\(QS25_{1}^{(0)}\)

Description

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

Phase Portrait

Topological Invariants

TCSP	Fin Sep	Inf Sep
\(25\)	\(42\)	\(211110\)

Example

The quadratic differential system

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

with parameters: \(e = 0.1\)

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 \(AA^s_4\) and \(AA^n_4\) 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 \(11S5\) 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 names \(27\) and \(30\) 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 \(11\) 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 \(A15\) 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 \(PP13\) in {J. Llibre and H. X. Ou}, Quadratic systems with two invariant real straight lines and an invariant parabola, Electron. J. Qual. Theory Differ. Equ. { bf 2025}, Paper No. 66, 54 pp.; MR5018064
With name \(1S01\) 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 \(Ric. 17\) 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 \(Fig 5.24 S^2_{9,1}\) 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 \(Fig 5.24 S^2_{9,1}\)): The system has 1 limit cycle.
With name \(V3\) 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.; MR3566296
With name \(S^2_{9,1}\) 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 \(V14\) 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 name \(Fig10 7\) 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 \(Fig 1.3 j\), \(Fig 1.7 a\), \(Fig 1.7 s\), \(Fig 1.21 b\) and \(Fig 1.31 a\) 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 \(Fig 1.7 s\)): wrong picture, it has 5 infinite singularities
With name \(V1\) 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 \(QS27_{1}^{(1)}\), by means of a bifurcation of type \(B\), we reach the neighbor \(QS23_{1}^{(0)}\).
Through the border \(QS27_{2}^{(1)}\), by means of a bifurcation of type \(B\), we reach the neighbor \(QS23_{1}^{(0)}\).
Through the border \(QS87_{1}^{(1)}\), by means of a bifurcation of type \(C\), we reach the neighbor \(QS15_{1}^{(0)}\).
Through the border \(QS45_{1}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS13_{1}^{(0)}\).
Through the border \(QS25_{3}^{(1)}\), by means of a bifurcation of type \(D\), we reach the neighbor \(QS25_{1}^{(0)}\).
Through the border \(QS25_{4}^{(1)}\), by means of a bifurcation of type \(D\), we reach the neighbor \(QS25_{2}^{(0)}\).
Through the border \(QS38_{4}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS10_{2}^{(0)}\).
Through the border \(QS38_{6}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS10_{3}^{(0)}\).
Through the border \(QS38_{7}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS10_{3}^{(0)}\).
Through the border \(QS38_{10}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS10_{5}^{(0)}\).
Through the border \(QS38_{11}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS10_{5}^{(0)}\).
Through the border \(QS38_{12}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS10_{5}^{(0)}\).
Through the border \(QS38_{14}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS10_{6}^{(0)}\).
Through the border \(QS38_{20}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS10_{9}^{(0)}\).
Through the border \(QS38_{21}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS10_{10}^{(0)}\).
Through the border \(QS38_{24}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS10_{12}^{(0)}\).
Through the border \(QS38_{30}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS10_{14}^{(0)}\).
Through the border \(QS38_{31}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS10_{14}^{(0)}\).
Through the border \(QS38_{32}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS10_{15}^{(0)}\).
Through the border \(QS38_{33}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS10_{16}^{(0)}\).
Through the border \(QS85_{1}^{(1)}\), by means of a bifurcation of type \(C\), we reach the neighbor \(QS19_{1}^{(0)}\).
Through the border \(QS85_{3}^{(1)}\), by means of a bifurcation of type \(C\), we reach the neighbor \(QS19_{3}^{(0)}\).

Comments

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 \(QS25_{1}^{(0)}\) could potentially exhibit up to two limit cycles (or a compound double limit cycle) bifurcating from the focus.