QS87₁⁽¹⁾

Phase Portrait

Topological Invariants

TCSP	Fin Sep	Inf Sep
$87$	$4$	$211110$

Example

\[\begin{cases} \dot{x} = x+a \, y+e \, y^{2}+d \, (m \, x+g \, y+x \, y) \\ \dot{y} = m \, x+g \, y+x \, y \end{cases}\]

with parameters: $a = 2, \quad e = 1, \quad d = -2, \quad m = 2, \quad g = -3$

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 name $A1S1$ 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 names $21$ and $46$ in {A. Ferragut and C. Valls}, Phase portraits of Abel quadratic differential systems of the second kind, Dyn. Syst. { bf 33} (2018), no.~4, 581--601; MR3869849

With name $68$ 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 names $Fig1a$, $Fig2a (I)$ and $Fig2a (III)$ in {J. W. Reyn}, Phase portraits of quadratic systems with finite multiplicity one, Nonlinear Anal. { bf 28} (1997), no.~4, 755--778; MR1420390

With name $U^1_{C4}$ 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 $hp01 Fig 2.62$ in {X. Huang}, Qualitative analysis or certain nonlinear differential equations, {Ph.D. U. Delft}, (1996).

With name $1S1$ 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 names $h19$ and $E30$ in {B. Coll, A. Gasull and J. Llibre}, Quadratic systems with a unique finite rest point, emph{Publ. Mat.}, textbf{32} (1988), 199--259.

With name $Fig10 8$ 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 name $hp01 Fig. 33$ in {J. W. Reyn and X. H. Huang}, Separatrix configurations of quadratic systems with finite multiplicity three and a $M^0_{1,1$ type of critical point at infinity}, Report U. Delft (1997?).

\(QS87_{1}^{(1)}\)

Description

Example

The phase portrait appears in the following papers

Bifurcations in codimension 0