\(QS19_{2}^{(0)}\)

Description

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

Phase Portrait

Topological Invariants

TCSP	Fin Sep	Inf Sep
\(19\)	\(11\)	\(111111\)

Example

The quadratic differential system

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

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 \(AA^n_7\) 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 names \(2S2\) and \(11S14\) 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)}Note (for name \(11S14\)): The system has limit cycles with distribution \((1,0)\).
With name \(30\) 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 names \(1S09\) and \(1S10\) in {J. C. Artés and L. Cairó}, Phase portraits of quadratic differential systems with a weak focus and a (1,1) SN, {Preprint} (2026).Note (for name \(1S09\)): The system has limit cycles with distribution \((1,0)\).
With names \(Fig 5.37 S^2_{11,2}\) and \(Fig 5.136 S^2_{11,2}\) 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 \(Fig 5.37 S^2_{11,2}\)): The system has limit cycles with distribution \((1,0)\).Note (for name \(Fig 5.136 S^2_{11,2}\)): The system has limit cycles with distribution \((1,0)\).
With name \(S^2_{11,2}\) 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 \(V33\), \(V34\) and \(V55\) 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 \(V33\)): The system has limit cycles with distribution \((1,0)\).Note (for name \(V55\)): The system has limit cycles with distribution \((1,1)\).
With names \(V33\) and \(V34\) 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.Note (for name \(V33\)): The system has limit cycles with distribution \((1,0)\).
With names \(Fig 1.24 a\), \(Fig 1.24 f\), \(Fig 1.24 m\) and \(Fig 1.24 n\) 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 \(Fig 1.24 f\)): The system has limit cycles with distribution \((1,0)\).Note (for name \(Fig 1.24 m\)): The system has limit cycles with distribution \((1,1)\).
With names \(V12\) and \(V15\) 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.Note (for name \(V15\)): The system has limit cycles with distribution \((1,0)\).
With name \(V19\) 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.

Neighbours of Codimension 1

Through the border \(QS22_{3}^{(1)}\), by means of a bifurcation of type \(B\), we reach the neighbor \(QS16_{1}^{(0)}\).
Through the border \(QS22_{2}^{(1)}\), by means of a bifurcation of type \(B\), we reach the neighbor \(QS16_{1}^{(0)}\).
Through the border \(QS85_{2}^{(1)}\), by means of a bifurcation of type \(C\), we reach the neighbor \(QS25_{3}^{(0)}\).
Through the border \(QS19_{3}^{(1)}\), by means of a bifurcation of type \(D\), we reach the neighbor \(QS19_{1}^{(0)}\).
Through the border \(QS19_{1}^{(1)}\), by means of a bifurcation of type \(D\), we reach the neighbor \(QS19_{3}^{(0)}\).
Through the border \(QS31_{10}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS5_{5}^{(0)}\).
Through the border \(QS31_{11}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS5_{5}^{(0)}\).
Through the border \(QS31_{12}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS5_{6}^{(0)}\).
Through the border \(QS31_{14}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS5_{7}^{(0)}\).

Comments

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 \(QS19_{2}^{(0)}\) could potentially exhibit up to three limit cycles (or compound double/triple arrangements) bifurcating from the focus.