Geodesic
Cubic differential systems with two affine real non-parallel invariant straight lines of maximal multiplicity
Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 3 (2015), pp. 79-101.

Voir la notice de l'article provenant de la source Math-Net.Ru

In this article we classify all differential real cubic systems possessing two affine real non-parallel invariant straight lines of maximal multiplicity. We show that the maximal multiplicity of each of these lines is at most three. The maximal sequences of multiplicities: $m(3,3;1)$, $m(3,2;2)$, $m(3,1;3)$, $m(2,2;3)$, $m_\infty(2,1;3)$, $m_\infty(1,1;3)$ are determined. The normal forms and the corresponding perturbations of the cubic systems which realize these cases are given. 
@article{BASM_2015_3_a6,
     author = {Olga Vacara\c{s}},
     title = {Cubic differential systems with two affine real non-parallel invariant straight lines of maximal multiplicity},
     journal = {Buletinul Academiei de \c{S}tiin\c{t}e a Republicii Moldova. Matematica},
     pages = {79--101},
     publisher = {mathdoc},
     number = {3},
     year = {2015},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/BASM_2015_3_a6/}
}
TY  - JOUR
AU  - Olga Vacaraş
TI  - Cubic differential systems with two affine real non-parallel invariant straight lines of maximal multiplicity
JO  - Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica
PY  - 2015
SP  - 79
EP  - 101
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/BASM_2015_3_a6/
LA  - en
ID  - BASM_2015_3_a6
ER  - 
%0 Journal Article
%A Olga Vacaraş
%T Cubic differential systems with two affine real non-parallel invariant straight lines of maximal multiplicity
%J Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica
%D 2015
%P 79-101
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/BASM_2015_3_a6/
%G en
%F BASM_2015_3_a6
Olga Vacaraş. Cubic differential systems with two affine real non-parallel invariant straight lines of maximal multiplicity. Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 3 (2015), pp. 79-101. http://geodesic.mathdoc.fr/item/BASM_2015_3_a6/

[1] Artes J., Grünbaum B., Llibre J., “On the number of invariant straight lines for polynomial differential systems”, Pacific J. of Math., 184:2 (1998), 207–230 | DOI | MR | Zbl

[2] Bujac C., “One subfamily of cubic systems with invariant lines of total multiplicity eight and with two distinct real infinite singularities”, Bul. Acad. Ştiinţe Repub. Mold., Mat., 2015, no. 1(77), 48–86 | MR

[3] Bujac C., Vulpe N., “Cubic systems with invariant straight lines of total multiplicity eight and with three distinct infinite singularities”, Qual. Theory Dyn. Syst., 14:1 (2015), 109–137 | DOI | MR | Zbl

[4] Bujac C., Vulpe N., “Cubic systems with invariant lines of total multiplicity eight and with four distinct infinite singularities”, Journal of Mathematical Analysis and Applications, 423:2 (2015), 1025–1080 | DOI | MR | Zbl

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

[6] Cozma D., Integrability of cubic systems with invariant straight lines and invariant conics, Ştiinţa, Chişinău, 2013, 240 pp.

[7] Cozma D., Şubă A., “The solution of the problem of center for cubic differential systems with four invariant straight lines”, An. Ştiinţ. Univ. “Al. I. Cuza”(Iaşi), 44, suppl. (1998), 517–530 | MR | Zbl

[8] Suo Guangjian, Sun Jifang, “The $n$-degree differential system with $(n-1)(n+1)/2$ straight line solutions has no limit cycles”, Proc. of Ordinary Differential Equations and Control Theory, Wuhan, 1987, 216–220 (in Chinese) | MR

[9] Kooij R., “Cubic systems with four line invariants, including complex conjugated lines”, Math. Proc. Camb. Phil. Soc., 118:1 (1995), 7–19 | DOI | MR | Zbl

[10] Llibre J., Vulpe N., “Planar cubic polynomial differential systems with the maximum number of invariant straight lines”, Rocky Mountain J. Math., 36:4 (2006), 1301–1373 | DOI | MR | Zbl

[11] Lyubimova R. A., “About one differential equation with invariant straight lines”, Differential and integral equations (Gorky Universitet), 1984, no. 8, 66–69; Differential and integral equations (Gorky Universitet), 1997, no. 1, 19–22 (in Russian)

[12] Mironenko V. I., Linear dependence of functions along solutions of differential equations, Beloruss. Gos. Univ., Minsk, 1981, 104 pp. (in Russian) | MR

[13] Puţuntică V., Şubă A., “The cubic differential system with six real invariant straight lines along two directions”, Studia Universitatis, 2008, no. 8(13), 5–16 | MR

[14] Puţuntică V., Şubă A., “The cubic differential system with six real invariant straight lines along three directions”, Bul. Acad. Ştiinţe Repub. Mold., Mat., 2009, no. 2(60), 111–130 | MR | Zbl

[15] Repeşco V., “Cubic systems with degenerate infinity and straight lines of total parallel multiplicity six”, ROMAI J., 9:1 (2013), 133–146 | MR | Zbl

[16] Rychkov G. S., “The limit cycles of the equation $u(x+1)du=(-x+ax^2+bxu+cu+du^2)dx$”, Differentsial'nye Uravneniya, 8:12 (1972), 2257–2259 (in Russian) | MR | Zbl

[17] Şubă A., Cozma D., “Solution of the problem of the center for cubic differential system with three invariant straight lines in generic position”, Qual. Th. of Dyn. Systems, 6 (2005), 45–58 | DOI | MR | Zbl

[18] Şubă A., Repeşco V., Puţuntică V., “Cubic systems with invariant affine straight lines of total parallel multiplicity seven”, Electron. J. Diff. Equ., 2013 (2013), 274, 22 pp. http://ejde.math.txstate.edu/ | DOI | MR | Zbl

[19] Şubă A., Vacaraş O., “Cubic differential systems with an invariant straight line of maximal multiplicity”, Annals of the University of Craiova. Mathematics and Computer Science Series, 42:2 (2015), 427–449 | MR

[20] Zhang Xikang, “Number of integral lines of polynomial systems of degree three and four”, J. of Nanjing University, Math. Biquartely, 10 (1993), 209–212 | MR