\(QS10_{13}^{(0)}\)

Description

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

Phase Portrait

Topological Invariants

TCSP	Fin Sep	Inf Sep
\(10\)	\(4422\)	\(321101\)

Example

The quadratic differential system

\[\begin{cases} \dot{x} = y+3 \, x^{2}+11 \, x \, y/8 \\ \dot{y} = e^{2} \, x/5-e \, y+x^{2}+3 \, x \, y+5 \, y^{2}/4+a \, (y+3 \, x^{2}+11 \, x \, y/8) \end{cases}\]

with parameters: \(e = 0.2, \quad a = -0.017\)

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 \(S^2_{10,13}\) in {J. C. Artés, R. E. Kooij and J. Llibre}, Structurally stable quadratic vector fields, Mem. Amer. Math. Soc. { bf 134} (1998), no.~639, viii+108 pp.; MR1432139

Neighbours of Codimension 1

Through the border \(QS11_{15}^{(1)}\), by means of a bifurcation of type \(B\), we reach the neighbor \(QS8_{4}^{(0)}\).
Through the border \(QS11_{16}^{(1)}\), by means of a bifurcation of type \(B\), we reach the neighbor \(QS8_{4}^{(0)}\).
Through the border \(QS38_{26}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS25_{2}^{(0)}\).
Through the border \(QS38_{27}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS25_{2}^{(0)}\).
Through the border \(QS38_{28}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS25_{3}^{(0)}\).
Through the border \(QS10_{4}^{(1)}\), by means of a bifurcation of type \(D(a)\), we reach the neighbor \(QS10_{7}^{(0)}\).
Through the border \(QS10_{20}^{(1)}\), by means of a bifurcation of type \(D\), we reach the neighbor \(QS10_{14}^{(0)}\).

Comments

This particular phase portrait holds a significant personal and historical value for me. It was the final codimension-zero configuration discovered within the global classification. Moreover, it occupies an exceptionally small region of the parameter space, and analyzing its neighboring systems, they proved also to be remarkably challenging. Furthermore, it remains as the unique non-trivial structurally stable phase portrait for which a bifurcation from a system with a center has not yet been rigorously demonstrated.