Additional files for the article

Phase portraits of quadratic differential systems with a weak focus and a $\binom{0}{2} SN$

Joan C. Artés and Laurent Cairó

As part of the project to obtain all topologically distinct phase portraits of the quadratic differential systems, we present here another family of codimension 2, {\bf Qwf1SN02}, which corresponds to cases with a weak focus of order 1 and an infinite saddle-node formed by the coalescence of an infinite singularity with another infinite singularity. As usual in these cases, the normal form that contains these systems also contains other more degenerate ones: we therefore study the complete closure $\overline { \textbf{Qwf1SN02}}$. The bifurcation diagram for this class, constructed in the appropriate parameter space, which is the 3-dimensional real projective space, is quite rich in its complexity and gives 295 subsets with 159 phase portraits for $\overline {\textbf{Qwf1SN02}}$, $25$ having limit cycles and $10$ with centers. The phase portraits are always represented in the Poincaré disc. The bifurcation set is formed by an algebraic set of finite or infinite singularity bifurcations and by a non-algebraic set of points corresponding to global separators that have separatrix connections or double limit cycles. Algebraic invariants are needed to construct the algebraic part of the bifurcation set; symbolic calculations are needed to process certain fairly complex invariants; and numerical calculations are needed to determine the position of the non-algebraic bifurcation set of connections.

The complete three dimensional bifurcation diagram cannot be viewed by projection on the paper due to its complexity. There are some slices where the number of phase portraits is so large that it has not been possible to display all of them on the slice as this would yield a very crowded picture with too small portraits.

Thus, we have decided to open a web page where we include all those files that one cannot include in the paper, so that the reader may download freely for better understanding and possibly further research. We will keep this page for as long as possible but presumably not indefinitely.

We do not add here the paper due to the copyright, but we place only the extra files, both in the original form (either Mathematica or Corel Draw) with some helpful comments. The original form is given in zipped form to reduce space.

Please note that some of the files are very large and may take a lot to download. They are consequently even bigger once unzipped. We have added the size of the files both zipped and unzipped so to warn you before downloading.

Mathematica file with the comitants (15318Kb/125815Kb)

Corel file containing the topologically distinct phase portraits   (298Kb/862Kb).

P4 files containing all the P4 files with an example of every phase portrait. The unzip generates a tree of directories for the different slices  (298Kb).

Geogebra file containing real pictures of the algebraic bifurcations in all slices  (59Kb).

Mathematica file with the 3-dimensional pictures of the slices (778Kb/1054Kb)