Electron. J. Differential Equations, Vol. 2026 (2026), No. 28, pp. 1-47.
Codimension in planar polynomial differential systems
Joan C. Artes, Jaume Llibre, Dana Schlomiuk, Nicolae Vulpe
Abstract:
The topological classification of large families of polynomial differential systems
remains an extremely difficult problem. Even for the simplest nonlinear case,
the quadratic systems, this problem is still open. In recent years however,
significant progress has been achieved on a related challenge: the topological
classification modulo limit cycles of the family QS of quadratic differential
systems. This progress was enabled by introducing new tools, such as the concepts of
the global geometrical and topological configurations of singularities, and by first
classifying QS with respect to these notions.
In this work, we extend Sotomayor's notion of codimension, originally based on
topological equivalence, so as to allow for other equivalence relations including
the geometric one. We provide a rigorous, improved definition of codimension that
could be used efficiently for obtaining the classification of the phase portraits of
QS modulo limit cycles, and not just for small codimensions as it usually occurs in the
present literature. This new definition of codimension can be extended to arbitrary
polynomial differential systems, and other differential systems.
In this work, we apply the new concept of codimension by assigning a codimension to each
one of the 207 global topological configurations of singularities of systems in
QS. This tool is of great help for assigning codimensions even to phase portraits
modulo limit cycles. This concept is a potent tool in the topological classification
problem modulo limit cycles in QS.
Submitted November 9, 2025. Published April 16, 2026.
Math Subject Classifications: 58K45, 34C05, 34A34.
Key Words: Quadratic vector fields; infinite and finite singularities; affine invariant polynomials; Poincare compactification; configuration of singularities;
topological equivalence relation
DOI: 10.58997/ejde.2026.28
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Joan C. Artés
Departament de Matemátiques
Universitat Autónoma de Barcelona
08193 Bellaterra, Barcelona, Spain
email: joancarles.artes@uab.cat
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Jaume Llibre
Departament de Matemátiques
Universitat Autónoma de Barcelona
08193 Bellaterra, Barcelona, Spain
email: jaume.llibre@uab.cat
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Dana Schlomiuk
Département de Mathématiques et de Statistiques
Université de Montréal, Canada
email: dana.schlomiuk@umontreal.ca
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Nicolae Vulpe
State University of Moldova
Vladimir Andrunachievici Institute of Mathematics and Computer Science
5 Academiei str, Chişinäu, MD-2028, Moldova
email: nvulpe@gmail.com
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