Abstract
In Artés et al. (Geometric configurations of singularities of planar polynomial differential systems. A global classification in the quadratic case. Birkhäuser, Basel, 2019) the authors proved that there are 1765 different global geometrical configurations of singularities of quadratic differential systems in the plane. There are other 8 configurations conjectured impossible, all of them related with a single configuration of finite singularities. This classification is completely algebraic and done in terms of invariant polynomials and it is finer than the classification of quadratic systems according to the topological classification of the global configurations of singularities, the goal of this article. The long term project is the classification of phase portraits of all quadratic systems under topological equivalence. A first step in this direction is to obtain the classification of quadratic systems under topological equivalence of local phase portraits around singularities. In this paper we extract the local topological information around all singularities from the 1765 geometric equivalence classes. We prove that there are exactly 208 topologically distinct global topological configurations of singularities for the whole quadratic class. The 8 global geometrical configurations conjectured impossible do not affect this number of 208. From here the next goal would be to obtain a bound for the number of possible different phase portraits, modulo limit cycles.
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Artés, J.C., Llibre, J., Schlomiuk, D., Vulpe, N.: Geometric configurations of singularities of planar polynomial differential systems [A global classification in the quadratic case]. To be published by Birkhäuser (2019)
Artés, J.C., Llibre, J., Vulpe, N.I.: Singular points of quadratic systems: a complete classification in the coefficient space \(\mathbb{R}^{12}\). Int J. Bifurc. Chaos 18, 313–362 (2008)
Baltag, V.A., Vulpe, N.I.: Affine-invariant conditions for determining the number and multiplicity of singular points of quadratic differential systems. Izv. Akad. Nauk Respub. Moldova Mat. 1, 39–48 (1993)
Chicone, C., Jinghuang, Tian: On general properties of quadratic systems. Am. Math. Mon. 89(3), 167–178 (1982)
Coppel, W.A.: A survey of quadratic systems J. Differential Equations 21966293–304
Dumortier, F., Llibre, J., Artés, J.C.: Qualitative theory of planar differential systems. Universitext, Springer–Verlag, New York–Berlin, ISBN: 3-540-32893-9 (2008)
Grace, J.H., Young, A.: The Algebra of Invariants. Stechert, New York (1941)
Olver, P.J.: Classical Invariant Theory. London Mathematical Society Student Texts, vol. 44. Cambridge University Press, Cambridge (1999)
Reyn, J.: Phase portraits of planar quadratic systems. Mathematics and Its Applications, p. xvi+334. Springer, New York (2007)
Schlomiuk, D., Vulpe, N.I.: The full study of planar quadratic differential systems possessing a line of singularities at infinity. J. Dynam. Diff. Equ. 20, 737–775 (2008)
Vulpe, N.: Characterization of the finite weak singularities of quadratic systems via invariant theory. Nonlinear Anal. 74(4), 6553–6582 (2011)
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Joan C. Artés and Jaume Llibre are partially supported by the Ministerio de Economia, Industria y Competitividad, Agencia Estatal de Investigación grant MTM2016-77278-P (FEDER), the Agència de Gestió d’Ajuts Universitaris i de Recerca grant 2017 SGR 1617, and the European project Dynamics-H2020-MSCA-RISE-2017-777911. Dana Schlomiuk is supported by NSERC. The fourth author is supported by the grant 12.839.08.05F from SCSTD of ASM and partially by NSERC.
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Artés, J.C., Llibre, J., Schlomiuk, D. et al. Global Topological Configurations of Singularities for the Whole Family of Quadratic Differential Systems. Qual. Theory Dyn. Syst. 19, 51 (2020). https://doi.org/10.1007/s12346-020-00372-7
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DOI: https://doi.org/10.1007/s12346-020-00372-7
Keywords
- Quadratic vector fields
- Infinite and finite singularities
- Affine invariant polynomials
- Poincaré compactification
- Configuration of singularities
- Topological equivalence relation