Pretty Physics Pictures

That's right - this post has nothing to do with kayaking. In my free time, I am working on my Ph.D. in theoretical physics at CSU. I am studying chaos. In the course of my research, I often recreate figures from papers that I read, and the results are sometimes interesting to see. Here is a collection of the more exciting pictures.

KAM torii:
These figures are recreations of those shown in "Amplitude Instability and Ergodic Behavior for Conservative Nonlinear Oscillator Systems" by Walker and Ford, Phys. Rev. 1969. Conservative systems are a huge pain in the butt.

These figures are Poincare sections of 4d torii. Without going into too much depth, each closed loop is a numerically solved solution to the Henon-Helies system. Each frame below shows the intersections of these trajectories with a 2d plane in this 4d phase space. With higher and higher E, more of these torii "break" and start to wander through the spaces between unbroken torii in a chaotic manner.


I asked my friend Zan what these figures reminded him of, inkblot-style. He said "fear". KAM torii inspire fear, and rightly so.

Henon map:
Here are a couple pictures of the Henon attractor (the shape that the Henon map settles down to). The Henon map is a 2d map invented in order to study chaotic mappings. Check the wikipedia article for more info. One interesting thing is that there is an unstable fixed point on the upper right edge, and the attractor appears to be fractal about that point (although it's not a very interesting looking fractal, it is self-similar). These figures are almost exact duplicates of the ones in Henon's original paper, but with more iterations.

The whole attractor

Zoomed in on the upper right edge

Zoomed in on the upper line in the previous figure. Notice that the upper line here is composed of three lines...

Zoom in on the upper line from the previous picture. Notice that that upper line was actually three closely spaced lines, and if we were to keep zooming in, the upper line of the three lines is itself composed of three lines, and the upper one of those is composed of three lines, and so on. This is self-similarity.

The little boxes indicating where we were going to zoom into are just kind of sketched in. I didn't actually calculate exactly where the box should be, pixel by pixel, it's just intended to give an idea of what region the next picture will contain.

Bifurcation Diagram:
Here's a bifurcation diagram for the logistic map. At different values of some parameter (lambda), different solutions are stable in the logistic map. The horizontal axis is lambda, the vertical axis is a trace of the different values that the system visits.

The whole picture

Zoom in on lambda = 3.5 to lambda = 4.

Fractals:

This was a warmup for me, a re-introduction to programming. The following pictures are different sections of the Mandlebrot set. I recommend the wikipedia article on this subject for a full description.
Wiki fractals article
Wiki Mandlebrot set article

The full Mandlebrot set

I think this was off the first lobe, but I don't actually remember.

This is in the valley between the main cardioid and the first lobe. It's the one I use for wallpaper.

This is above the main cardioid, near the imaginary axis.

This is below the main blob, and to the right. I am basically copying a figure I saw on the wiki page somewhere.

Here are a couple new images that I have made. I used inverse iteration to quickly sketch out many julia sets, and made little movies of how the julia sets change as c is moved around on the complex plane.

Here is one of them. It's saved as an animated gif. Click on the image below to see the animation.