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This towering rain cloud glowering on the horizon and chasing off a flock of birds is actually the Henon map's Mandelbrot set as it truly appears.

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

Maps of two variables don't have the neat property that the dynamics can be entirely characterized, and all of the attractors discovered, solely by iterating a small set of critical points. Therefore, to create a parameter-plane map that is truly analogous to the Mandelbrot Set, one must sample a large number of points.

This image used 225 dynamic-plane samples per parameter value -- more than that per pixel, since the image is antialiased. The sample points were derived by taking a regular 15x15 grid of points spanning the interval -1 to 1 on each axis, then squaring the magnitude of each coordinate. The result is distorted grid spaced more closely near 0,0 and more widely toward the boundary of the square. This grid is then displaced to be centered on the map's sole interesting fixed point, and scaled down further by a complicated formula that I devised to produce decent results even as the Julia sets change size in a parameter-dependent way.

The numerous layers capture details of the dynamics:

• the outer white-blue-black gradient colors parameters where all samples escape;
• the large red triangle colors parameters where at least one sample converged to an attractive fixed point;
• the large orange stripe colors parameters where at least one sample converged to an attractive 2-cycle;
• the yellow band corresponds to the existence of attractive 4-cycles;
• the greenish "horn" corresponds to the existence of attractive 3-cycles;
• the violet strips and blobs correspond to the existence of attractive cycles of period 5 or higher; and
• the pink corresponds to the existence of an aperiodic (with these dynamics, usually strange) attractor.

Some of the stripes overlap other areas; this occurs for parameter-space regions in which the system has multiple finite attractors. In my explorations of the map's dynamic plane images, I've determined that there exist parameter points for which it has six finite attractors, and I know of no theoretical limit. Contrast this with the dynamics of complex maps where there provably can be no more attractors than the map has critical points.

All of the gradients are based on the greatest Lyapunov exponent of all of the points meeting the requisite fate; for example, the red area's gradient corresponds to the greatest Lyapunov exponent of all the sample points that converge to an attracting fixed point. For the periodic components, paler/brighter corresponds to a more negative Lyapunov exponent, and therefore to a stabler attractor. In the "chaos zone", the exponent is positive; purpler shades correspond to lower values and pinker shades to higher ones, so the strange attractors get "more strange" in the pinker regions. The outside area has pale blue, dark blue, and then black as the Lyapunov exponent grows. (Divergent points have a positive Lyapunov exponent.)

A couple of features of the image are worth noting. One is the narrow horizontal band that is vertically centered. This corresponds to b = 0 in the above, and therefore to the partial derivative of y(x,y) with respect to each of x and y being zero. So one row of the Jacobian is zero, and so its determinant is zero, and so the Lyapunov exponent is minus infinity.

Another is the olive "horn" with a purple strip. The 3-cycles in the olive area seem to exist entirely independently of the "main" attractor that results from bifurcation of the map's one interesting fixed point. For b less than -1 this 3-cycle is unstable, however. The purple strip is where the 3-cycle has bifurcated into 6, 12, or whatever; the very narrow bright band is where the 3-cycle has bifurcated all the way to strangeville. The horn goes straight through the main chaos zone and continues into the "outside" area.

There are other similar "horns" here and there corresponding to higher periods, and other protrusions and blobs indicating attracting periodic cycles exist can be found in the "outside" area near the chaos zone.

Dark streaks, bands, and curves can be seen in the outside area that don't seem to contain any Mandelbrot points, as well.

A lot of the features lie along parabolas. I was able to determine the exact shape of the main red triangular region; it is bounded below by b = -1, above by b = 1, and to the sides by the parabolas a = (b - 1)² and a = -(b - 1)²/4. This was proven by analyzing the Jacobian at the "interesting" fixed point to determine where this is stable.

The "tattered" look in the top right part of the "cloud" and other gaps, streaks, and omissions are the result of the sample grid sometimes entirely missing a basin of attraction. Even more sample points, or a more stochastic distribution of them, would be needed to improve this.

Zooms and some Julia set images forthcoming.

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

Detailed stats:
Name: Nimbus
Date: January 30, 2009
Fractal: Henon/Phoenix Mandelbrot
Location: Whole Set
Depth: Very Shallow
Min Iterations: 1
Max Iterations: 1000
Layers: 7
Anti-aliasing: 3x3, threshold 0.10, depth 1
Preparation time: 2 hours
Calculation time: 9 hours (2GHz dual-core Athlon XP)