\(QS16_{1}^{(0)}\)
Description
Phase Portrait
Topological Invariants
| TCSP | Fin Sep | Inf Sep |
|---|---|---|
| \(16\) | \(11\) | \(11\) |
Example
The quadratic differential system
\[\begin{cases} \dot{x} = 2 \, x \, y+a \, (-1+2 \, x^{2}-y^{2}) \\ \dot{y} = 1-x^{2}+y^{2} \end{cases}\]
with parameters: \(a = 0.2\)
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 \(A V01\) and \(A V04\) 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 \(A V04\)): The system has limit cycles with distribution \((1,0)\).
- With name \(5\) in {J. C. Artés, J. Llibre and N. Vulpe}, Quadratic systems with a rational first integral of degree 2: A complete classification in the coefficient space $ R^{12$}, emph{Rend. Circ. Mat. Palermo}, textbf{56}, no. 3 (2007), 417--444.
- With names \(1\) and \(2\) 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 \(1\)): The system has limit cycles with distribution \((1,0)\).
- With name \(A1=A3\) in {J. C. Artés and J. Llibre}, Phase portraits for quadratic systems having a focus and one anti-saddle, emph{Rocky Mountain J. Math.}, textbf{24} (1994), 875--889.
- With name \(AA^n_2\) 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 \(2S3\) and \(11S12\) 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 \(11S12\)): The system has limit cycles with distribution \((1,0)\).
- With name \(3\) in {A. Belfar and R. Benterki}, Qualitative dynamics of quadratic systems exhibiting reducible invariant algebraic curve of degree 3, Palest. J. Math. { bf 11} (2022), Special Issue II, 1--12; MR4447008
- With name \(Chap 2 14\) in {B. Imane and B. Souad}, Global phase portraits of quadratic differential systems exhibiting an invariant algebraic curve or an algebraic cubic first integral, {Ph.D. Universite Mohamed el Bachir}, (2020).
- With names \(3\), \(6\) and \(13\) in {R. Benterki and J. Llibre}, Phase portraits of quadratic polynomial differential systems having as solution some classical planar algebraic curves of degree 4, Electron. J. Differential Equations { bf 2019}, Paper No. 15, 25 pp.; MR3919655
- With name \(2\) in {R. Benterki and A. Belfar}, Global phase portraits of quadratic polynomial differential systems having as solution some classical planar algebraic curves of degree 6, Tatra Mt. Math. Publ. { bf 81} (2022), 129--144; MR4519365
- With name \(E2\) in {L. Cairó and J. Llibre}, Phase portraits of quadratic polynomial vector fields having a rational first integral of degree 2. Nonlinear Anal. 67 (2007), no. 2, 327–348.
- With name \(Qin b\) 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 \(Qin b\)): The system has limit cycles with distribution \((1,0)\).
- With names \(82\) and \(119\) 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 \(4.6b\) and \(4.41b\) 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 \(6.3\) and \(5.6\) 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.Note (for name \(6.3\)): missed arrow
- With names \(P10\), \(P11\) and \(P17\) 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 \(P10\)): The system has limit cycles with distribution \((1,0)\).
- With name \(34\) 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 \(R4\) in {J. Llibre and J. Yu}, Global phase portraits of quadratic systems with an ellipse and a straight line as invariant algebraic curves, Electron. J. Differential Equations { bf 2015}, No. 314, 14 pp.; MR3441696Note (for name \(R4\)): separatrix is orbit
- With name \(3\) in {J. Llibre, M. Messias and A. C. Reinol}, Normal forms and global phase portraits of quadratic and cubic integrable vector fields having two nonconcentric circles as invariant algebraic curves, Dynamical Systems, DOI: 10.1080/14689367.2016.1263600
- With name \(J\) 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 name \(7\) in {M. Ndiaye and H. J. Giacomini}, Quadratic systems equivalent by domains to a linear one: global phase portraits, Extracta Math. { bf 15} (2000), no.~1, 97--119; MR1792982
- With names \(1S16\) and \(1S17\) 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 \(1S17\)): The system has limit cycles with distribution \((1,0)\).
- With names \(Fig 5.7 S^2_{4,1}\), \(Fig 5.58 S^2_{4,1}\), \(Fig 5.59 S^2_{4,1}\) and \(Fig 5.104 S^2_{4,1}\) 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 \(Fig 5.7 S^2_{4,1}\)): The system has limit cycles with distribution \((1,0)\).Note (for name \(Fig 5.58 S^2_{4,1}\)): The system has limit cycles with distribution \((1,0)\).Note (for name \(Fig 5.59 S^2_{4,1}\)): The system has limit cycles with distribution \((1,1)\).Note (for name \(Fig 5.104 S^2_{4,1}\)): The system has limit cycles with distribution \((1,0)\).Note (for name \(Fig 5.104 S^2_{4,1}\)): The system has limit cycles with distribution \((1,1)\).
- With name \(S IV 6\) 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.; MR4793966
- With name \(S^2_{4,1}\) 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 \(V49\), \(V50\) and \(V60\) 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 \(V49\)): The system has limit cycles with distribution \((1,0)\).Note (for name \(V50\)): The system has limit cycles with distribution \((1,1)\).
- With name \(V49\) 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 \(V49\)): The system has limit cycles with distribution \((1,0)\).
- With names \(Fig 1.24 h\), \(Fig 1.24 i\) and \(Fig 1.27 a\) 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 \(Fig 1.24 h\)): The system has limit cycles with distribution \((1,0)\).Note (for name \(Fig 1.27 a\)): The system has limit cycles with distribution \((1,1)\).
- With names \(V4\) and \(V6\) 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 \(V6\)): The system has limit cycles with distribution \((1,0)\).
- With name \(2.5L10\) 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.
Neighbours of Codimension 1
- Through the border \(QS84_{1}^{(1)}\), by means of a bifurcation of type \(C\), we reach the neighbor \(QS23_{1}^{(0)}\).
- Through the border \(QS16_{1}^{(1)}\), by means of a bifurcation of type \(D\), we reach the neighbor \(QS16_{1}^{(0)}\).
- Through the border \(QS30_{1}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS3_{1}^{(0)}\).
- Through the border \(QS30_{2}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS3_{1}^{(0)}\).
- Through the border \(QS22_{3}^{(1)}\), by means of a bifurcation of type \(B\), we reach the neighbor \(QS19_{2}^{(0)}\).
- Through the border \(QS22_{4}^{(1)}\), by means of a bifurcation of type \(B\), we reach the neighbor \(QS19_{3}^{(0)}\).
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 \(QS16_{1}^{(0)}\) could potentially exhibit up to three limit cycles (or compound double/triple arrangements) bifurcating from the focus.