\(QS13_{1}^{(0)}\)

Description

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

Phase Portrait

Topological Invariants

TCSP	Inf Sep
\(13\)	\(110110\)

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 names \(PP25\) and \(PP36\) in {J. C. Artés, J. Llibre and Huaxin Ou}, Quadratic systems with two invariant straight lines and an invariant hyperbola, {Preprint} (2026).
• With names \(4\) and \(7\) 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 \(A25\) 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 \(Fig 7 9\) in {L. Cairó and J. Llibre}, Phase portraits of Families VII and VIII of the Quadratic Systems. Axioms. No. 12(756), (2023), 18pp.
• With names \(20\) and \(59\) 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 \(4.13a\) and \(4.34b\) 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 \(5.4\), \(5.5\) and \(5.16\) 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.
• With name \(Fig 1 A1-V2\) in {J. C. Artés, L. Cairó and J. Llibre}, New Exploration of Phase portraits Classification of QuadraticPolynomial Differential Systems based on Invariant Theory. Applied Math. No. 1(0), (2025), 24pp.
• With names \(55\) and \(58\) 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 \(III.1\) 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 \(P05\) 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 \(Fig. 4 b\) 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 \(P07\) in {J. Llibre and C. Valls}, Global phase portraits for the Abel quadratic polynomial differential equations of second kind with $Z_2$-symmetries, Canad. Math. Bull. { bf 61} (2018), no.~1, 149--165; MR3746481
• With name \(S20\) 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 names \(Fig1 E^1E^1M^-1_04,1\), \(Fig1 E^-1E^1M^1_04,1\), \(Fig2 E^1M^-1_02,1M^1_02,1(a)\) and \(Fig2 E^1M^-1_02,1M^1_02,1(c)\) in {J. W. Reyn}, Phase portraits of quadratic systems without finite critical points, Nonlinear Anal. { bf 27} (1996), no.~2, 207--222; MR1389478
• With name \(Ric. 2\) 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 \(Fig 5 P3B\), \(Fig 7 P1C\), \(Fig 9 P3E\), \(Fig 11 P2F\), \(Fig 13 P1G\), \(Fig 15 P3H\), \(Fig 17 P2I\), \(Fig 19 P2j\), \(Fig 21 P3K\) and \(Fig 23 P2L\) in {A. M. Travaglini}, Integrability and geometryof quadratic differential systems with invariant hyperbolas, {Ph. D., Uni. de Sao Paulo} (2026).
• With name \(1\) in {A. Gasull, S. Li Ren and J. Llibre}, Chordal quadratic systems, emph{Rocky Mountain J. Math.}, textbf{16}, no. 4 (1986), 751--782.
• With name \(S^2_{8,1}\) 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 names \(Fig 1.3 k\), \(Fig 1.7 u\), \(Fig 1.7 aa\), \(Fig 1.21 f\) and \(Fig 1.31 j\) in {J. W. Reyn and R. E. Kooij}, Phase portraits of non-degenerate quadratic systems with finite multiplicity two, Differential Equations Dynam. Systems { bf 5} (1997), no.~3-4, 355--414; MR1660222
• With name \(P5\) 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

Neighbours of Codimension 1

• Through the border \(QS14_{1}^{(1)}\), by means of a bifurcation of type \(B\), we reach the neighbor \(QS12_{1}^{(0)}\).
• Through the border \(QS14_{2}^{(1)}\), by means of a bifurcation of type \(B\), we reach the neighbor \(QS12_{1}^{(0)}\).
• Through the border \(QS13_{1}^{(1)}\), by means of a bifurcation of type \(D\), we reach the neighbor \(QS13_{1}^{(0)}\).
• Through the border \(QS45_{1}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS25_{1}^{(0)}\).
• Through the border \(QS45_{2}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS25_{2}^{(0)}\).
• Through the border \(QS45_{3}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS25_{3}^{(0)}\).