\(QS19_{1}^{(0)}\)

Description

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

Phase Portrait

Topological Invariants

TCSP	Fin Sep	Inf Sep
\(19\)	\(11\)	\(211011\)

Example

The quadratic differential system

\[\begin{cases} \dot{x} = 2 \, x \, y \\ \dot{y} = 3-x^{2}/3+3 \, y^{2}+a \, y \end{cases}\]

with parameters: \(a = 0.5\)

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 \(AA^n_5\) 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 \(11S9\) 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 \(Yablonskii\) 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 \(Yablonskii\)): The system has limit cycles with distribution \((1,0)\).
With name \(118\) 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 name \(7\) 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 \(1S08\) 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 names \(Fig 5.34 S^2_{11,1}\) and \(Fig 5.136 S^2_{11,1}\) 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 \(Fig 5.34 S^2_{11,1}\)): The system has limit cycles with distribution \((1,0)\).Note (for name \(Fig 5.136 S^2_{11,1}\)): The system has limit cycles with distribution \((1,0)\).
With name \(S^2_{11,1}\) 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 \(V32\) and \(V35\) 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}.Note (for name \(V35\)): The system has limit cycles with distribution \((1,0)\).
With names \(V3\) and \(V32\) in {J. Llibre and D. Schlomiuk}, Geometry of quadratic differential systems with a weak focus of third order, emph{Canad. J. of Math.}, textbf{56}, no. 2 (2004), 310--343.
With names \(Fig 1.21 c\), \(Fig 1.24 c\) and \(Fig 1.24 d\) 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; MR1660222Note (for name \(Fig 1.21 c\)): The system has limit cycles with distribution \((1,0)\).Note (for name \(Fig 1.24 c\)): The system has limit cycles with distribution \((1,0)\).
With name \(V7\) 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.

Neighbours of Codimension 1

Through the border \(QS22_{1}^{(1)}\), by means of a bifurcation of type \(B\), we reach the neighbor \(QS16_{1}^{(0)}\).
Through the border \(QS85_{1}^{(1)}\), by means of a bifurcation of type \(C\), we reach the neighbor \(QS25_{1}^{(0)}\).
Through the border \(QS19_{2}^{(1)}\), by means of a bifurcation of type \(D\), we reach the neighbor \(QS19_{2}^{(0)}\).
Through the border \(QS31_{1}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS5_{1}^{(0)}\).
Through the border \(QS31_{2}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS5_{1}^{(0)}\).
Through the border \(QS31_{7}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS5_{3}^{(0)}\).
Through the border \(QS31_{8}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS5_{4}^{(0)}\).
Through the border \(QS31_{13}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS5_{6}^{(0)}\).

Comments

This is one of three, the only phase portraits for which the configuration (3,1) of limit cycles has been proved to exist.
This phase portrait appears in J. Llibre and D. Schlomiuk (emph{Canad. J. of Math.}, textbf{56}, no. 2 (2004), 310--343) featuring a weak focus of third order. Since the portrait is of codimension 0, a configuration structurally equivalent to \(QS19_{1}^{(0)}\) could potentially exhibit up to three limit cycles (or compound double/triple arrangements) bifurcating from the focus.