QS74₅⁽¹⁾

Phase Portrait

Topological Invariants

TCSP	Fin Sep	Inf Sep
$74$	$432$	$111110$

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: $a = 2.1, \quad e = -1, \quad d = -3, \quad m = 1, \quad g = 1$

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 $Fig 10 3$ in {Y. Bolaños, J. Llibre and C. Valls}, Phase portraits of quadratic Lotka-Volterra systems with a Darboux invariant in the Poincaré disc, Commun. Contemp. Math. { bf 16} (2014), no.~6, 1350041, 23 pp.; MR3277950

With name $4.4b$ 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.Note (for name $4.4b$): wrong witdh

With names $4,4b$, $3,4(h2)$ and $3,5(h2)$ in {D. Schlomiuk and N. Vulpe}, Global classification of the planar Lotka--Volterra differential systems according to their configurations of invariant straight lines, emph{J. Fixed Point Theory Appl.}, { bf 8}, no. 1 (2010), 177--245.

With name $Fig1.e$ in {J. Llibre and C. Valls}, Global dynamics of a system coming from the study of a static star, Differ. Equ. Dyn. Syst. { bf 32} (2024), no.~2, 607--617; MR4721747

With name $Fig. 13$ 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 $U^1_{C22}$ 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 names $bn13 Fig 2.45$, $fn11 Fig 2.47$ and $an44 Fig 2.51$ in {X. Huang}, Qualitative analysis or certain nonlinear differential equations, {Ph.D. U. Delft}, (1996).

With names $bn13 Fig. 16$, $fn11 Fig. 18$ and $an28 Fig. 22$ 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?).

With names $Fig3.1 B1$, $Fig3.1 B4$, $Fig3.1 C1$, $Fig3.1 C2$, $Fig3.3 II$, $Fig3.3 V$ and $Fig3.3 VI$ in {J. W. Reyn}, Phase portraits of a quadratic system of differential equations occurring frequently in applications, emph{Nieuw Arch. Wisk. (4)}, textbf{5}, no. 2 (1987), 107--151.

\(QS74_{5}^{(1)}\)

Description

Example

The phase portrait appears in the following papers

Bifurcations in codimension 0