\(QS10_{14}^{(0)}\)
Description
Phase Portrait
Topological Invariants
| TCSP | Fin Sep | Inf Sep |
|---|---|---|
| \(10\) | \(4432\) | \(211210\) |
Example
The quadratic differential system
\[\begin{cases} \dot{x} = -x \, (1-x+2 \, y)-e \, y \\ \dot{y} = y \, (1-y)+a \, (-x \, (1-x+2 \, y)-e \, y) \end{cases}\]
with parameters: \(e = 0.5, \quad a = 0\)
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 \(PP45a\), \(PP45b\) and \(PP45c\) in {J. C. Artés, J. Llibre and Huaxin Ou}, Quadratic systems with two invariant straight lines and an invariant hyperbola, {Preprint} (2026).
- With name \(24\) 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 \(PP12\) 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 \(PP12\) in {J. Llibre and H. X. Ou}, Quadratic systems with two invariant real straight lines and an invariant hyperbola, {Preprint} (2026).
- With names \(Fig. 23\) and \(Fig. 24 f\) 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 \(S17\) in {M. C. Mota, R. D. S. Oliveira and A. M. Travaglini}, The interplay among the topological bifurcation diagram, integrability and geometry for the family { bf QSH(D)}, Geom. Dedicata { bf 217} (2023), no.~6, Paper No. 95, 42 pp.; MR4631488
- With name \(Ric. 16\) 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 \(Fig 21 P2K\) in {A. M. Travaglini}, Integrability and geometryof quadratic differential systems with invariant hyperbolas, {Ph. D., Uni. de Sao Paulo} (2026).
- With name \(S^2_{10,14}\) 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
Neighbours of Codimension 1
- Through the border \(QS11_{16}^{(1)}\), by means of a bifurcation of type \(B\), we reach the neighbor \(QS8_{4}^{(0)}\).
- Through the border \(QS11_{18}^{(1)}\), by means of a bifurcation of type \(B\), we reach the neighbor \(QS8_{4}^{(0)}\).
- Through the border \(QS76_{8}^{(1)}\), by means of a bifurcation of type \(C\), we reach the neighbor \(QS1_{2}^{(0)}\).
- Through the border \(QS38_{30}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS25_{1}^{(0)}\).
- Through the border \(QS38_{31}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS25_{1}^{(0)}\).
- Through the border \(QS38_{29}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS25_{3}^{(0)}\).
- Through the border \(QS10_{3}^{(1)}\), by means of a bifurcation of type \(D\), we reach the neighbor \(QS10_{10}^{(0)}\).
- Through the border \(QS10_{6}^{(1)}\), by means of a bifurcation of type \(D\), we reach the neighbor \(QS10_{12}^{(0)}\).
- Through the border \(QS10_{20}^{(1)}\), by means of a bifurcation of type \(D\), we reach the neighbor \(QS10_{13}^{(0)}\).
- Through the border \(QS10_{21}^{(1)}\), by means of a bifurcation of type \(D\), we reach the neighbor \(QS10_{16}^{(0)}\).
- Through the border \(QS74_{13}^{(1)}\), by means of a bifurcation of type \(C\), we reach the neighbor \(QS5_{6}^{(0)}\).