The video clips illustrating complex dynamics typically have a few MB and a duration of about 30s. They were produced with FFmpeg, combining images made with Mandel. For each video, two links are given: clicking play will open the video in a new browser tab or window. Or click right to save the file to your hard disk, and open it with a video player, e.g., VLC. This is recommended for large file size in particular. Clicking show shall open a small pop-up box to play the video, possibly in reduced size. (When you are closing the pop-up before the video is completed, please stop playback first.)

Mandelbrot set and quadratic polynomials

The Mandelbrot set M is the connectedness locus of complex quadratic polynomials fc(z) = z2 + c . The filled Julia set Kc contains those values z that are not going to ∞ under iteration. The following videos illustrate bifurcations of Kc or self-similarity properties of M, respectively. See Mandel demos 2, 5, 6, 7 for a more extensive explanation.

Bifurcation of Julia sets

Approaching the airplane root with Fatou-Lavaurs translation (1/σ ~ time): play — show.
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Airplane primitive parabolic implosion (c ~ time, 1/σ accelerated): play — show.
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Rabbit satellite bifurcation: play — show.
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Bifurcation of preperiodic points and rays at c = γM(5/12) : play — show.
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Similarity and self-similarity of the Mandelbrot set

Slideshow of Feigenbaum scaling: play — show.
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Slideshow of rescaled limbs converging to the Lavaurs elephant: play — show.
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Embedded Julia set similar to Misiurewicz Julia set: play — show (24 MB).
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Embedded Julia set similar to Siegel Julia set: play — show (30 MB).
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Nodes converging to a Misiurewicz point within an embedded Julia set: play — show (42 MB).
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Nested structures at a node, converging to another Misiurewicz point: play — show (43 MB).
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Local similarity, zooming into both planes, dynamic plane rescaled: play — show (24 MB).
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Local similarity, the rescaled Julia set is changing with the parameter c: play — show.
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Renormalization, the Julia set is changing with the parameter c without rescaling: play — show (24 MB).
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Slideshow of asymptotic similarity at a = γM(9/56) with the scale γ=1: play — show.
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Slideshow of asymptotic similarity at a = γM(9/56) with the scale γ=3/2: play — show.
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Slideshow of asymptotic similarity at a = γM(5/12) with the scale γ=2: play — show.
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Visualization of the Thurston algorithm

Recapture turns basilica into airplane, one step per second: play — show.
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Left Dehn twist turns rabbit into airplane, one step per second: play — show.
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Right Dehn twist turns rabbit into corabbit, one step per second: play — show.
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