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2015 From topological to geometric equivalence in the classification of singularities at infinity for quadratic vector fields
J.C. Artés, J. Llibre, D. Schlomiuk, N. Vulpe
Rocky Mountain J. Math. 45(1): 29-113 (2015). DOI: 10.1216/RMJ-2015-45-1-29

Abstract

In the topological classification of phase portraits no distinctionsare made between a focus and a node and neither are they madebetween a strong and a weak focus or between foci of differentorders. These distinctions are however important in the production oflimit cycles close to the foci in perturbations of the systems. Thedistinction between the one direction node and the two directionsnode, which plays a role in understanding the behavior of solutioncurves around the singularities at infinity, is also missing in thetopological classification.

In this work we introduce the notion of \textit{geometricequivalence relation of singularities} which incorporates theseimportant purely algebraic features. The \textit{geometric}equivalence relation is finer than the \textit{topological} one andalso finer than the \textit{qualitative equivalence relation}introduced in \cite{J_L}. We also list all possibilities we have for finite and infinite singularities, taking into consideration thesefiner distinctions, and introduce notation for each one of them.%Our%long term goal is to use this finer and deeper equivalence relation to classify%the quadratic family according to their different \textit{geometric%configurations of singularities}, finite and infinite.

In this work we give the classification theorem andbifurcation diagram in the 12-dimensional parameter space,using the \textit{geometric equivalence relation}, of the class of quadratic systemsaccording to the configuration of singularities at infinity ofthe systems. Our classification theorem, stated in termsof invariant polynomials, is an algorithm for computingthe \textit{geometric configurations} of infinite singularities for anyfamily of quadratic systems, in any normal form.%The theorem we give also%contains a bifurcation diagram, done in the 12-parameter space, of%the \textit{geometric configurations} of singularities at infinity,%and this bifurcation set is algebraic in the parameter space. To%determine the bifurcation diagram of configurations of singularities%at infinity for any family of quadratic systems, given in any normal%form, becomes thus a simple task using computer algebra%calculations.

Citation

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J.C. Artés. J. Llibre. D. Schlomiuk. N. Vulpe. "From topological to geometric equivalence in the classification of singularities at infinity for quadratic vector fields." Rocky Mountain J. Math. 45 (1) 29 - 113, 2015. https://doi.org/10.1216/RMJ-2015-45-1-29

Information

Published: 2015
First available in Project Euclid: 7 April 2015

zbMATH: 1318.34045
MathSciNet: MR3334204
Digital Object Identifier: 10.1216/RMJ-2015-45-1-29

Subjects:
Primary: 34A34, 34C05, 58K45

Rights: Copyright © 2015 Rocky Mountain Mathematics Consortium

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Vol.45 • No. 1 • 2015
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