Phase portraits of integrable quadratic systems with an invariant parabola and an invariant straight line

Llibre, Jaume; Fronza da Silva, Maurício

doi:10.1016/j.crma.2018.12.008

Ordinary differential equations

Phase portraits of integrable quadratic systems with an invariant parabola and an invariant straight line
[Portraits de phase de systèmes quadratiques intégrables avec une parabole et une ligne droite invariantes]

Présenté par : the Editorial Board

Llibre, Jaume ¹ ; Fronza da Silva, Maurício ²

¹ Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain
² Departamento de Matemática, Universidade Federal de Santa Maria, 97110-820, Santa Maria, RS, Brazil

Comptes Rendus. Mathématique, Tome 357 (2019) no. 2, pp. 143-166.

Résumé
Abstract

Nous classifions les portraits de phase des systèmes différentiels polynomiaux quadratiques ayant une parabole invariante, une ligne droite invariante et une intégrale première de Darboux produite par ces deux invariants.

We classify the phase portraits of the quadratic polynomial differential systems having an invariant parabola, an invariant straight line, and a Darboux first integral produced by these two invariant curves.

Reçu le : 2018-11-19
Accepté le : 2018-12-16
Publié le : 2019-01-31

DOI : 10.1016/j.crma.2018.12.008

Affiliations des auteurs :

Llibre, Jaume ¹ ; Fronza da Silva, Maurício ²

@article{CRMATH_2019__357_2_143_0,
     author = {Llibre, Jaume and Fronza da Silva, Maur{\'\i}cio},
     title = {Phase portraits of integrable quadratic systems with an invariant parabola and an invariant straight line},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {143--166},
     publisher = {Elsevier},
     volume = {357},
     number = {2},
     year = {2019},
     doi = {10.1016/j.crma.2018.12.008},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2018.12.008/}
}

TY  - JOUR
AU  - Llibre, Jaume
AU  - Fronza da Silva, Maurício
TI  - Phase portraits of integrable quadratic systems with an invariant parabola and an invariant straight line
JO  - Comptes Rendus. Mathématique
PY  - 2019
SP  - 143
EP  - 166
VL  - 357
IS  - 2
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2018.12.008/
DO  - 10.1016/j.crma.2018.12.008
LA  - en
ID  - CRMATH_2019__357_2_143_0
ER  -

%0 Journal Article
%A Llibre, Jaume
%A Fronza da Silva, Maurício
%T Phase portraits of integrable quadratic systems with an invariant parabola and an invariant straight line
%J Comptes Rendus. Mathématique
%D 2019
%P 143-166
%V 357
%N 2
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2018.12.008/
%R 10.1016/j.crma.2018.12.008
%G en
%F CRMATH_2019__357_2_143_0

Llibre, Jaume; Fronza da Silva, Maurício. Phase portraits of integrable quadratic systems with an invariant parabola and an invariant straight line. Comptes Rendus. Mathématique, Tome 357 (2019) no. 2, pp. 143-166. doi : 10.1016/j.crma.2018.12.008. http://www.numdam.org/articles/10.1016/j.crma.2018.12.008/

Bibliographie
Cité par

[1] Artés, J.C.; Llibre, J. Quadratic Hamiltonian vector fields, J. Differ. Equ., Volume 107 (1994), pp. 80-95

[2] Artés, J.C.; Llibre, J.; Vulpe, N. Complete geometric invariant study of two classes of quadratic systems, Electron. J. Differ. Equ., Volume 2012 (2012) no. 9, pp. 1-35

[3] Bautin, N.N. On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type, Amer. Math. Soc. Transl., Volume 30 (1952), pp. 181-196

[4] Date, T. Classification and analysis of two-dimensional homogeneous quadratic differential equations systems, J. Differ. Equ., Volume 32 (1979), pp. 311-334

[5] Dulac, H. Détermination et integration d'une certaine classe d'équations différentielle ayant par point singulier un centre, Bull. Sci. Math. Sér. (2), Volume 32 (1908), pp. 230-252

[6] Dumortier, F.; Llibre, J.; Artés, J.C. Qualitative Theory of Planar Differential Systems, Universitext, Springer-Verlag, 2006

[7] Kalin, Y.F.; Vulpe, N.I. Affine-invariant conditions for the topological discrimination of quadratic Hamiltonian differential systems, Differ. Equ., Volume 34 (1998) no. 3, pp. 297-301

[8] Kapteyn, W. On the midpoints of integral curves of differential equations of the first degree, Nederl. Akad. Wetensch. Verslag. Afd. Natuurk. Konikl. Nederland (1911), pp. 1446-1457 (in Dutch)

[9] Kapteyn, W. New investigations on the midpoints of integrals of differential equations of the first degree, Nederl. Akad. Wetensch. Verslag Afd. Natuurk., Volume 20 (1912), pp. 1354-1365 (in Dutch)

[10] Korol, N.A. The integral curves of a certain differential equation, Minsk. Gos. Ped. Inst. Minsk (1973), pp. 47-51 (in Russian)

[11] Liapunov, M.A. Problème général de la stabilité du mouvement, Ann. Math. Stud., vol. 17, Princeton University Press, Princeton, NJ, USA, 1947

[12] Llibre, J.; da Silva, M.F. Global phase portraits of Kukles differential systems with homogeneous polynomial nonlinearities of degree 6 having a center and their small limit cycles, Int. J. Bifurc. Chaos, Volume 26 (2016) (25 p)

[13] Llibre, J.; Yu, J. Phase portraits of quadratic systems with an ellipse and a straight line as invariant algebraic curves, Electron. J. Differ. Equ., Volume 314 (2015) (14 p)

[14] Lunkevich, V.A.; Sibirskii, K.S. Integrals of a general quadratic differential system in cases of a center, Differ. Equ., Volume 18 (1982), pp. 563-568

[15] Lyagina, L.S. The integral curves of the equation $y^{'} = (a x^{2} + b x y + c y^{2}) / (d x^{2} + e x y + f y^{2})$ , Usp. Mat. Nauk, Volume 6-2 (1951) no. 42, pp. 171-183 (in Russian)

[16] Markus, L. (Ann. Math. Stud.), Volume vol. 45, Princeton University Press, Princeton, NJ, USA (1960), pp. 185-213

[17] Newton, T.A. Two dimensional homogeneous quadratic differential systems, SIAM Rev., Volume 20 (1978), pp. 120-138

[18] Poincaré, H. Mémoire sur les courbes définies par les équations différentielles, J. Math. (Œuvres d'Henri Poincaré), Volume 37 (1881), pp. 375-422

[19] Reyn, J.W. Phase Portraits of Planar Quadratic Systems, Mathematics and Its Applications, vol. 583, Springer, New York, 2007

[20] Schlomiuk, D. Algebraic particular integrals, integrability and the problem of the center, Trans. Amer. Math. Soc., Volume 338 (1993), pp. 799-841

[21] Sibirskii, K.S.; Vulpe, N.I. Geometric classification of quadratic differential systems, Differ. Equ., Volume 13 (1977), pp. 548-556

[22] Vdovina, E.V. Classification of singular points of the equation $y^{'} = (a_{0} x^{2} + a_{1} x y + a_{2} y^{2}) / (b_{0} x^{2} + b_{1} x y + b_{2} y^{2})$ by Forster's method, Differ. Equ., Volume 20 (1984), pp. 1809-1813 (in Russian)

[23] Vulpe, N.I. Affine-invariant conditions for the topological discrimination of quadratic systems with a center, Differ. Equ., Volume 1 (1983), pp. 273-280

[24] Ye, Y. Qualitative Theory of Polynomial Differential Systems, Shanghai Scientific & Technical Publishers, Shanghai, 1995 (in Chinese)

[25] Ye, Y. et al. Theory of Limit Cycles, Transl. Math. Monogr., vol. 66, Amer. Math. Soc., Providence, RI, USA, 1984

[26] Ye, W.Y.; Ye, Y. On the conditions of a center and general integrals of quadratic differential systems, Acta Math. Sin. (Engl. Ser.), Volume 17 (2001), pp. 229-236

[27] Żołądek, H. Quadratic systems with center and their perturbations, J. Differ. Equ., Volume 109 (1994), pp. 223-273

Cité par Sources :