QS76₁⁽¹⁾

Phase Portrait

Topological Invariants

TCSP	Fin Sep	Inf Sep
$76$	$441$	$411111$

Example

\[\begin{cases} \dot{x} = y \, (1+x)+a \, x^{2} \\ \dot{y} = d+e \, y+x^{2}-2 \, x \, y \end{cases}\]

with parameters: $a = -0.1, \quad e = -0.1, \quad d = -0.000001$

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 names $V08$, $V14$, $V61$ and $10S3$ in {J. C. Artés and L. Cairó}, Phase portraits of quadratic differential systems with a weak focus and a (1,1) SN, {Preprint} (2026).Note (for name $V14$): The system has 1 limit cycle.Note (for name $V61$): The system has $2$ limit cycles.Note (for name $10S3$): The system has $d$ limit cycles.

With name $U^1_{C5}$ 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 $ap01 Fig 2.63$ in {X. Huang}, Qualitative analysis or certain nonlinear differential equations, {Ph.D. U. Delft}, (1996).

With names $1S2$ and $1S3$ in {J. C. Artés, J. Llibre and D. Schlomiuk}, The geometry of quadratic differential systems with a weak focus of second order, emph{ Internat. J. Bifur. Chaos Appl. Sci. Engrg.}, textbf{16} (2006), {3127--3194}.Note (for name $1S3$): The system has 1 limit cycle.

With name $1S2$ in {J. Llibre and D. Schlomiuk}, Geometry of quadratic differential systems with a weak focus of third order, emph{Canad. J. of Math.}, textbf{56}, no. 2 (2004), 310--343.

With name $ap01 Fig. 34$ 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?).

Comments

This phase portrait appears in J. Llibre and D. Schlomiuk (emph{Canad. J. of Math.}, textbf{56}, no. 2 (2004), 310--343) featuring a weak focus of third order. Given that the portrait is of codimension 1, hyperbolic limit cycles can be generated without breaking its other unstable features. However, multiple or compound limit cycle configurations are not guaranteed, as they might be incompatible with the pre-existing unstable properties of the system.

\(QS76_{1}^{(1)}\)

Description

Example

The phase portrait appears in the following papers

Bifurcations in codimension 0

Comments