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Where does the complexity come from?

Part 2 - Julia Sets

Introduction

We saw on the Mandelbrot Complexity how we can iterate the basic equation z → z2 + c to generate a series of equations that border the region where the iteration converges. These equations define closed curves, points in the areas outside these curves definitely diverge, the areas inside may converge on an attractor or they may not. It assigning colours to the non-convergent areas between these curves that enables us to generate the spectacular images associated with the Mandelbrot set, the set itself usually being shown in a solid colour like black.

When we come to look at the Julia sets the system gets even more complicated. The equation being iterated hasn't changed, bur whereas with the Mandelbrot we always start of with Z0 = 0, with the Julia sets Z0 can take any value. The equations that determine the contours are now:

These equations are a lot more complicated that the equivalent Mandelbrot ones. You can see how they are related by substituting Z = 0 into these equations, which will regenerate the equations listed on the Mandelbrot Complexity page.

As we now have 4 independent variables, Zreal, Zimaginary, Creal and Cimaginary, the contours being define are really three dimensional surfaces in a 4-dimensinal space. The Julia set images displayed around the web, including on this site, are two dimensional slices constructed by picking a single 'c' value and varying 'z'. There are lots of other potential 2-dimensional slices through this object, some of which can be explored in my interactive mandelbrot generator. There are also some example images here and here showing non-traditional slices.

Arractors

We saw on the mandelbrot complexity page that each 'c' value has an infinite set of 'z' values where iterating z → z → z2 + c a number of times will eventually return to 'z', the first few being sumarised in this table. In the mandelbrot set 'z' starts out as 0 and the only 'c' value where 0 is a stable point is 0. When constructing the mandelbrot images the black areas are those where successive iterations converge on one of these stable points (or cycles) and the coloured areas those when the iteration diverges (sometimes described as converging on the point at imfinity). You can see what type of attractor z = 0 converges on in this image:

Orbit Map

This image was generated using the interactive orbit map on the mandelbrot complexity page, with the number of iterations set to 1500.

The large red region shows the domain of 'c' values where starting the iteration at Z0 = 0 converges on a single point, the large blue circle to the left is the 'c' region where Z0 = 0 converges on a 2-cycle. The two big yellow circles are the three cycles, and the three cyan circles the three 4-cycles. The other circles can be identified using the interactive version, see if you can find the ninety nine 10-cycles.

When we draw the Julia sets we choose a single 'c' value and plot the convergence as a function of Z0.

(c) John Whitehouse 2016