Additional files for the article

Quadratic differential systems with a weak focus of first order and a finite saddle-node

Joan C. Artés and Carles Trullàs

Planar quadratic differential systems occur in many areas of applied mathematics. Although more than one thousand papers were written on these systems, a complete understanding of this class is still missing. Classical problems, and in particular, Hilbert's 16th problem, are still open for this class. One of the goals of recent researchers is the topological classification of quadratic systems. As this attempt is not possible in the whole class due to the large number of parameters (twelve, but, after affine transformations and time rescaling, we arrive at families with five parameters, which is still a large number), many subclasses are considered and studied. Specific characteristics are taken into account and this implies a decrease in the number of parameters, which makes possible the study.

In this article we perform a global study (modulo islands) of the class Qwf1sn which is the closure within real quadratic differential systems, of the family Qwf1sn of all such systems which have a weak focus of first order and a finite saddle-node. The bifurcation diagram for this class, done in the adequate parameter space which is the 3-dimensional real projective space RP^3, is quite rich in its complexity since yields 399 subsets with 192 topologically distinct phase portraits for the clousure of Qwf1sn, 146 of which have a representative in Qwf1sn. It can be shown that some of these parts have at least two limit cycles.

The phase portraits are always represented in the Poincaré disc. The bifurcation set is formed by an algebraic set of bifurcations of singularities, finite or infinite and by a set of bifurcations which we suspect to be analytic corresponding to global separatrices which have connections, or double limit cycles. Algebraic invariants were needed to construct the algebraic part of the bifurcation set, symbolic computations to deal with some quite complex invariants and numerical calculations to determine the position of the analytic 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. We have produced complete pictures for those slices (or part of them) each one on one sheet, but some need an A0 format page to see the phase portraits at reasonably well.

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) and the more compact JPG,  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 (9355Kb)

PDF file containing the phase portraits of all the regions (this includes repetitions)  (3491Kb).

Zip file containing all the P4 files with an example of every phase portrait  (39Kb).

PDF file containing real pictures of the algebraic bifurcations in all slices  (1254Kb).

Mathematica file with the 3-dimensional pictures of the slices (596Kb)