Additional files for the article
The geometry of quadratic polynomial differential systems with a weak focus and an invariant straight line
Joan C. Artés, Jaume Llibre and Dana Schlomiuk
In this article we make a global study of
the class $QW^I$ of all real quadratic differential systems which have
a weak focus and invariant straight lines of total multiplicity at
least two. As it turns out, these conditions imply that all the systems
in $QW^I$, having a weak focus at some point necessarily have a weak
focus of order one at that point.
In the clousure of this family we find some systems having a center.
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 151
subsets with 99 phase portraits for the whole class, 73 of them
specifically for quadratic vector fields with a weak focus of first
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 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 containing most of the calculations (927Kb/5.009Kb)
From this file, we recommend to open the Section "Study of the 3-dimensional region", subsection "Complete bifurcation map".
Corel Draw files containing 2-dimensional views of the slices without phase portraits (97Kb/137Kb).
Corel Draw files containing 2-dimensional views of the slices with phase portraits (504Kb/549Kb).
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 containing most of the calculations (927Kb/5.009Kb)
From this file, we recommend to open the Section "Study of the 3-dimensional region", subsection "Complete bifurcation map".
Corel Draw files containing 2-dimensional views of the slices without phase portraits (97Kb/137Kb).
Corel Draw files containing 2-dimensional views of the slices with phase portraits (504Kb/549Kb).
