Oasis

[Pauldelbrot]

A Matchmaker Julia set with twi attractors, one of period 22 (browns and greens) and one of period 7 (blues). This latter is rather unusual: the basin forms disconnected lakes, like you get in a minibrot cardioid, but the parameter value at first blush appears to be from a (somewhat oddly-shaped) bud in the Mandelbrot. In fact, there is a period-7 bud and a period-7 minibrot cardioid fused and the parameter point is from where the latter intersects with a different bud. A period-1 component buds the period-7 bud and a nearby period-11 bud, whose nose sports a period-22 bud that intersects the cardioid that is fused with the first bud. (Confused yet?)

A problem with consistently assigning an identity to an attracting basin becomes apparent here: starting at this fractal's parameter value and moving in a particular closed curve in the Mandelbrot, first the period-22 cycle disappears, then the period-7 one's lakes fuse into seven petals as the "neck" joining cardioid and bud is traversed. Then, going through the root of the bud, the petals fuse and one gets a basin of period 1, with a dust-like Julia set. Then going through the root of the period-11 bud, this basin splits into 11-petaled structures. The petals develop pinches and become beads entering the period-22 bud. Then finally, dense quadratic dust-Julias appear and condense into lakes of a period-7 basin.

Notice something? The attractors switched roles after one circuit. The lakes became the "plants" in Oasis.

This becomes important in attempts to assign a) a consistent color scheme to buds in Mandelbrot views and b) a consistent color scheme to basins in Julia animations as the parameter value changes, in systems with multiple critical points.

I've done some research into this matter. More images will be posted relating to results.

First, a forthcoming (but very slow to calculate) image named Stargate will display the period-7 cardioid and the parameter point of this Julia set image.

Second, an animation from a new map-family called Supernova will be forthcoming that shows a similar Julia morph in which two basins exchange roles.

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

Detailed statistics:
Name: Oasis
Date: February 6, 2009
Fractal: Matchmaker Julia set
Location: a = 0.53256 + 0.31163i; b = 0.22090040939 - 1.0672743081i
Depth: Very Shallow
Min Iterations: 250
Max Iterations: 4153
Layers: 2
Anti-aliasing: 3x3, threshold 0, depth 2
Preparation time: 10 minutes
Calculation time: 6 hours (2GHz dual-core Athlon XP)