## Movies created in some pde2path demosHome |Tutorials and Demos | Applications| Backlog |

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.

A simple warmup BD for the 1D Allen-Cahn (AC) equation, with imperfect bifurcations due to a nonhomogeneous Dirichlet BC at the right. See AC-tutorial, and demos/acsuite/ac1D/bdmovieimperf.m. |

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 demos/acsuite/ac3D/bdmovie3Dws.m. |

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). |