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

Dana Schlomiuk, Nicolae Vulpe

Buletinul Academiei de Stiente a Republicii Moldova. Matematica Number 1(56), 2008, Pages 1–57 ISSN 1024–7696

Abstract

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. Our Main 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.

Phase portraits (68)

• \(QS1_1^{(0)}\).
• \(QS2_3\).
• \(QS5_3^{(0)}\).
• \(QS6_2\).
• \(QS5_{16}^{(0)}\).
• \(QS10_{5}^{(1)}\). \(QS10_{21}^{(1)}\).
• \(QS11_{20}^{(1)}\).
• \(QS11_{25}^{(2)}\).
• \(QS12_{1}^{(0)}\).
• \(QS13_{1}^{(0)}\).
• \(QS13_{1}^{(1)}\).
• \(QS14_{2}^{(0)}\).
• \(QS14_{1}^{(2)}\).
• \(QS16_{1}^{(0)}\).
• \(QS17_1\).
• \(QS18_1\).
• \(QS23_{1}^{(0)}\).
• \(QS24_1\).
• \(QS32_{5}^{(3)}\).
• \(QS33_1\).
• \(QS38_{33}^{(1)}\).
• \(QS38_{10}^{(2)}\). \(QS38_{32}^{(2)}\).
• \(QS39_{46}^{(3)}\).
• \(QS47_{1}^{(2)}\).
• \(AAD_{1}\). Provisional name for a phase portrait of class (51) and codimension 3.
• \(AAD_{2}\). Provisional name for a phase portrait of class (51) and codimension 3.
• \(QS52_{1}^{(5)}\).
• \(QS60_2\).
• \(QS64_{1}^{(3)}\).
• \(QS67_{1}^{(4)}\).
• \(QS68_{1}^{(4)}\).
• \(QS69_{1}^{(4)}\).
• \(QS70_{1}^{(5)}\).
• \(QS71_{1}^{(4)}\).
• \(QS72_{1}^{(5)}\).
• \(QS74_{5}^{(1)}\).
• \(QS76_{10}^{(1)}\).
• \(QS84_{1}^{(0)}\).
• \(QS88_{1}^{(2)}\).
• \(QS90_1\).
• \(QS95_{1}^{(3)}\).
• \(QS100_{16}^{(4)}\).
• \(QS123_{1}^{(4)}\).
• \(QS124_{1}^{(3)}\).
• \(QS125_{1}^{(3)}\).
• \(QS126_{1}^{(3)}\).
• \(QS129_{1}^{(3)}\).
• \(QS133_2\).
• \(CCD_{1}\). Provisional name for a phase portrait of class (134) and codimension 3.
• \(QS134_{4}^{(2)}\).
• \(QS137_{1}^{(3)}\).
• \(QS138_{1}^{(3)}\).
• \(QS138_{1}^{(4)}\).
• \(QS144_{1}^{(4)}\).
• \(QS145_{1}^{(5)}\).
• \(QS149_{1}^{(4)}\).
• \(QS158_{1}^{(5)}\).
• \(QS160_{1}^{(5)}\).
• \(QS161_{1}^{(4)}\).
• \(QS162_{1}^{(5)}\).
• \(QS163_{1}^{(5)}\).
• \(QS164_{1}^{(5)}\).
• \(QS165_{1}^{(5)}\).
• \(QS167 _{1}^{(6)}\).
• \(QS168_{1}^{(6)}\).
• \(QS169_{1}^{(6)}\).

Comments

• Type of paper: Bifurcation diagram with global algebraic invariants.

• Professors Schlomiuk and Vulpe have several papers classifying phase portraits according to the number and multiplicity of their invariant straight lines. Some of them classify the configurations of invariant straight lines, and others their phase portraits.

• There are some few repetitions of phase portraits in this paper but this is due that the same topological phase portrait may appear with a different geometric set of invariant straight lines.

• Even thought the authors claim in the abstract that they have found 69 topologically different portraits, the real number is 68.