Planar quadratic vector fields with invariant lines of total multiplicity at least five

By Dana Schlomiuk, Nicolae Vulpe. 2004

49 citations recorded by the source corpus.

In this article we consider the action of the real affine group and time rescaling on real planar quadratic differential systems. We construct a system of representatives of the orbits of systems with at least five invariant lines, including the line at infinity and including multiplicities. For each orbit we exhibit its configuration. We characterize in terms of algebraic invariants and comitants and also geometrically, using divisors of the complex projective plane, the class of real quadratic differential systems with at least five invariant lines. These conditions are such that no matter how a system may be presented, one can verify by using them whether the system has or does not have at least five invariant lines and to check to which orbit (or family of orbits) it belongs.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/29d113c189f24c5b0d5f769056bda410249a8c3a

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In Computational Theory and Mathematics

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In Statistical and Nonlinear Physics

Citations in the corpus

Global classification of the planar Lotka–Volterra differential systems according to their configurations of invariant straight lines (2010, methodology): “Example 2.3. Let QSL i be the family of all nondegenerate quadratic differential systems possessing invariant straight lines (including the line at infinity) of total multiplicity i with i ∈{ 2, 3, 4, 5, 6}. The families QSL i with i ∈{ 4, 5, 6} were classified topologically using their algebro-geom”
Integrals and Phase Portraits of Planar Quadratic Differential Systems With Invariant Lines of at Least Five Total Multiplicity (2008, background): “4) { ẋ = 2xy, ẏ = 1− x2 + y2 [18] =⇒ 4.”
On the Darboux Integrability of Polynomial Differential Systems (2012, background): “In [39] Schlomiuk and Vulpe gave a definition for the algebraic multiplicity of the line at infinity for a planar vector field using a limit inside the definition.”