One class of quadratic di®erential systems with two invariant lines
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BUJAC, Cristina; SCHLOMIUK, Dana; VULPE, Nicolae. One class of quadratic di®erential systems with two invariant lines. In: Mathematics and IT: Research and Education. 1-3 iulie 2021, Chişinău. Chișinău, Republica Moldova: 2021, pp. 17-18.
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Mathematics and IT: Research and Education 2021
Conferința "Mathematics and IT: Research and Education "
Chişinău, Moldova, 1-3 iulie 2021

One class of quadratic di®erential systems with two invariant lines

Pag. 17-18

Bujac Cristina1, Schlomiuk Dana2, Vulpe Nicolae1
1 Vladimir Andrunakievich Institute of Mathematics and Computer Science,
2 Université de Montréal
Disponibil în IBN: 29 iunie 2021


Consider the family of real planar polynomial differential systemsformulap(x; y); q(x; y) 2 R[x; y], n = maxfdeg(p); deg(q)g; n = 2 - quadratic systems. At the beginning of this century prof. Dana Schlomiuk initiated an extensive project for determining all possible configurations of invariant lines for the whole family of quadratic systems. Consider a real planar polynomial differential system (1). We call configuration of invariant straight lines of this system, the set of (complex) invariant straight lines (which may have real coefficients), including the line at infinity of the system, each endowed with its own multiplicity and together with all the real singular points of this system located on these invariant straight lines, each one endowed with its own multiplicity. We denote by QSLi the family of all non-degenerate quadratic differential systems possessing invariant straight lines (including the line at infinity) of total multiplicity i with i 2 f1; 2; 3; 4; 5; 6g. For any quadratic system on the affine plane the line at infinity is invariant. If this is the only invariant line and if it is of multiplicity 1 and in addition the system is non-degenerate, then the system belongs to QSL1. Up to now the families QSLi with i 2 f4; 5; 6g were classified topologically using their algebraic geometric structures and they were all proven to be Liouvillian integrable. Another family of systems possessing invariant lines is formed by the Lotka-Volterra systems, that have systems with two real invariant lines intersecting at a finite point. This family was classified by using the previous classifications of QSLi with i 2 f4; 5; 6g and clearly contains a part of QSL3. Another part of QSL3 is QSL2p 3 formed by quadratic systems that have two lines intersecting at infinity (parallel lines) and have the total multiplicity of their invariant lines equal to 3. In this work we classify the family QSL2p 3 of quadratic systems that have two parallel lines and possess invariant lines of total multiplicity at least 3. This family clearly contains QSL2p 3 . First we determine affine invariant criteria for a system (1) to belong to the class QSL2p 3 .