\(QS5_{6}^{(1)}\)

Description

Topological configuration of singularities: \(s,a,a,a;S,S,N\)

Phase Portrait

Topological Invariants

TCSP	Fin Sep	Inf Sep
\(5\)	\(4321\)	\(110110\)

The quadratic differential system

\[\begin{cases} \dot{x} = x \, (-1+x-4 \, y) \\ \dot{y} = 3 \, y \, (1+3 \, x/2-y)+e \end{cases}\]

with parameters: \(e = 0.1\)

has the following phase portrait done with P4. If you want, you may download the P4 file here.

With name \(3,1(b2)\) 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 \(Fig2.c\) 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 \(Fig. 38 e\) 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 \(U^1_{D35}\) 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 \(V16\) in {J. C. Artés, J. Llibre and D. Schlomiuk}, The geometry of quadratic polynomial differential systems with a weak focus and an invariant straight line, emph{ Internat. J. Bifur. Chaos Appl. Sci. Engrg.}, textbf{20}, no. 11 (2010), 3627--3662.
With names \(Fig 2B VI\), \(Fig 2B VIIa\), \(Fig 2D Xic\), \(Fig 2D XII\) and \(Fig 2D XIII\) 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.

Bifurcation interface between the codimension 0 portrait \(QS5_{2}^{(0)}\) and \(QS5_{3}^{(0)}\).

This phase portrait appears in J. C. Artés, J. Llibre and D. Schlomiuk (emph{ Internat. J. Bifur. Chaos Appl. Sci. Engrg.}, textbf{20}, no. 11 (2010), 3627--3662) featuring a weak focus of first order. Consequently, a configuration structurally equivalent to \(QS5_{6}^{(1)}\) could potentially exhibit an additional limit cycle bifurcating from the focus.