\(QS25_{3}^{(0)}\)

Description

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

Phase Portrait

Topological Invariants

TCSP	Fin Sep	Inf Sep
\(25\)	\(41\)	\(311101\)

Example

The quadratic differential system

has the following phase portrait done with P4. If you want, you may download the P4 file here. Since the image is not clear enough, we have added a ZOOM of it.

The phase portrait appears in the following papers

• With names \(AA^s_3\) and \(AA^n_3\) 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 \(2S4\) and \(11S3\) 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 \(11S3\)): The system has 1 limit cycle.
• With names \(1S03\) and \(1S04\) 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 \(1S03\)): The system has 1 limit cycle.
• With names \(Fig 5.21 S^2_{9,3}\) and \(Fig 5.129 S^2_{9,3}\) 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 \(Fig 5.21 S^2_{9,3}\)): The system has 1 limit cycle.Note (for name \(Fig 5.129 S^2_{9,3}\)): The system has 1 limit cycle.
• With names \(V5\) and \(V9\) 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 \(V9\)): The system has 1 limit cycle.
• With name \(S^2_{9,3}\) 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 \(V1\) and \(V8\) 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 \(V8\)): The system has 1 limit cycle.
• With name \(V1\) 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 \(Fig 1.3 a\), \(Fig 1.3 b\), \(Fig 1.3 c\), \(Fig 1.3 d\), \(Fig 1.7 e\) and \(Fig 1.7 f\) 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 \(Fig 1.3 b\)): The system has \(d\) limit cycles.Note (for name \(Fig 1.3 c\)): The system has \(2\) limit cycles.Note (for name \(Fig 1.3 d\)): The system has 1 limit cycle.Note (for name \(Fig 1.7 e\)): The system has 1 limit cycle.
• With name \(V11\) 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 \(V9\) 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 \(QS27_{5}^{(1)}\), by means of a bifurcation of type \(B\), we reach the neighbor \(QS23_{1}^{(0)}\).
• Through the border \(QS27_{6}^{(1)}\), by means of a bifurcation of type \(B\), we reach the neighbor \(QS23_{1}^{(0)}\).
• Through the border \(QS45_{3}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS13_{1}^{(0)}\).
• Through the border \(QS25_{5}^{(1)}\), by means of a bifurcation of type \(D\), we reach the neighbor \(QS25_{2}^{(0)}\).
• Through the border \(QS25_{2}^{(1)}\), by means of a bifurcation of type \(D\), we reach the neighbor \(QS25_{2}^{(0)}\).
• Through the border \(QS38_{1}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS10_{1}^{(0)}\).
• Through the border \(QS38_{2}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS10_{1}^{(0)}\).
• Through the border \(QS38_{3}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS10_{1}^{(0)}\).
• Through the border \(QS38_{5}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS10_{2}^{(0)}\).
• Through the border \(QS38_{9}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS10_{4}^{(0)}\).
• Through the border \(QS38_{15}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS10_{7}^{(0)}\).
• Through the border \(QS38_{18}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS10_{8}^{(0)}\).
• Through the border \(QS38_{25}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS10_{12}^{(0)}\).
• Through the border \(QS38_{28}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS10_{13}^{(0)}\).
• Through the border \(QS38_{29}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS10_{14}^{(0)}\).
• Through the border \(QS85_{2}^{(1)}\), by means of a bifurcation of type \(C\), we reach the neighbor \(QS19_{2}^{(0)}\).

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 \(QS25_{3}^{(0)}\) could potentially exhibit up to three limit cycles (or compound double/triple arrangements) bifurcating from the focus.