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

Dana Schlomiuk ; Nicolae Vulpe

Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 1 (2008), pp. 27-83

Résumé

In this article we consider the class $\mathbf{QSL}_4$ of all real quadratic differential systems $\dfrac{dx}{dt}=p(x,y)$, $\dfrac{dy}{dt}=q(x,y)$ with $\mathrm{gcd}(p,q)=1$, 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 action.