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The Volterra-Lotka Mandelbrot set.

Very slow to calculate. But it sure beats that crude diagram on page 126 of The Beauty of Fractals!

Nonadaptive antialiasing means every pixel has nine parameter pairs sampled. For each parameter pair, 100 dynamic-plane pairs were sampled, randomly distributed within an 0.5x0.5 square centered on the fixed point (1,1).

The outside pink-to-white gradient is escaping points, each pixel colored by average smoothed iterations of the 900 sampled points.

The orange-to-black gradient colors the points that display periodic behavior after a while. The color is by Lyapunov exponent: black means a not very stable cycle, and the paler oranges indicate a high degree of stability. In the big orange component at left, (1,1) is attracting; the brighter the orange the stabler it is. In the smaller, wedge-shaped orange components, (1,1) repels but there's an attracting cycle that typically rings (1,1).

The solid dark red regions pierced by these wedges exhibit "limit circle" attractors in the dynamics: the attractor is a closed curve, typically ringing (1,1), and points on the attractor rotate around it by a fixed irrational angle. Similar to a Siegel disk, only it's just a curve rather than a solid disk.

The purple regions exhibit strange attractors: the orbits are ergodic and irregular. The Lyapunov exponent is positive, and the shade of purple depends on it. The brighter the purple, the higher the exponent and the more chaotic the attractor.

Only three layers are used: one for the points where all samples escape, one for the points where some samples go to a strange attractor, and one for the points where some samples go to a periodic attractor. The latter two have some tiny specks of overlap, where the colors are combined additively.

The limit circle points with no other kind of attractor get colored by none of the layers; the background color was set to dark red.

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

Detailed statistics:

Name: Launch
Date: January 11, 2010
Fractal: Volterra-Lotka Mandelbrot Set
Location: First quadrant
Depth: Very shallow
Min Iterations: 1
Max Iterations: 10000
Layers: 3
Anti-aliasing: 3x3, threshold 0, depth 1
Preparation time: 1 hour
Calculation time: 25 hours (2.5GHz dual-core E5200)

x and y are the dynamic plane coordinates. Discretization as described in TBOF removes t and the other four can all be set to 1 without loss in generality (and then the fixed point is forced to be at (1,1)). The discretization adds two new parameters, h and p, and these are the axes of the image above. Real, not complex, numbers.


There are features, along the edges; zooms forthcoming.


I've posted on recently: Vortex.

The border is where they vary chaotically. They tend to a fixed point in the main orange area. One or both go to infinity in the pink area.

Very close. (0.75,0.75) to (1.25,1.25).

The discretization is a combination of the Euler method and the Heun method:





with the functions





being the functions in the differential equations being solved numerically. TBOF uses instead of p in the above, but there's no rho key on my keyboard and I doubt my compiler would accept it as a variable name in source code, so ...