Additional
files for
the
article
Global phase
portraits
of quadratic
polynomial
differential
systems
with
a semi-elemental
triple node
Joan C. Artés, Alex C. Rezende and Regilene D. S. Oliveira
In this paper we classify all the quadratic vector fields which have a semi-elemental triple node, i.e. it is a singular point whose determinant of the Jacobian is zero but the trace is not, and it is topologically equivalent to a node.
The bifurcation
diagram
for
this
class,
done in the
adequate
parameter
space
which
is
the
3-dimensional real space,
is
quite rich
in its
complexity
and yields
63 subsets
with
28 phase
portraits
for
the
whole
class.
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 3-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
large
and 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
each
one
on
one
sheet,
but
they
all
need
an
A0 (or
A1) format
page to
see
the
phase
portraits
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
of time 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
(2.160Kb/16.100Kb)
From
this
file,
we
recommend
to
open the
Section
"Needed invariants and slices" and take a
look at the different slices of the bifurcation diagram in order
to better understand the their transition. In the following file we
present two animations where this transition can be seen, and the
"movement" of the singularity of the red surface (the cusp point) along
the slices.
Animations Mathematica
file (3.660Kb/44.500Kb)
Corel
Draw
files containing
2-dimensional views
of the
slices
without
phase
portraits
(61,2Kb/93,7Kb).
JPG
files
containing
2-dimensional views
of the
slices
without
phase
portraits
(902Kb/1.026Kb). 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
(244Kb/276Kb). Attention
should
be
paid
as these
four
files and also
the
one
corresponding
to k=3
(slice3pp) 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 13 from
the
paper).
They
have
been
calculated
numerically
and included
in a sequence
of Excel sheets.
Mathematica
file containing
2-dimensional views
of the
part
of the
slices where we find a non-algebraic bifurcation
surface
(376Kb/10.300Kb). Nice
to
see
them
in a row.
Tornem
a la pàgina
principal
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