Chaos Fractals Connections

Chaos Fractals Connections

The main idea behind chaos theory is that even simple deterministic systems can sometimes produce completely unpredictable results. This happens when a deterministic system has sensitivity to initial conditions: basically, that if we take two starting states which are very similar to each other, over time the states will diverge and re-converge in unpredictable ways.

the original Lorenz pattern

I think pool tables are an accessible example, if not entirely correct. Very small changes in the position of the cue ball, or of the power or angle behind the cue stick, can produce very large differences in the way the rack of numbered balls breaks over the table.

Lorenz Pattern

Now the odd thing that people noticed about deterministic systems is that even though the development of the system from any particular state is unpredictable, if we graph it and watch it develop over a long enough time it will start to look like it is almost following some kind of pattern. e.g. the original Lorenz pattern

It will loop back and around over the same territory in the same basic kind of way, even though it never falls back into exactly its own track. And moreover, if we choose a different starting state it will develop completely differently than the first state (sensitivity to initial conditions), but it will still appear to be following that same basic pattern.

Deterministic System

So, we cannot predict what state the system will be in at time t, but we can (apparently) predict the general shape the states of the system move through over time.

What’s happening here is that the deterministic system is following an orbit. Literally, this is an orbit just like the Moon makes around the Earth. Now when we talk about orbits, we are talking about some kind of attraction — in the Moon’s case, gravity — that keeps the state from flying off to infinity. That attraction is defined by an attractor: a point or curve that the system tries to conform to, which sets the basic pattern of movement. In the Moon’s case the attractor is a point (or maybe very small circle, or rather a helix because system is moving through space) at the center of gravity of the Earth/Moon system. The Moon moves around that attractor in something resembling an ellipse, a nice, smooth, predictable curve, because the attractor is itself everywhere continuous, and the Moon has no difficulty conforming to it.

Fractal

But in chaotic deterministic systems, the attractor is not a point or a simple, smooth, continuous curve. The attractor — often called a strange attractor or sometimes a fractal — could be an infinite set of unconnected points, or a smooth curve with mathematical discontinuities, or a curve that is fully connected but discontinuous everywhere. The term fractal in this sense (not to be confused with the idea of a fractal image, discussed below) refers to a space of fractional dimensions: not 1-dimensional or 2- dimensional or n-dimensional, but 1.23, or 2.78 or any positive non-integer number.

The attractor cannot fit in X dimensions, but also cannot fill up X+1 dimensions. Any system that tries to conform to such a curve will behave chaotically, for the same reason that trying to roll a tennis ball on a pebbly beach ends up with the ball bouncing and jumping all over the place; the pebbly surface is something more than 2 dimensional and something less than 3. Physical systems cannot be discontinuous, obviously, even if their mathematical representations can be, and so we get chaotic results.

Mandelbrot

So now we have these strange attractors underlying deterministic systems, and the question is; how do we visualize them? There are two regular solutions (or rather, two variations of a general solution), both of which requires a decent amount of computing power:

Repeatedly sample a large range of starting states to see which starting states seem to be permanent orbits and which fly off into infinity. This works well for more difficult equations, and is used to produce fractal images like the Mandelbrot set.

Repeatedly apply the system rules, either systematically, or by plugging hordes of random numbers into the equations (which is often used to create fractal leaves and such, shown second).

So, a fractal image is a visual representation of a strange attractor (or fractal space) that defines the orbit of a deterministic system that behaves chaotically.

Daniel Melhem, PhD CEO of Dimtech