If bifurcation diagrams (BDs)
become complicated (many branches, or long snaking
branches, and many bifurcations), then for presentations it may be useful
to create movies illustrating the BD by moving a marker through the BD and
plotting the associated solution. Moreover, Hopf orbits in 2D
and 3D can often best be visualized by movies. Here we collect a number
of movies generated in the demos. This proceeds by some ad-hoc scripts,
and if you plan to make a movie, we suggest to copy the respective demo
script to your working directory and modify it there.
These movies are generated from the data written to disk for creating
figures! In particular, we generally write rather few points from the BD
to disk (default p.file.smod=10),
and hence have only few points on the branches available for the
movies. If you want to create smoother movies, or do not care about
disk space anyway, write more solution data to disk, e.g., set
|A simple warmup BD for the 1D Allen-Cahn (AC) equation,
with imperfect bifurcations
due to a nonhomogeneous Dirichlet BC at the right. See
|Two snaking branches of localized rolls in the 1D
Swift-Hohenberg (quadratic-cubic) equation, see PF-tutorial (§3.2), and demos/pftut/sh/bdmov1D.m
|The 2D SH equation on a 'medium size' (quite small actually) domain allowing hexagon patterns; BD illustrating some secondary branches,
see PF-tutorial (§3.8), and demos/pftut/sh/bdmov2D.m
|The 3D SH equation on a long and slender bar,
meant to illustrate localized 3D patterns, generated by initial guess;
blue and red branch are BCCs and tubes, respectively; see PF-tutorial (§3.8), and demos/pftut/sh/bdmov3D.m
|A 'wandering boundary spot', meant to illustrate
3D mesh-adaption (slightly dragging behind the spot) by trullekrul.
See Trullekrul-tutorial and
|The 2D Schnakenberg reaction diffusion system on a long
and slender rectangle, meant to illustrate 'beans' (orange) connecting the
stripes (blue) and hexagons (red), and containing bifurcation points to
branches (magenta) of hexagons embedded in stripes; see PF-tutorial (§4.2), and demos/pftut/schnakpat/bdmov2D.m
|The AC equation on a sphere yields branch points from
the trivial branch of high multiplicity. Except for the primary (spatially
homogeneous) bifurcating branch (magenta)
these are all unstable, but the problem is
interesting from the symmetry perspective. See PF-tutorial (§6.2), and demos/pftut/acS/bdmovacS.m.
|In demos/pftut/schnaktor/ we consider the
Schnakenberg RD system on tori, see also PF-tutorial (§6.4). Here we find stable solutions with stripes inside and spots outside.
|For Hopf branches we can create movies showing
orbits (at fixed parameter values). In hopfdemos/cgl/cmds2dsq.m (and sqmov.m) we
consider the complex Ginzburg-Landau (cGL) equation on a square.
The 2nd BP then has multiplicity 3, and modulo symmetry we find 3 isotropy
classes of Hopf orbits: edge oscillations (blue), vertex oscillations (black)
and rotations (red). The magenta branch is from the third BP.
|In case of O(2) symmetry (for instance a disk domain),
primary Hopf branches come as standing (blue) or rotating (dark red) (spiral)
waves, while secondary bifurcations may lead to meandering (light red) spirals.
In hopfdemos/cgldisk/cmds2d.m we compute such branches, and, e.g.,
illustrate the spiral tip motion. In the movie, the last solution
is stable, and computed by Hopf bifurcation from the the rotating wave
as a steady solution in a comoving frame, hence the meandering spiral
is also plotted in a coming frame.
See also Hopf tutorial (§6.2).