Taffy

[Pauldelbrot]

This is a mosaic of six Henon map Julia sets.

The Henon map is this map of two real variables:
x(x,y) = y + 1 - ax²
y(x,y) = bx

Maps of two real variables don't have the neat property that every point on a basin boundary is on the boundaries of all basins.

In all six images, green points escape and yellowish points converge on the mapping's sole "interesting" fixed point or on the results of that fixed point bifurcating one or more times.

Counter-clockwise from bottom left:

• An image from the period-3 "horn" in the Henon Mandelbrot (see "Nimbus", posted right before this one) where it overlaps the triangular parameter-plane region in which that fixed point is stable. Purple points go to a period-3 attractor. The 3-cycle's points are found in the darkest regions of the three largest purple regions, close to the fixed point, which is located in the dark region in the largest yellow area. Observe that though points on the boundary of the green basin are limit points of both the yellow and purple basins, plenty of points exist bordering yellow and purple but nowhere near any green.
• Further along the horn, the attracting fixed point has bifurcated and we have an attracting 2-cycle as well as the 3-cycle. The 2-cycle's points are in the darkest parts of the dark "coffee swirls" in the central yellow region. The white bands divide the two phases of the attractor; if the map is iterated two steps at a time, the 2-cycle becomes two attracting fixed points with separate basins and the white bands separate these.
• Further still, the "main" attractor has gone strange. This is from where, in the "Nimbus" image, the horn crosses the pink chaos region. The attractor itself is the dark coathanger shape within the yellow region; again, the purple areas comprise the basin of the attracting 3-cycle associated with the horn.
• Further still, where the horn leaves the main body of the "Nimbus" Mandelbrot set, there is only the basin of infinity and that of the 3-cycle.
• Further along the horn, now some distance from the main body of the M-Set. The 3-cycle is still present, though its basin has continued shrinking around it, and the iterations in the "outside" area are lower. Everything is much less folded up.
• The big taffy-pull image shows another parameter point with three attractors. Aside from infinity, there is a strange attractor and an 8-cycle. The yellow region contains the strange attractor (black curves) and the brown region contains the 8-cycle (in the centers of the eight darkened areas). The basins interleave fractally, without the sharp boundaries found in the first image of these six.

The last image is not from the period-3 horn, but rather from the upper-central part of the Nimbus parameter-space image, at the upper-left of the pink chaos area. There is a purplish "blob" indicating a 5-or-higher-period attracting cycle embedded in the chaos there, and part of it is blended with the pink instead of on top of it, indicating the coexistence of the strange attractor and the periodic one in a part of the parameter space. It is with a parameter from this region that the top-left image was produced.

The Nimbus image can be a useful guide to finding especially-interesting Henon Julia sets, in much the way the classic Mandelbrot Set can be a useful guide to finding interesting classic Julia sets. It just has to use aggressive aggregate sampling of the dynamic plane per parameter point to capture enough information about the dynamics to serve as a guide.

The other thing done aggressively here being anti-aliasing. The banded and swirled structures, especially when there are lots of closely-spaced parallel lines, come out pretty ugly without it. With it, though, images that take seconds to generate as 640x480 previews take tens of minutes at 1024x768 and over an hour at 2048x1536.

One thing that does not occur even with heavy dynamic-plane sampling is a correspondence of shapes and forms between the parameter-space images and the dynamic plane.

Note: Julia set images of the same map iterated with x, y, a, and b complex, in the x-plane, are commonly called Phoenix images and look very different from these real-real x-y plane images.

Individual PNGs of the six images are available upon request. The top left is available in 2048x1536, the others in 1024x768.

Freely redistributable and usable subject to the Creative Commons Attribution license, version 3.0.

Detailed stats:
Name: Taffy
Date: January 30, 2009
Fractal: Henon Julia sets
Location: Various
Depth: Very Shallow
Min Iterations: 1
Max Iterations: Various, usually 1000
Layers: Various, usually 3
Anti-aliasing: 3x3, threshold 0, depth 2
Preparation time: 1 hour
Calculation time: Various, totaling about 3 hours (2GHz dual-core Athlon XP)