Strange Attractors

That looks pretty good, I was viewing the wikipedia page for fractals yesterday and I saw the image for a strange attractor. I remembered seeing it a long time ago so I clicked on it then saw some stuff referencing the de jong attractors. I implemented them in JS then Java(none are actually included here), but then I saw a post about a quadratic map strange attractor. The algorithm is: double x = values[0] + values[1] * lx + values[2] * lx * lx + values[3] * lx * ly + values[4] * ly + values[5] * ly * ly; double y = values[6] + values[7] * lx + values[8] * lx * lx + values[9] * lx * ly + values[10] * ly + values[11] * ly * ly; lx = x; ly = y; The values are taken from a A-Y character string where A is -1.1 and Y is 1.2. There is a lot more info and cool things(I got most of my codes from this one, but it explains how to search for your own chaotic codes(some just go to infinity, only 1.6 percent of the A-Y ones are chaotic)) http://sprott.physics.wisc.edu/fractals/booktext/sabook.pdf

I like that you compute the velocity of the system and show it in color.

I'm going to assume you don't know how or why these are generated.

If you take a system of differential equations, e.g. the double pendulum, it is in general impossible to solve analytically. A double pendulum will move in chaotic and non repeating motion. Luckily, we have very good numerical approximations the can model the system provided you have enough computational power.

Unfortunately a numerical approximation doesn't tell you much about the general behaviour of a system. One method for probing a system's behaviour is the strange attractor. If you simulate the system, and measure its position and velocity at regular intervals, you can plot those points. This is the strange attractor. The fact that doing this can produce distinct patterns is extremely surprising, especially when you consider that we're modeling chaotic systems.

Each point in the strange attractor is the position of the system in phase space after an "orbit". There is no way to create these without running the simulation repeatedly. At first you will get a set of seemingly random points, but as time goes on the pattern emerges.

Unfortunately you can't interpolate values without damaging the attractor. For systems with an analytical solution, you will get periodic motion which means there is a finite number of points in the attractor. A simple pendulum has only one point. Even for chaotic systems, you don't get continuous lines. After enough cycles the plot will look nearly continuous, but strange attractors are fractals.

Tl;dr a strange attractor is a discrete set of points, it doesn't make sense to think of it as a continuous function.

These are all rendered at approximately 2kx2k, the more samples and lower resolution the less grainy it is but the higher resolutions you can see more details. These are all rendered with 100 million iterations and only take a few seconds. I might try lines but I suspect it might mess up the perceived patterns in the final image. I will try the line thing soonish and post again if it works.