Sensitive dependence on initial conditions

This next demo highlights the butterfly effect. These curves show the same pendulum as before, this version has the time axis wrapped around the center of the screen and the exponential of the angle as the radius. All the pendulums start at a very similar initial position and for some driving functions they diverge wildly and others they all stay pretty close.

Ensemble #2

This is more of what I was thinking of when I started down this route.

These are ensembles of particle paths in a chaotic dynamical system. Not only are these ensemble pictures beautiful, but they highlight some interesting mathematics and physics.

But what I’ve really been after is a better source of semi regular noise. The sinusoidal waves and interpolation between random points have been good, but for really entrancing fluid random motion, I need a chaotic semi-periodic source to feed into the other visualizations.

Ensemble #1

I’m continuing with the envelope series, this time with an ensemble of states of a damped driven pendulum. One of the “classic” chaotic systems. This one isn’t in a chaotic regime, but striking none the less.

 

WebGL buffer fractal renders

Simple fractal with a webGL based renderer.

Fractals are abstract mathematical sets, and there are multiple methods to render them. I have some examples based on the “chaos game” and some based on the escape time algorithm.

To teach myself about webGL frame buffers and rendering to off screen targets, I wanted to try out rendering based on repeatedly rendering the entire image and merging it through each of the functions. It works well for subjects in the center of the screen, but it breaks down quickly as portions of the set stray off screen, which makes it tough to use as a general purpose renderer.

Playing around with straight lines

I’ve been messing around with various simple rendering techniques. This is a random walk branching off at right angles. To add a little more interest I added random motion through the space over time. Unlike my past demos, the motion is just a random walk which makes the motion very jittery and chaotic.

A little free time

With the holidays my workload has backed off a little and I’ve been checking out chrome experiments for some inspiration. One of those was Ablaze js. One of the first graphic tricks I used on my TRS-80 color computer on the high resolution black and white mode 256×192 pixel screen was drawing curves with string art style lines. I never made it past just filling in along the sides of the screens. Seeing ablaze brought me back to those days.

So string art is my take on doing something similar. I wanted to use the same string art style and the brilliant colors of the original. However I wanted it to continue to evolve and look good indefinitely. I also wanted to use the differential equation solver as the source for the lines.

The first thing I thought about was the movement of the lines. I started with a particle system moving in a force field. The particles are moving in a central force field with a moving center. Overlaid on that is a uniform magnetic field so that the particles travel in tight loops.

Once I had the particle motion nailed down I experimented with several attempts to get something that looked good. My first attempt was to just draw lines between randomly generated particles. That started out with a nice initial distribution, but got unruly as the system unfolded. Next I tried similar particles, but those eventually drifted too far apart. Adding code to force them back together created some strange wedge shapes. Eventually, I settled on drawing a tangent line segment centered on the current location.

I tried a few simple color schemes before I settled on the blue and gold. If I spend some more time on it, I plan on making the colors independent for each particle and dependent on the velocity, but this looks good enough for now.