Solution of the center problem for cubic systems with a~bundle of three invariant straight lines
Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 1 (2003), pp. 91-101.

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

For cubic differential system with three invariant straight lines which pass through the same point it is proved that a singular point with purely imaginary eigenvalues (weak focus) is a center if and only if the focal values g2j+1, j=1,5, vanish.
@article{BASM_2003_1_a9,
     author = {Alexandru \c{S}ub\u{a}},
     title = {Solution of the center problem  for cubic systems with a~bundle of three invariant  straight lines},
     journal = {Buletinul Academiei de \c{S}tiin\c{t}e a Republicii Moldova. Matematica},
     pages = {91--101},
     publisher = {mathdoc},
     number = {1},
     year = {2003},
     language = {en},
     url = {https://geodesic-test.mathdoc.fr/item/BASM_2003_1_a9/}
}
TY  - JOUR
AU  - Alexandru Şubă
TI  - Solution of the center problem  for cubic systems with a~bundle of three invariant  straight lines
JO  - Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica
PY  - 2003
SP  - 91
EP  - 101
IS  - 1
PB  - mathdoc
UR  - https://geodesic-test.mathdoc.fr/item/BASM_2003_1_a9/
LA  - en
ID  - BASM_2003_1_a9
ER  - 
%0 Journal Article
%A Alexandru Şubă
%T Solution of the center problem  for cubic systems with a~bundle of three invariant  straight lines
%J Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica
%D 2003
%P 91-101
%N 1
%I mathdoc
%U https://geodesic-test.mathdoc.fr/item/BASM_2003_1_a9/
%G en
%F BASM_2003_1_a9
Alexandru Şubă. Solution of the center problem  for cubic systems with a~bundle of three invariant  straight lines. Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 1 (2003), pp. 91-101. https://geodesic-test.mathdoc.fr/item/BASM_2003_1_a9/

[1] Amel'kin V. V., Lukashevich N. A., Sadovskii A. P., Nonlinear oscillations in second-order systems, Beloruss. Gos. Univ., Minsk, 1982, 208 pp. (in Russian) | MR

[2] Chavarriga J., Giné J., “Integrability of cubic systems with degenerate infinity”, Differential Equations and Dynamical Systems, 6:4 (1998), 425–438 | MR | Zbl

[3] Chavarriga J., Llibre J., “Intvariant algebraic curves and rational first integrals for planar polynomial vector fields”, Journal of Differential Equations, 169 (2001), 1–16 | DOI | MR | Zbl

[4] Cherkas L. A., Romanovskii V. G., Żoła̧dek H., “The centre conditions for a certain cubic system”, Differential Equations Dynam. Systems, 5:3–4 (1997), 299–302 | MR | Zbl

[5] Christopher C., Llibre J., “Integrability via invariant algebraic curves for planar polynomial differential systems”, Ann. Differential Equations, 16:1 (2000), 5–19 | MR | Zbl

[6] Cozma D., Şubă A., “Partial integrals and the first focal value in the problem of centre”, NoDEA Nonlinear Differential Equations Appl., 2:1 (1995), 21–34 | DOI | MR | Zbl

[7] Cozma D., Şubă A., “The solution of the problem of centre for cubic differential systems with four invariant straight lines”, Ann. Scientific of the University “Al. I. Cuza” (Iaşi, Romania), s.I.a., Math., 44 (1998), 517–530 | MR | Zbl

[8] Cozma D., Şubă A., “Solution of the problem of the centre for a cubic differential system with three invariant straight lines”, Qualitative Theory of Dynamical Systems. Universitat de Lleida (Spaine), 2:1 (2001), 129–145 | DOI | MR

[9] Daniliuc V. I., Şubă A. S., “Distinguishing the cases of the centre and focus for cubic systems with six parameters”, Izv. Akad. Nauk Moldav. SSR, Mat., 1990, no. 3, 18–21 (in Russian) | MR

[10] Dulac H., “Détermination et intégration d'une certaine classe d'équations différentielles ayant pour point singuliere un centre”, Bull. sci. Math., 32 (1908), 230–252

[11] Lloyd N. G., Pearson J. M., Romanovsky V. G., “Computing integrability conditions for a cubic differential system”, J. Computers and Mathematics with Applications, 32:10 (1996), 99–107 | DOI | MR | Zbl

[12] Romanovskii V. G., Şubă A., “Center of some cubic systems”, Annals of Differential Equation (China), 17:4 (2001), 363–376 | MR

[13] Differential Equations, 33:2 (1997), 236–244 | MR

[14] Differential Equations, 36:1 (2000), 113–119 | DOI | MR | Zbl

[15] Sadovskii A. P., “On conditions for a center and focus for nonlinear oscilation equations”, Differ. Uravn., 15:9 (1979), 1716–1719 (in Russian) | MR

[16] Sibirskii K. S., “On the number of limit cycles in the neighbourhood of a singular point”, Differential Equations, 1 (1965), 36–47 | MR

[17] Differential Equations, 32:7 (1996), 884–892 | MR | Zbl

[18] Şubă A. S., “On the Liapunov quantities of two-dimensional autonomous system of differential equations with a critical point of centre or focus type”, Bulletin of Baia Mare University (România). Mathematics and Informatics, 13:1–2 (1998), 153–170 | MR

[19] Şubă A. S., Cozma D., “Solution of the problem of the center for cubic systems with two homogeneous and one non-homogeneous invariant straight lines”, Bulletin of Academy of Sciences of Rep. Moldova. Math., 1999, no. 1(29), 37–44 | MR | Zbl

[20] Żołądek H., “On algebraic solutions of algebraic Pfaff equations”, Stud. Math., 114 (1995), 117–12