The bifurcation diagram for this class, done in the adequate parameter space which is the 3-dimensional real projective space, is quite rich in its complexity and yields 373 subsets with 126 phase portraits for the whole class, 95 of them specifically for quadratic vector fields with a weak focus of second order. Thus, the paper contains pictures with all the phase portraits, sketches of the bifurcations surfaces in three dimensions and slices of the parameter space where one can see curves obtained by intersecting the bifurcations surfaces with the slices. On some slices all phase portraits are included, on others we only drew their labels of these portraits which can be found in the pictures.

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 they all 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 containing most of the calculations (3.514Kb/16.677Kb)

From this file, we recommend to open the Section "Study of the 3-dimensional region", subsection "Complete bifurcation map" and make a mathematica film of the set of slices, either on the real plane, or on the compactified plane.

Mathematica file containing a 3-dimensional view of bifurcation surface S₁  (61.172Kb/296.122Kb).

Mathematica file containing a 3-dimensional view of bifurcation surface S₂  (3.863Kb/18.116Kb).

Mathematica file containing a 3-dimensional view of bifurcation surface S₅ (4.150Kb/25.446Kb).

Mathematica file containing a 3-dimensional view of bifurcation surface S₆  (4.247Kb/21.678Kb).

In each one of these files, one can see a series of 3-dimensional images of the different bifurcation surfaces from different viewpoints. They are done with the minimum precision, enough to be understood but the user can freely change the parameter k on top of the file to redo the computations with higher precision if he so decides. Note but that the amount of computing time may be huge. Also a film with the series of images is recommended to view the surfaces from different view points in a row.

Corel Draw files containing 2-dimensional views of the slices without phase portraits  (400Kb/643Kb). They include a decomposition in 4 pieces of the main slice.

JPG files containing 2-dimensional views of the slices without phase portraits  (1.053Kb/1.156Kb). Nice to see all of them in a row with  LVIEW or similar program.

Corel Draw files containing 2-dimensional views of the slices with phase portraits  (1.044Kb/1.281Kb). They include a decomposition in 4 pieces of the main slice. Attention should be paid as these four files and also the one corresponding to a=0 (pslic24p) are very large and must be printed in format greater than A4 (up to A0).

All the pictures of the bifurcation diagram contained in the last three sets of files are not quantitatively correct but they are qualitatively right, and the shapes of the bifurcations have been modified in order to make the subsets of the parameter space more visible, otherwise some of them would be too small to be seen.

We have also produced a set of images quantitatively accurate, and close to the most interesting part of the bifurcation diagram (see Figure 56 from the paper). They have been calculated numerically and included in a sequence of Excel sheets.

Excel files containing 2-dimensional views of the part of the slices (1.002Kb/1.156Kb). Nice to see them in a row.