\(QS23_{1}^{(1)}\)
Description
Phase Portrait
Topological Invariants
| TCSP | Fin Sep | Inf Sep |
|---|---|---|
| \(23\) | \(40\) | \(11\) |
Example
The quadratic differential system
\[\begin{cases} \dot{x} = y+3 \, x^{2}-10 \, x \, y \\ \dot{y} = x^{2}+2 \, a \, x \, y+(a^{2}-1) \, y^{2}+d+e \, y \end{cases}\]
with parameters: \(e = 0.00, \quad d = 0.0000000001, \quad a = -1.679\)
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 name \(6\) 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; MR4286006
- With name \(7S1\) in {J. C. Artés, Hebai Chen, L. M. Ferrer and Man Jia}, Quadratic vector fields in class $I$, Dyn. Syst. { bf 40} (2025), no.~2, 191--222; MR4906437
- With name \(P20\) 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.
- With name \(A7S3\) in {J. C. Artés, L. Cairó and J. Llibre}, Phase portraits of the family IV of the quadratic polynomial differential systems, Qual. Theory Dyn. Syst. { bf 24} (2025), no.~2, Paper No. 66, 34 pp.; MR4860323
- With name \(Fig. 28\) in {J. Llibre and C. Pantazi}, Global phase portraits of the quadratic systems having a singular and irreducible invariant curve of degree 3. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 33 (2023), no. 1, Paper No. 2350003, 54 pp.
- With name \(63\) in {J. Llibre and X. Zhang}, Topological phase portraits of planar semi-linear quadratic vector fields, Houston J. Math. { bf 27} (2001), no.~2, 247--296; MR1874098
- With name \(U^1_{D1}\) 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.
- With name \(7S1\) 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 \(Fig 3 b\) 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 names \(Fig 1.7 k\) and \(Fig 2.20 b\) 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 \(Fig 1.7 k\)): missed orbit
- With name \(P60\) 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.
Bifurcations in codimension 0
- Bifurcation interface between the codimension 0 portrait \(QS23_{1}^{(0)}\) and \(QS23_{1}^{(0)}\).