\(QS8_{2}^{(1)}\)

Description

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

Phase Portrait

Topological Invariants

TCSP	Fin Sep	Inf Sep
\(8\)	\(4411\)	\(31\)

The quadratic differential system

\[\begin{cases} \dot{x} = 1-x^{2}-2 \, y^{2} \\ \dot{y} = y \, (30 \, x+29)/10 \end{cases}\]

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

With name \(1\) in {A. Ferragut, J. D. García-Saldaña and C. Valls}, Phase portraits of Abel quadratic differential systems of second kind with symmetries, Dyn. Syst. { bf 34} (2019), no.~2, 301--333; MR3941199
With name \(U^1_{D3}\) 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 \(V1\) 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 \(Fig. 1 1\), \(Fig. 1 3\), \(Fig. 3 1\), \(Fig. 3 2\), \(Fig. 3 6\) and \(Fig. 3 7\) in {P. C. Carri\~ao, M. E. S. Gomes and A. A. G. Ruas}, Planar quadratic vector fields with finite saddle connection on a straight line (convex case), Qual. Theory Dyn. Syst. { bf 6} (2005), no.~2, 187--204; MR2420856Note (for name \(Fig. 3 2\)): The system has limit cycles with distribution \((1,0)\).Note (for name \(Fig. 3 6\)): The system has limit cycles with distribution \((1,0)\).

Bifurcation interface between the codimension 0 portrait \(QS8_{1}^{(0)}\) and \(QS8_{1}^{(0)}\).

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 \(QS8_{2}^{(1)}\) could potentially exhibit an additional limit cycle bifurcating from the focus.