The bifurcation diagram of the configurations of invariant lines of total multiplicity exactly three in quadratic vector fields

Cristina Bujac ; Dana Schlomiuk ; Nicolae Vulpe

Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 1 (2023), pp. 42-77.

Résumé

We denote by $QS{L}_3$ the family of quadratic differential systems possessing invariant straight lines, finite and infinite, of total multiplicity exactly three. In a sequence of papers the complete study of quadratic systems with invariant lines of total multiplicity at least four was achieved. In addition three more families of quadratic systems possessing invariant lines of total multiplicity at least three were also studied, among them the Lotka-Volterra family. However there were still systems in $QS{L}_3$ missing from all these studies. The goals of this article are: to complete the study of the geometric configurations of invariant lines of $QS{L}_3$ by studying all the remaining cases and to give the full classification of this family modulo their configurations of invariant lines together with their bifurcation diagram. The family $QS{L}_3$ has a total of 81 distinct configurations of invariant lines. This classification is done in affine invariant terms and we also present the bifurcation diagram of these configurations in the 12-parameter space of coefficients of the systems. This diagram provides an algorithm for deciding for any given system whether it belongs to $QS{L}_3$ and in case it does, by producing its configuration of invariant straight lines.

Bibliographie

[1] J. C. Artés, J. Llibre, D. Schlomiuk and N. Vulpe, Geometric configurations of singularities of planar polynomial differential systems [A global classification in the quadratic case], Birkha̋user/Springer, Cham, 2021, xii+699 pp. | MR | Zbl

[2] J. C. Artés, J. Llibre, D. Schlomiuk and N. Vulpe, “Invariant conditions for phase portraits of quadratic systems with complex conjugate invariant lines meeting at a finite point”, Rend. Circ. Mat. Palermo, Series 2, 23 pp. | DOI | MR

[3] J. C. Artés, J. Llibre, D. Schlomiuk and N. Vulpe, “Global Topological Configurations of Singularities for the Whole Family of Quadratic Differential Systems”, Qualitative Theory of Dynamical Systems, 19:51 (2020), 32 pp. | MR | Zbl

[4] V. A. Baltag and N. Vulpe, “Total multiplicity of all finite critical points of the polynomial differential system”, Differ. Equ. Dyn. Syst., 5 (1997), 455–471 | MR | Zbl

[5] C. Bujac, D. Schlomiuk and N. Vulpe, “On families $QSL_{\geq 2}$ of quadratic systems with invariant lines of total multiplicity at least 2”, Qual. Theory Dyn. Syst., 21:4 (2022), 133, 68 pp. | DOI | MR | Zbl

[6] C. Christopher, J. Llibre and J. V. Pereira, “Multiplicity of invariant algebraic curves in polynomial vector fields”, Pacific J. Math., 229:1 (2007), 63–117 | DOI | MR | Zbl

[7] J. H. Grace and A. Young, The algebra of invariants, Stechert, New York, 1941 | MR

[8] P. J. Olver, Classical Invariant Theory, London Mathematical Society student texts, 44, Cambridge University Press, 1999 | MR | Zbl

[9] D. Schlomiuk and N. Vulpe, “Planar quadratic differential systems with invariant straight lines of at least five total multiplicity”, Qual. Theory Dyn. Syst., 5:1 (2004), 135–194 | DOI | MR | Zbl

[10] D. Schlomiuk and N. Vulpe, “Geometry of quadratic differential systems in the neighborhood of infinity”, J. Differential Equations, 215:2 (2005), 357–400 | DOI | MR | Zbl

[11] D. Schlomiuk and N. Vulpe, “Planar quadratic differential systems with invariant straight lines of total multiplicity four”, Nonlinear Anal., 68:4 (2008), 681–715 | DOI | MR | Zbl

[12] D. Schlomiuk and N. Vulpe, “Integrals and phase portraits of planar quadratic differential systems with invariant lines of at least five total multiplicity”, Rocky Mountain J. Math., 38:6 (2008), 2015–2075 | DOI | MR | Zbl

[13] D. Schlomiuk and N. Vulpe, “Integrals and phase portraits of planar quadratic differential systems with invariant lines of total multiplicity four”, Bul. Acad. Ştiinţe Repub. Mold. Mat., 2008, no. 1 (56), 27–83 | MR | Zbl

[14] D. Schlomiuk and N. Vulpe, “Integrals and phase portraits of planar quadratic differential systems with invariant lines of at least five total multiplicity”, Rocky Mountain J. Math., 38:6 (2008), 2015–2075 | DOI | MR | Zbl

[15] D. Schlomiuk and N. Vulpe, “The full study of planar quadratic differential systems possessing a line of singularities at infinity”, J. Dynam. Differential Equations, 20:4 (2008), 737–775 | DOI | MR | Zbl

[16] D. Schlomiuk and N. Vulpe, “Global classification of the planar Lotka–Volterra differential systems according to their configurations of invariant straight lines”, J. Fixed Point Theory Appl., 8:1 (2010), 177–245 | DOI | MR | Zbl

[17] D. Schlomiuk and N. Vulpe, “Global topological classification of Lotka–Volterra quadratic differential systems”, Electron. J. Differential Equations, 2012 (2012), 64, 69 pp. | MR | Zbl

[18] K. S. Sibirskii, Introduction to the algebraic theory of invariants of differential equations, Translated from Russian, Nonlinear Science: Theory and Applications, Manchester University Press, Manchester, 1988 | MR | Zbl

[19] D. Schlomiuk and X. Zhang, “Quadratic differential systems with complex conjugate invariant lines meeting at a finite point”, J. Differential Equations, 265:8 (2018), 3650–3684 | DOI | MR | Zbl

[20] Guangjian Suo and Yongshau Chen, “The real quadratic system with two conjugate imaginary straight line solutions”, Ann. Diff. Eqns., 2 (1986), 197–207 | MR | Zbl