Classification of quadratic systems with a~symmetry center and simple infinite singular points
Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 1 (2003), pp. 102-119.

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We classify the family of planar quadratic differential systems with a center of symmetry and two invariant straight lines according to the topology of their phase portraits. The case of the existence of simple infinite singular points is only considered. For each of the obtained distinct topological classes we give necessary and sufficient conditions in terms of algebraic invariants and comitants. The program was implemented for computer calculations.
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Mircea Lupan; Nicolae Vulpe. Classification of quadratic systems with a~symmetry center and simple infinite singular points. Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 1 (2003), pp. 102-119. https://geodesic-test.mathdoc.fr/item/BASM_2003_1_a10/

[1] John Wiley Sons, New York, 1973 | Zbl

[2] Argemi J., “Sur les points singuliers multiples de systémes dinamiques dans $R^2$”, Annali di matematica pura ed applicata, 4 ser., 79 (1968), 35–70 | DOI | MR

[3] Babaeva R. A., Burlak E. M., Vulpe N. I., “About the number, multiplicities and types of the singular points of some differential system”, Izvestiya AN Respubliki Moldova. Matematika, 1992, no. 3(9), 3–8 (in Russian) | MR | Zbl

[4] Berlinskii A. N., “Qualitative stady of a differential equation”, Diff. uravn., 2:3 (1966), 353–360 (in Russian) | MR

[5] D. Boularas, Iu. Calin, L. Timochouk, N. Vulpe, $T$-comitants of qudratic systems: A study via the translation invariants, Report No 96-90, Delft University of Technology, Faculty of Technical Mathematics and Informatics, 1996; URL: ftp://ftp.its.tudelft.nl/publications/tech-reports/1996/DUT-TWI-96-90.ps.gz

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

[7] Date T., “Classification and analysis of two-dimensional real homogeneous quadratic differential equation systems”, J. of Diff. Equations, 32 (1979), 311–334 | DOI | MR | Zbl

[8] J. H. Grace, A. Young, The algebra of invariants, Stechert, New York, 1941

[9] M. N. Lupan, “Types of the critical points for quadratic system with a center of symmetry”, Buletinul Academiei de Ştiinţe a Moldovei, 1998, no. 1(26), 69–80 | MR

[10] M. Lupan, N. Vulpe, “Quadratic Systems with Dicritical Points”, Buletinul Academiei de Ştiinţe a Moldovei, 1994, no. 3(16), 52–60 (in Russian) | MR | Zbl

[11] Lyagina L. S., “The integral curves of the equation $y'=\frac{ax^2+bxy+cy^2}{dx^2+exy+fy^2}$”, Uspehi matem. nauk, 6:2(42) (1951), 171–183 (in Russian) | MR | Zbl

[12] Markus L., “Quadratic differential equations and non-associative algebras”, Ann. Math. Studies, 45, 1960, 185–213 | MR | Zbl

[13] Newton T. A., “Two-dimensional homogeneous quadratic differential systems”, SIAM Review, 20:1 (1978), 120–138 | DOI | MR | Zbl

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

[15] D. Schlomiuk, N. Vulpe, Geometry of quadratic differential systems in the neighbourhood of the line at infinity, Report No 2701, Centre de recherches mathématiques et Département de Mathématiques et de Statistiques, Université de Montréal, 2001

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

[17] Suo Guangjian, “The algebraic critical cycles and bifurcations of limit cycles for the system $\dot x=a+\sum_{i+j=2}a_{ij}x^iy^j$, $\dot y=b+\sum_{i+j=2}b_{ij}x^iy^j$”, Journal of Math. Res. and Expos., 2:2 (1982), 69–75 | MR

[18] Suo Guangjian, “Existence of at most two limit cycles for the system of differential equations $\frac{dx}{dt}=a+\sum_{i+j=2}a_{ij}x^i x^j$, $\frac{dy}{dt}=b+\sum_{i+j=2} b_{ij}x^i x^j$”, TUDelft, 27, Report 90-03, 1990, 572–577

[19] Vdovina E. V., “The singular points classification of the equation $y'=\frac{a_0x^2+a_1xy+a_2y^2}{b_0x^2+b_1xy+b_2y^2}$ by Forster's method”, Diff. uravneniya, 20:10 (1984), 1809–1813 (in Russian) | MR | Zbl

[20] N. I. Vulpe, A. Yu. Likhovetskii, “Coefficient conditions for topological classification of quadratic systems in the Darboux form”, Diff. uravn. i matem. fizika, Matem. issledov., 106, 1989, 34–49 (in Russian) | MR | Zbl

[21] Vulpe N. I., Sibirskii K. S., “Affine classification of the quadratic systems”, Diff. uravneniya, 10:12 (1974), 2111–2124 (in Russian) | MR | Zbl

[22] Vulpe N. I., Sibirskii K. S., “Affine invariant coefficient conditions for topological differentiation of the quadratic systems”, Matem. issled., 10, no. 3(37), Shtiintsa, Cishinau, 1975, 15–28 (in Russian) | MR

[23] Vulpe N. I., Sibirskii K. S., “Geometrical classification of quadratic systems”, Diff. uravneniya, 13:5 (1977), 803–814 (in Russian) | MR | Zbl

[24] Ye Yan-Qian and others, Theory of Limit Cycles, Translations of Mathematical Monographs, 66, AMS, 1986