$QS99_{4}^{(2)}$

Description

Topological configuration of singularities: $sn,a;(1,1)SN,S,N$

Phase Portrait

Topological Invariants

TCSP	Fin Sep	Inf Sep
$99$	$41$	$211111$

Example

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.

The phase portrait appears in the following papers

With name $U^2_AC,20$ in {J. C. Artés, M. C. Mota and A. C. Rezende}, Structurally unstable quadratic vector fields of codimension two: families possessing a finite saddle-node and an infinite saddle-node, Electron. J. Qual. Theory Differ. Equ. { bf 2021}, Paper No. 35, 89 pp.; MR4252667
With names $1S6$, $1S9$, $1S10$ and $1.10L1$ in {J. C. Artés and C. Trullàs}, Quadratic Differential Systems with a Weak Focus of First-Order and a Finite Saddle-Node, {International Journal of Bifurcation and Chaos, Vol. 36, No. 6 (2026) 2630013 (99 pages)}Note (for name $1S9$): The system has 1 limit cycle.Note (for name $1S10$): The system has $2$ limit cycles.Note (for name $1.10L1$): The system has $d$ limit cycles.
With names $2S03$, $2S04$, $2S14$ and $2.10L1$ 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 $2S03$): The system has 1 limit cycle.Note (for name $2S14$): The system has $2$ limit cycles.Note (for name $2.10L1$): The system has $d$ limit cycles.
With name $cn25 Fig 2.44$ in {X. Huang}, Qualitative analysis or certain nonlinear differential equations, {Ph.D. U. Delft}, (1996).
With names $1.2L3$ and $1.2L4$ 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 $1.2L4$): The system has 1 limit cycle.
With name $1.2L3$ 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 $cn21 Fig. 15$ 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 $V78$, $V80$, $V107$, $V109$, $10S1$ and $10S2$ in {J. C. Artés, M. C. Mota and A. C. Rezende}, Quadratic differential systems with a finite saddle-node and an infinite saddle-node $(1, 1)SN$ - $({ rm B)$}, Internat. J. Bifur. Chaos Appl. Sci. Engrg. { bf 31} (2021), no.~9, Paper No. 2130026, 110 pp.; MR4291723Note (for name $V78$): The system has 1 limit cycle.Note (for name $V107$): Invariant 7 must be a numberNote (for name $V107$): The system has $2$ limit cycles.Note (for name $V109$): The system has $3$ limit cycles.Note (for name $10S1$): The system has $d$ limit cycles.Note (for name $10S2$): The system has $d+1$ limit cycles.

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.

\(QS99_{4}^{(2)}\)

Description

Example

The phase portrait appears in the following papers

Comments