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 right-click to save the file to your hard disk, and open it with a video player. Clicking show opens a small pop-up box to play the video in reduced size.

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.
X


Airplane primitive parabolic implosion (c ~ time, 1/σ accelerated): play — show.
X


Rabbit satellite bifurcation: play — show.
X


Bifurcation of preperiodic points and rays at c = γM(5/12) : play — show.
X

Similarity and self-similarity of the Mandelbrot set

Slideshow of Feigenbaum scaling: play — show.
X


Slideshow of rescaled limbs converging to the Lavaurs elephant: play — show.
X


Embedded Julia set similar to Misiurewicz Julia set: play — show (24 MB).
X


Embedded Julia set similar to Siegel Julia set: play — show (30 MB).
X


Nodes converging to a Misiurewicz point within an embedded Julia set: play — show (42 MB).
X


Nested structures at a node, converging to another Misiurewicz point: play — show (43 MB).
X


Local similarity, zooming into both planes, dynamic plane rescaled: play — show (24 MB).
X


Local similarity, the rescaled Julia set is changing with the parameter c: play — show.
X


Renormalization, the Julia set is changing with the parameter c without rescaling: play — show (24 MB).
X


Slideshow of asymptotic similarity at a = γM(9/56) with the scale γ=1: play — show.
X


Slideshow of asymptotic similarity at a = γM(9/56) with the scale γ=3/2: play — show.
X


Slideshow of asymptotic similarity at a = γM(5/12) with the scale γ=2: play — show.
X


Zoom illustrating both asymptotic and local similarity: play — show (42 MB).
X

Visualization of the Thurston algorithm

Recapture turns Basilica into Airplane, one step per second: play — show.
X


Left Dehn twist turns Rabbit into Airplane, one step per second: play — show.
X


Right Dehn twist turns Rabbit into Corabbit, one step per second: play — show.
X


Eigenvalues of the Thurston pullback or pushforward are shown in a slideshow, such that the color indicates the quadrant of the value of the characteristic polynomial on the square [-1, 1] × [-i, i]. Compare this paper by Xavier Buff, Adam L. Epstein, and Sarah Koch.
For two sequences of centers with periods 5, 7, ..., 35 converging geometrically to the Misurewicz point with angle 5/12, most eigenvalues accumulate on a small circle, and the leading eigenvalue converges as well: play — show.
X

For a sequence of centers with periods 5, 8, ..., 62 converging to the Airplane root, eigenvalues accumulate on the unit circle: play — show.
X

Cubic polynomials:

A Misiurewicz point in S3: play — show (28 MB).
X