\(QS10_{16}^{(0)}\)

Description

Topological configuration of singularities: \(s,s,a,a;S,N,N\)

Phase Portrait

Topological Invariants

TCSP	Fin Sep	Inf Sep
\(10\)	\(4433\)	\(111111\)

Example

The quadratic differential system

has the following phase portrait done with P4. If you want, you may download the P4 file here.

The phase portrait appears in the following papers

• With name \(1\) in {J. C. Artés, J. Llibre and N. Vulpe}, Quadratic systems with a rational first integral of degree 2: A complete classification in the coefficient space $ R^{12$}, emph{Rend. Circ. Mat. Palermo}, textbf{56}, no. 3 (2007), 417--444.Note (for name \(1\)): missed arrow
• With names \(PP03a\), \(PP03b\), \(PP03c\), \(PP03d\), \(PP03e\), \(PP03f\), \(PP03g\), \(PP03h\), \(PP03i\), \(PP03j\) and \(PP03k\) in {J. C. Artés, J. Llibre and Huaxin Ou}, Quadratic systems with two invariant straight lines and an invariant hyperbola, {Preprint} (2026).
• With names \(Chap 2 2\) and \(Chap 3 11\) in {B. Imane and B. Souad}, Global phase portraits of quadratic differential systems exhibiting an invariant algebraic curve or an algebraic cubic first integral, {Ph.D. Universite Mohamed el Bachir}, (2020).
• With names \(1\), \(9\) and \(25\) in {A. Belfar and R. Benterki}, Qualitative dynamics of five quadratic polynomial differential systems exhibiting five classical cubic algebraic curves, Rend. Circ. Mat. Palermo (2) { bf 72} (2023), no.~1, 393--420; MR4543844
• With name \(9\) in {R. Benterki and A. Belfar}, Phase portraits of two classes of quadratic differential systems exhibiting as solutions two cubic algebraic curves, Demonstr. Math. { bf 56} (2023), no.~1, Paper No. 20220218, 16 pp.; MR4592893
• With name \(A01\) in {C. A. Buzzi and D. J. Tonon}, Quadratic planar systems with two parallel invariant straight lines, Qual. Theory Dyn. Syst. { bf 7} (2009), no.~2, 295--316; MR2486677
• With name \(H1\) in {L. Cairó and J. Llibre}, Phase portraits of quadratic polynomial vector fields having a rational first integral of degree 2. Nonlinear Anal. 67 (2007), no. 2, 327–348.
• With name \(35\) in {B. Coll, A. Ferragut and J. Llibre}, Phase portraits of the quadratic systems with a polynomial inverse integrating factor, Internat. J. Bifur. Chaos Appl. Sci. Engrg. { bf 19} (2009), no.~3, 765--783; MR2533481
• With names \(4.1a\), \(4.3a\) and \(4.9b\) in {D. Schlomiuk and N. Vulpe}, Integrals and phase portraits of planar quadratic differential systems with invariant lines of total multiplicity four, emph{Bul. Acad. c{S}tiin c{t}e Repub. Mold. Mat.}, { bf 1 (56)} (2008), 27--83.
• With names \(6.1\), \(5.1\) and \(5.3\) in {D. Schlomiuk and N. Vulpe}, Integrals and phase portraits of planar quadratic differential systems with invariant lines of at least five total multiplicity, emph{Rocky Mountain J. Math.}, textbf{38}, no. 6 (2008), 2015--2075.Note (for name \(6.1\)): missed arrowsNote (for name \(5.3\)): All phase portraits in this paper use a criteria that draws infinite arcs wide or thin. All infinite archs must be considered separatrices and be drawn wide
• With names \(4,1a\), \(4,3a\), \(4,9b\), \(5,1\), \(5,3\), \(6,1\) and \(3,1(c3)\) in {D. Schlomiuk and N. Vulpe}, Global classification of the planar Lotka--Volterra differential systems according to their configurations of invariant straight lines, emph{J. Fixed Point Theory Appl.}, { bf 8}, no. 1 (2010), 177--245.
• With name \(11\) in {A. Ferragut, J. D. García-Saldaña and C. Valls}, Phase portraits of Abel quadratic differential systems of second kind with symmetries, Dyn. Syst. { bf 34} (2019), no.~2, 301--333; MR3941199
• With name \(Fig1.d\) in {J. Llibre and C. Valls}, Global dynamics of a system coming from the study of a static star, Differ. Equ. Dyn. Syst. { bf 32} (2024), no.~2, 607--617; MR4721747
• With name \(PP10\) in {J. Llibre and H. X. Ou}, Quadratic systems with two invariant real straight lines and an invariant parabola, Electron. J. Qual. Theory Differ. Equ. { bf 2025}, Paper No. 66, 54 pp.; MR5018064
• With name \(PP06\) in {J. Llibre and H. X. Ou}, Quadratic systems with two invariant real straight lines and an invariant hyperbola, {Preprint} (2026).
• With name \(V\) in {T. Li and J. Llibre}, Phase portraits of separable quadratic systems and a bibliographical survey on quadratic systems, Expo. Math. { bf 39} (2021), no.~4, 540--565; MR4340762
• With name \(P01\) in {J. Llibre and R. D. S. Oliveira}, Phase portraits of quadratic polynomial vector fields having a rational first integral of degree 3, Nonlinear Anal. { bf 70} (2009), no. 12, 6378--6379.
• With name \(Fig. 24 h\) in {J. Llibre and C. Pantazi}, Global phase portraits of the quadratic systems having a singular and irreducible invariant curve of degree 3. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 33 (2023), no. 1, Paper No. 2350003, 54 pp.
• With names \(Fig. 1 b\) and \(Fig. 1 d\) in {J. Llibre, W. F. Pereira and C. G. Pessoa}, Phase portraits of Bernoulli quadratic polynomial differential systems, Electron. J. Differential Equations { bf 2020}, Paper No. 48, 19 pp.; MR4102990
• With name \(S1\) in {M. C. Mota, R. D. S. Oliveira and A. M. Travaglini}, The interplay among the topological bifurcation diagram, integrability and geometry for the family { bf QSH(D)}, Geom. Dedicata { bf 217} (2023), no.~6, Paper No. 95, 42 pp.; MR4631488
• With name \(1\) in {M. Ndiaye and H. J. Giacomini}, Quadratic systems equivalent by domains to a linear one: global phase portraits, Extracta Math. { bf 15} (2000), no.~1, 97--119; MR1792982Note (for name \(1\)): The quality of the pictures in this article is very low.
• With name \(Ric. 3\) in {J. C. Artés, J. Llibre, D. Schlomiuk and N. Vulpe}, Global analysis of Riccati quadratic differential systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg. { bf 34} (2024), no.~1, Paper No. 2450004, 46 pp.; MR4701478
• With names \(Fig 9 P2E\), \(Fig 11 P1F\), \(Fig 15 P2H\), \(Fig 21 P1K\), \(Fig 23 P1L\) and \(Fig 25 P1M\) in {A. M. Travaglini}, Integrability and geometryof quadratic differential systems with invariant hyperbolas, {Ph. D., Uni. de Sao Paulo} (2026).
• With name \(S^2_{10,16}\) in {J. C. Artés, R. E. Kooij and J. Llibre}, Structurally stable quadratic vector fields, Mem. Amer. Math. Soc. { bf 134} (1998), no.~639, viii+108 pp.; MR1432139
• With name \(Portrait 29\) in {J. C. Artés, J. Llibre and N. Vulpe}, Quadratic systems with an integrable saddle: A complete classification in the coefficient space $ mathbb{R^{12}$}, emph{Nonlinear Anal.}, textbf{75}, no. 14 (2012), 5416--5447.
• With name \(P1\) in {J. C. Artés, J. Llibre and N. Vulpe}, Quadratic systems with a rational first integral of degree three: a complete classification in the coefficient space $ Bbb R^{12$}, Rend. Circ. Mat. Palermo (2) { bf 59} (2010), no.~3, 419--449; MR2745521
• With names \(Fig 2A III\), \(Fig 2B IV\), \(Fig 2B VIII\), \(Fig 2B X\), \(Fig 2B XI\), \(Fig 2C II\), \(Fig 2C IV\), \(Fig 2C VI\), \(Fig 2D IV\), \(Fig 2D VII\), \(Fig 2D VIII\) and \(Fig 2D X\) in {J. W. Reyn}, Phase portraits of a quadratic system of differential equations occurring frequently in applications, emph{Nieuw Arch. Wisk. (4)}, textbf{5}, no. 2 (1987), 107--151.

Neighbours of Codimension 1

• Through the border \(QS11_{20}^{(1)}\), by means of a bifurcation of type \(B\), we reach the neighbor \(QS8_{4}^{(0)}\).
• Through the border \(QS76_{10}^{(1)}\), by means of a bifurcation of type \(C\), we reach the neighbor \(QS1_{1}^{(0)}\).
• Through the border \(QS38_{33}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS25_{1}^{(0)}\).
• Through the border \(QS10_{21}^{(1)}\), by means of a bifurcation of type \(D\), we reach the neighbor \(QS10_{14}^{(0)}\).
• Through the border \(QS10_{5}^{(1)}\), by means of a bifurcation of type \(D(a)\), we reach the neighbor \(QS10_{15}^{(0)}\).
• Through the border \(QS74_{5}^{(1)}\), by means of a bifurcation of type \(C\), we reach the neighbor \(QS5_{3}^{(0)}\).

Comments

• This portrait is conjectured that cannot have limit cycles in resenhas