Integrals and phase portraits of planar quadratic differential systems with invariant lines of total multiplicity four
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SCHLOMIUK, Dana; VULPE, Nicolae. Integrals and phase portraits of planar quadratic differential systems with invariant lines of total multiplicity four. In: Buletinul Academiei de Ştiinţe a Moldovei. Matematica. 2008, nr. 1(56), pp. 27-83. ISSN 1024-7696.
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Buletinul Academiei de Ştiinţe a Moldovei. Matematica
Numărul 1(56) / 2008 / ISSN 1024-7696

Integrals and phase portraits of planar quadratic differential systems with invariant lines of total multiplicity four

Pag. 27-83

Schlomiuk Dana, Vulpe Nicolae
 
Institute of Mathematics and Computer Science ASM
 
Disponibil în IBN: 6 decembrie 2013


Rezumat

In this article we consider the class QSL4 of all real quadratic differential systems dx dt = p (x, y), dy dt = q(x, y) with gcd(p, q) = 1, having invariant lines of total multiplicity four and a finite set of singularities at infinity. We first prove that all the systems in this class are integrable having integrating factors which are Darboux functions and we determine their first integrals. We also construct all the phase portraits for the systems belonging to this class. The group of affine transformations and homotheties on the time axis acts on this class. OurMain Theorem gives necessary and sufficient conditions, stated in terms of the twelve coefficients of the systems, for the realization of each one of the total of 69 topologically distinct phase portraits found in this class. We prove that these conditions are invariant under the group action.

Cuvinte-cheie
quadratic differential system, affine invariant polynomial, configuration of invariant lines,

Poincar´e compactification, algebraic invariant curve

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