First integrals and phase portraits of planar polynomial differential cubic systems with invariant straight lines of total multiplicity eight

• Main Authors: Cristina Bujac, Nicolae Vulpe
• Format: Article
• Language: English
• Published: University of Szeged 2017-12-01
• Series: Electronic Journal of Qualitative Theory of Differential Equations
• Subjects: quadratic vector fields; infinite and finite singularities; affine invariant polynomials; poincaré compactification; configuration of singularities; geometric equivalence relation
• Online Access: http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1&paramtipus_ertek=publication&param_ertek=5916

In the article [C. Bujac, J. Llibre, N. Vulpe, Qual. Theory Dyn. Syst. 15(2016), 327–348] for the family of cubic differential systems with the maximum number of invariant straight lines, i.e. 9 (considered with their multiplicities), the all first integrals and phase portraits were constructed. Here we proceed this investigation for systems with invariant straight lines of total multiplicity eight. For such systems the classification according to the configurations of invariant lines in terms of affine invariant polynomials were done in [C. Bujac, Bul. Acad. Științe Repub. Mold. Mat. 75(2014), 102–105], [C. Bujac, N. Vulpe, J. Math. Anal. Appl. 423(2015), 1025–1080], [C. Bujac, N. Vulpe, Qual. Theory Dyn. Syst. 14(2015), 109–137], [C. Bujac, N. Vulpe, Electron. J. Qual. Theory Differ. Equ}. 2015, No. 74, 1–38], [C. Bujac, N. Vulpe, Qual. Theory Dyn. Syst. 16(2017), 1–30] and all possible 51 configurations were constructed. For each one of the 51 such classes we perform the corresponding first integral as well as its phase portrait.