$QS12_{1}^{(0)}$

Description

Topological configuration of singularities: $∅;N$

Phase Portrait

Topological Invariants

TCSP
$12$

Example

The quadratic differential system

\[\begin{cases} \dot{x} = 0 \\ \dot{y} = b - x^{2} + f \, y \end{cases}\]

with parameters: $b = -1, \quad f = 0$

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 $A V03$ in {J. C. Artés, C. Bujac, D. Schlomiuk and N. Vulpe}, Phase portraits of real quadratic differential systems possessing an invariant ellipse, {Preprint} (2026).
With name $15$ 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.
With name $A27$ 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 $PL2$ 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 names $Fig 7 10$, $Fig 7 26$ and $Fig 11 8$ in {L. Cairó and J. Llibre}, Phase portraits of Families VII and VIII of the Quadratic Systems. Axioms. No. 12(756), (2023), 18pp.
With name $6$ 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 $4.38b$ and $4.44a$ 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 name $5.30$ 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 names $Fig 1 A1-V1$ and $Fig 2 A-V2$ 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 name $AV8$ in {J. C. Artés, L. Cairó and J. Llibre}, Phase portraits of the family IV of the quadratic polynomial differential systems, Qual. Theory Dyn. Syst. { bf 24} (2025), no.~2, Paper No. 66, 34 pp.; MR4860323
With name $50$ 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 $43$ in {A. Ferragut and C. Valls}, Phase portraits of Abel quadratic differential systems of the second kind, Dyn. Syst. { bf 33} (2018), no.~4, 581--601; MR3869849
With name $15$ in {B. García, J. Llibre and J. S. Pérez del Río}, Phase portraits of the quadratic vector fields with a polynomial first integral, Rend. Circ. Mat. Palermo (2) { bf 55} (2006), no.~3, 420--440; MR2287071
With name $Fig 9 (c)$ in {M. R. A. Gouveia, J. Llibre and L. A. F. Roberto}, Phase portraits of the quadratic polynomial Liénard differential systems, Proc. Roy. Soc. Edinburgh Sect. A { bf 151} (2021), no.~1, 202--216; MR4202639
With name $P19$ 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 $c$ in {J. Llibre and T. Salhi}, Planar quadratic differential systems with invariants of the form $ax^2+bxy+cy^2+d x+ey+c_1t$, Bull. Iranian Math. Soc. { bf 50} (2024), no.~4, Paper No. 49, 16 pp.; MR4755552
With name $P10$ 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 $C$ in {J. Llibre, and C. Valls}, Global phase portraits of quadratic systems with a complex ellipse as invariant algebraic curve. Acta Math. Sin. (Engl. Ser.) 34 (2018), no. 5, 801–811.
With name $25$ 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 $25$): Since the singular points are not clear, there are 3 possible phase portraits, 16, 25 and 41. We denote each one with a different TSCP
With names $Fig1 M^1_04,3(a)$, $Fig1 C^0C^0M^1_04,1$, $Fig3 M^1_04,3(d)$, $Fig3 M^1_04,3(b)$ and $Fig3 E^1C^0_02,1C^0_02,1$ 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 $Ric. 28$ 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 name $S IV 5$ in {J. C. Artés, J. Llibre, D. Schlomiuk and N. Vulpe}, Abel quadratic differential systems of second kind, Electron. J. Differential Equations { bf 2024}, Paper No. 50, 38 pp.; MR4793966
With name $8$ 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 $S^2_{1,1}$ 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 name $11$ in {J. C. Artés and J. Llibre}, Quadratic Hamiltonian vector fields, emph{J. Differential Equations}, { bf 107} (1994), 80--95.
With names $Fig 1.7 w$, $Fig 1.31 g$, $Fig 1.37 a$, $Fig 2.17 c$, $Fig 2.20 h$, $Fig 2.22 a$ and $Fig 2.22 d$ 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 $P19$ 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 $QS44_{1}^{(1)}$, by means of a bifurcation of type $A$, we reach the neighbor $QS23_{1}^{(0)}$.
Through the border $QS14_{1}^{(1)}$, by means of a bifurcation of type $B$, we reach the neighbor $QS13_{1}^{(0)}$.
Through the border $QS14_{2}^{(1)}$, by means of a bifurcation of type $B$, we reach the neighbor $QS13_{1}^{(0)}$.

Neighbours of Codimension 9

Through the border $QS206_{1}^{(9)}$, by means of a bifurcation of type $deg$, we reach the neighbor $QS205_{1}^{(8)}$.

Comments

This is the most trivial quadratic phase portrait, and moreover the most common one. It is proved statistically in resenhas that 25,42% of quadratic systems have this portrait. In fact, the proportion is very similar to the cases with $QS23_{1}^{(0)}$, but since 2,1% of the cases of $QS23_{1}^{(0)}$ have limit cycles, this portrait takes the leadership.