institutt, Universitetet i Bergen.
2010 (English)In: International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, ISSN 0218-1274, Vol. 20, no 1, p. 63-70Article in journal (Refereed) Published
##### Abstract [en]

The limit cycle bifurcations of a ${\mathbb{Z}}_2$ equivariant quintic planar Hamiltonian vector field under $\mathbb{Z}_2$ equivariant quintic perturbation is studied. We prove that the given system can have at least 27 limit cycles. This is an improved lower bound on the possible number of limit cycles that can bifurcate from a quintic planar Hamiltonian system under quintic perturbation.

##### Place, publisher, year, edition, pages
World Scientific Publishing , 2010. Vol. 20, no 1, p. 63-70
##### Keyword [en]
Limit cycles, bifurcation theory, planar Hamiltonian systems, interval analysis
Mathematics
##### Identifiers
ISI: 000275981100004OAI: oai:DiVA.org:uu-103153DiVA, id: diva2:217594
Available from: 2009-05-14 Created: 2009-05-14 Last updated: 2017-12-13Bibliographically 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. p. vi+24
##### 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.
Mathematics
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)
##### Supervisors
Available from: 2009-09-17 Created: 2009-08-14 Last updated: 2009-09-21

#### Open Access in DiVA

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Johnson, Tomas

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##### By organisation
Analysis and Applied Mathematics
##### In the same journal
International Journal of Bifurcation and Chaos in Applied Sciences and Engineering
Mathematics

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Cite
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