$QS23_{1}^{(0)}$

Description

Topological configuration of singularities: $s,a;N$

Phase Portrait

Topological Invariants

TCSP	Fin Sep	Inf Sep
$23$	$41$	$21$

Example

The quadratic differential system

\[\begin{cases} \dot{x} = -y \, (1+2 \, x) \\ \dot{y} = -1+(x-1)^{2}+y^{2}-a \, y \, (1+2 \, x) \end{cases}\]

with parameters: $a = 0.1$

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 $A V02$ and $A V18$ in {J. C. Artés, C. Bujac, D. Schlomiuk and N. Vulpe}, Phase portraits of real quadratic differential systems possessing an invariant ellipse, {Preprint} (2026).Note (for name $A V18$): The system has 1 limit cycle.
With names $5$ and $7$ in {J. C. Artés, J. Llibre, D. Schlomiuk and N. Vulpe}, Invariant conditions for phase portraits of quadratic systems with complex conjugate invariant lines meeting at a finite point, Rend. Circ. Mat. Palermo (2) { bf 70} (2021), no.~2, 923--945; MR4286006Note (for name $5$): The system has 1 limit cycle.
With names $AA^s_1$ and $AA^n_1$ in {J. C. Artés, R. D. S. Oliveira and A. C. Rezende}, Structurally unstable quadratic vector fields of codimension two: families possessing either a cusp point or two finite saddle-nodes, J. Dynam. Differential Equations { bf 33} (2021), no.~4, 1779--1821; MR4333383
With name $2S1$ in {J. C. Artés and C. Trullàs}, Quadratic Differential Systems with a Weak Focus of First-Order and a Finite Saddle-Node, {International Journal of Bifurcation and Chaos, Vol. 36, No. 6 (2026) 2630013 (99 pages)}
With name $15$ 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 $17$ 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 $A13$ 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 $Fig 11 4$ 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 $Qin c$ in {M. Alberich-Carramiñana, A. Ferragut and J. Llibre}, Quadratic planar differential systems with algebraic limit cycles via quadratic plane, Cremona maps, Adv. Math. { bf 389} (2021), Paper No. 107924, 38 pp.; MR4290137Note (for name $Qin c$): The system has 1 limit cycle.
With names $V1$ and $V5$ in {J. C. Artés, Hebai Chen, L. M. Ferrer and Man Jia}, Quadratic vector fields in class $I$, Dyn. Syst. { bf 40} (2025), no.~2, 191--222; MR4906437Note (for name $V5$): The system has 1 limit cycle.
With names $14$ and $27$ 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 $4.2a$, $4.6a$, $4.27a$ and $4.41a$ 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 $5.2$ and $5.9$ 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 $P18$, $P19$ and $P21$ in {D. Schlomiuk and X. Zhang}, Quadratic differential systems with complex conjugate invariant lines meeting at a finite point, emph{J. Differential Equations}, { bf 265}, no. 8 (2018), 3650--3684.Note (for name $P19$): The system has 1 limit cycle.
With name $Fig 2 A-V5$ 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 $AV9$ and $AV11$ 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.; MR4860323Note (for name $AV11$): The system has 1 limit cycle.
With name $33$ 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 $11$ 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 $Fig 9 (a)$ 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 $L$ 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 names $9$, $62$ and $96$ in {J. Llibre and X. Zhang}, Topological phase portraits of planar semi-linear quadratic vector fields, Houston J. Math. { bf 27} (2001), no.~2, 247--296; MR1874098Note (for name $62$): The system has 1 limit cycle.
With names $Fig 2 a$ and $Fig 2 b$ in {M. Jia, H. B. Chen and H. Chen}, Bifurcation diagram and global phase portraits of a family of quadratic vector fields in class $I$, Qual. Theory Dyn. Syst. { bf 19} (2020), no.~2, Paper No. 64, 22 pp.; MR4109532
With name $1S12$ in {J. C. Artés and L. Cairó}, Phase portraits of quadratic differential systems with a weak focus and a (1,1) SN, {Preprint} (2026).
With name $Ric. 30$ 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 $Fig 5.6 S^2_{2,1}$ in {J. C. Artés, J. Llibre and A. C. Rezende}, Structurally unstable quadratic vector fields of codimension one, Birkhäuser/Springer, Cham, 2018, vi+267 pp.Note (for name $Fig 5.6 S^2_{2,1}$): The system has 1 limit cycle.
With names $V4$ and $V8$ in {J. C. Artés, R. D. S. Oliveira and A. C. Rezende}, Topological classification of quadratic polynomial differential systems with a finite semi-elemental triple saddle, Internat. J. Bifur. Chaos Appl. Sci. Engrg. { bf 26} (2016), no.~11, 1650188, 26 pp.; MR3566296Note (for name $V8$): The system has 1 limit cycle.
With names $S IV 9$, $Fig 3 a$ and $Fig 3 c$ 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.; MR4793966Note (for name $Fig 3 c$): The system has 1 limit cycle.
With name $S^2_{2,1}$ 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 $V39$ in {J. C. Artés, J. Llibre and D. Schlomiuk}, The geometry of quadratic differential systems with a weak focus of second order, emph{ Internat. J. Bifur. Chaos Appl. Sci. Engrg.}, textbf{16} (2006), {3127--3194}.
With names $Portrait 30$ and $Portrait 31$ 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 $Fig10 4$ in {P. de Jager}, Phase portraits for quadratic systems with a higher order singularity with two zero eigenvalues, emph{J. Differential Equations}, textbf{87} (1990), 169--204.
With names $Fig 1.7 i$, $Fig 1.7 l$, $Fig 1.7 o$, $Fig 1.31 c$, $Fig 2.17 a$, $Fig 2.20 a$, $Fig 2.20 c$, $Fig 2.20 d$, $Fig 2.20 e$, $Fig 2.20 g$ and $Fig 2.22 c$ 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; MR1660222Note (for name $Fig 1.7 l$): The system has 1 limit cycle.Note (for name $Fig 2.20 c$): The system has 1 limit cycle.
With name $V10$ in {J. C. Artés, A. C. Rezende and R. Oliveira}, Global phase portraits of quadratic polynomial differential systems with a semi-elemental triple node, emph{Internat. J. Bifur. Chaos Appl. Sci. Engrg.}, { bf 23}, no. 8 (2013), 1350140, 21 pp.
With name $V6$ in {J. C. Artés, J. Llibre and D. Schlomiuk}, The geometry of quadratic polynomial differential systems with a weak focus and an invariant straight line, emph{ Internat. J. Bifur. Chaos Appl. Sci. Engrg.}, textbf{20}, no. 11 (2010), 3627--3662.
With names $2.5L11$ and $2.5L13$ in {J. C. Artés, A. C. Rezende and R. D. S. Oliveira}, The geometry of quadratic polynomial differential systems with a finite and an infinite saddle-node (C), emph{Internat. J. Bifur. Chaos Appl. Sci. Engrg.}, textbf{25}, no. 3 (2015), 1530009, 111 pp.Note (for name $2.5L13$): The system has 1 limit cycle.

Neighbours of Codimension 1

Through the border $QS44_{1}^{(1)}$, by means of a bifurcation of type $A$, we reach the neighbor $QS12_{1}^{(0)}$.
Through the border $QS23_{1}^{(1)}$, by means of a bifurcation of type $D$, we reach the neighbor $QS23_{1}^{(0)}$.
Through the border $QS37_{1}^{(1)}$, by means of a bifurcation of type $A$, we reach the neighbor $QS8_{1}^{(0)}$.
Through the border $QS37_{2}^{(1)}$, by means of a bifurcation of type $A$, we reach the neighbor $QS8_{1}^{(0)}$.
Through the border $QS37_{3}^{(1)}$, by means of a bifurcation of type $A$, we reach the neighbor $QS8_{1}^{(0)}$.
Through the border $QS37_{4}^{(1)}$, by means of a bifurcation of type $A$, we reach the neighbor $QS8_{2}^{(0)}$.
Through the border $QS37_{5}^{(1)}$, by means of a bifurcation of type $A$, we reach the neighbor $QS8_{3}^{(0)}$.
Through the border $QS37_{6}^{(1)}$, by means of a bifurcation of type $A$, we reach the neighbor $QS8_{4}^{(0)}$.
Through the border $QS37_{7}^{(1)}$, by means of a bifurcation of type $A$, we reach the neighbor $QS8_{5}^{(0)}$.
Through the border $QS37_{8}^{(1)}$, by means of a bifurcation of type $A$, we reach the neighbor $QS8_{5}^{(0)}$.
Through the border $QS84_{1}^{(1)}$, by means of a bifurcation of type $C$, we reach the neighbor $QS16_{1}^{(0)}$.
Through the border $QS27_{1}^{(1)}$, by means of a bifurcation of type $B$, we reach the neighbor $QS25_{1}^{(0)}$.
Through the border $QS27_{2}^{(1)}$, by means of a bifurcation of type $B$, we reach the neighbor $QS25_{1}^{(0)}$.
Through the border $QS27_{3}^{(1)}$, by means of a bifurcation of type $B$, we reach the neighbor $QS25_{2}^{(0)}$.
Through the border $QS27_{4}^{(1)}$, by means of a bifurcation of type $B$, we reach the neighbor $QS25_{2}^{(0)}$.
Through the border $QS27_{5}^{(1)}$, by means of a bifurcation of type $B$, we reach the neighbor $QS25_{3}^{(0)}$.
Through the border $QS27_{6}^{(1)}$, by means of a bifurcation of type $B$, we reach the neighbor $QS25_{3}^{(0)}$.

Comments

This is the second most common one. It is proved statistically in resenhas that 25,42% of quadratic systems have this portrait which happens to be the same proportion as $QS12_{1}^{(0)}$, and the counting of cases is even greater here, but not conclusive. However, since 2,1% of the portraits this portrait have limit cycles, then this portrait without limit cycles stays second. Anyway, both together apply for more than 50% of the parameter space.
This phase portrait appears in J. C. Artés, J. Llibre and D. Schlomiuk (emph{ Internat. J. Bifur. Chaos Appl. Sci. Engrg.}, textbf{16} (2006), {3127--3194}) featuring a weak focus of second order. Since the portrait is of codimension 0, a configuration structurally equivalent to $QS23_{1}^{(0)}$ could potentially exhibit up to two limit cycles (or a compound double limit cycle) bifurcating from the focus.

\(QS23_{1}^{(0)}\)

Description

Example

The phase portrait appears in the following papers

Neighbours of Codimension 1

Comments