\(QS1_{3}^{(1)}\)

Description

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

Phase Portrait

Topological Invariants

TCSP	Fin Sep	Inf Sep
\(1\)	\(4441\)	\(221211\)

Example

The quadratic differential system

\[\begin{cases} \dot{x} = -1-5 \, x/2+x^{2}+2 \, x \, y \\ \dot{y} = e \, (x \, y-1-a)+1+5 \, y/2-2 \, x \, y-y^{2}+c \, (-1-5 \, x/2+x^{2}+2 \, x \, y) \end{cases}\]

with parameters: \(e = 0.1, \quad a = 0.1, \quad c = -0.028\)

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 name \(U^1_{D17}\) 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.Note (for name \(U^1_{D17}\)): The portrait obtained in Fig 6.41 is not D17 but D16. In fact the existence of D17 is not really proved in this paper, but the example given here using a rotated vector field works as proof.
With name \(7S4\) 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 name \(e^e\) in {A. Zegeling}, Quadratic systems with three saddles and one antisaddle, Delft University of Technology, Faculty of Technical Mathematics and Informatics, Report 80 (1989).

Bifurcations in codimension 0

Bifurcation interface between the codimension 0 portrait \(QS1_{2}^{(0)}\) and \(QS1_{3}^{(0)}\).

Comments

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.