$QS75_{2}^{(2)}$

Description

Topological configuration of singularities: $s,a,a;(1,1)SN,(0,2)SN$

Phase Portrait

Topological Invariants

TCSP	Fin Sep	Inf Sep
$75$	$441$	$2111$

The quadratic differential system

\[\begin{cases} \dot{x} = P_x(x,y) \\ \dot{y} = P_y(x,y) \end{cases}\]

has the following phase portrait done with P4.

With name $U^2_{BC,05}$ in {J. C. Artés}, Systems of class BC2, {Preprint} (2026).
With names $5S08$ and $5S09$ 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 $5S09$): The system has limit cycles with distribution $(0,1)$.
With names $bn03 Fig 2.34$, $fn03 Fig 2.36$, $an20 Fig 2.40$ and $an21 Fig 2.40$ in {X. Huang}, Qualitative analysis or certain nonlinear differential equations, {Ph.D. U. Delft}, (1996).
With name $1.5L2$ 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}.
With names $an01 Fig. 5$, $bn03 Fig. 6$, $fn03 Fig. 8$ and $an12 Fig. 12$ 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?).

This phase portrait appears in J. C. Artés, J. Llibre and D. Schlomiuk (emph{ Internat. J. Bifur. Chaos Appl. Sci. Engrg.}, textbf{16} (2006), {3127--3194}) featuring a weak focus of second order. Given that the portrait is of codimension 1, hyperbolic limit cycles can be generated without breaking its other unstable features. However, multiple limit cycle configurations are not guaranteed, as they might be incompatible with the pre-existing unstable properties of the system.