University of Groningen. Bifurcations in Hamiltonian systems Lunter, Gerard Anton

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1 University of Groningen Bifurcations in Hamiltonian systems Lunter, Gerard Anton IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from it. Please check the document version below. Document Version Publisher's PDF, also known as Version of record Publication date: 1999 Link to publication in University of Groningen/UMCG research database Citation for published version (APA): Lunter, G. A. (1999). Bifurcations in Hamiltonian systems: computing singularities by Gröbner bases Groningen: s.n. Copyright Other 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 the author(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons). Take-down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Downloaded from the University of Groningen/UMCG research database (Pure): For technical reasons the number of authors shown on this cover page is limited to 10 maximum. Download date:

2 Bifurcations in Hamiltonian systems Computing singularities by Grobner bases

3 Omslag: Cartoons: Druk: Babs Wijnstra Jeroen Gildemacher Universal Press, Veenendaal

4 Rijksuniversiteit Groningen Bifurcations in Hamiltonian systems Computing singularities by Grobner bases Proefschrift ter verkrijging van het doctoraat in de Wiskunde en Natuurwetenschappen aan de Rijksuniversiteit Groningen op gezag van de Rector Magnicus, dr. D.F.J. Bosscher, in het openbaar te verdedigen op vrijdag 10 december 1999 om uur door Gerard Anton Lunter geboren op 17 maart 1971 te Sneek.

5 Promotor: Prof.dr. H.W. Broer Co-promotor: Dr. G. Vegter ISBN

6 Contents 1 Introduction and overview Background Focus on qualitative results From qualitative to quantitative results Goal of research Chronology and details Results and discussion Reduction methods The planar reduction method The energy-momentum map method Standard bases Discussion - Reduction methods Discussion - Standard bases The plan I Applications 25 2 Method I: Planar reduction Introduction BCKV-restricted morphisms Details of the planar reduction method Overview Some notation Birkho normalization Reduction to planar 1 degree-of-freedom system Reduction to the central singularity Inducing the system from a versal deformation BCKV normal form Spring-pendulum in 1:2-resonance The system

7 Contents vi Reduction Dynamics and bifurcations Method II: The energy-momentum map Introduction Description of the method Birkho normalization Circle-equivariant vector elds The energy-momentum map Removing the -dependence Application to several resonances The 1:2 resonance The 1:3 resonance The 1:4 resonance Spring-pendulum in 1:2 resonance Bifurcation analysis of the 1:2-resonant normal form Pictures Inducing the system from the model II Theory 79 4 Birkho normalization Introduction Introduction to Hamiltonian mechanics Birkho normal form theorem Semisimple quadratic part, and resonance Second normalization Algorithms for the Birkho normal form Another formulation of Birkho's result Deprit's algorithm The exponential-map algorithm Singularity theory Overview Introduction: The nite dimensional case Deformations Functions and right-transformations Equivalence and versal deformations Applications BCKV normal form Maps and left-right transformations The tangent space

8 vii Contents 6 Grobner bases and Standard bases Introduction Algorithms and real numbers Motivation: Grobner bases Term orders for Grobner bases Basic question Rephrasing the basic question A criterion for Grobner bases Standard bases Overview Denitions Setup Normal form property The standard map theorem Normal form algorithm Reduced normal forms Instances of standard bases Grobner bases Standard bases for submodules Standard bases for subalgebras Standard bases for modules over subalgebras Left-Right tangent space The ring of formal power series Three approaches Existence of a normal form for formal power series Truncated formal power series Computing normalizing transformations Introduction Deformations of functions Finding a universal deformation The algorithm of Kas and Schlessinger Solving the innitesimal stability equation Application: The hyperbolic umbilic Deformations of maps Adaptation of Kas and Schlessinger's algorithm Solving the innitesimal stability equation Example of a LR-tangent space calculation A Appendix 167 A.1 Classication of term orders A.2 Proof of Proposition

9 Contents viii Summary 173 Samenvatting 177 Dankwoord 181 References 183

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