Sunday, January 2, 2000

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.

Saturday, January 1, 2000

Ice Cream Scoreboard

The Ice Cream Game is basically a rolling game. A full set of rules can be found in the newest Colorado guidebook; Whitewater of the Southwest Rockies. What the rules boil down to is that the person who rolls the most times during a run buys everyone else ice cream. To help with the shame and heckling associated with losing the Ice Cream Game, we will be keeping a running score of the people we paddle with, and provide a public area for degrading comments.

A note on the scores. The "ties" are ties for last place. Sometimes the rules have to be flexed a little, in confusing situations. The "Winning %" is calculated by taking the number of wins plus half the number of ties, and dividing by the total number of runs that person has been on. Basically, a tie is half a win and half a loss.

























NameWinsLossesTiesWinning %Most Recent Win/Loss
Leif Anderson133871Clear creek of the Ark
Natalie Kramer76852Clear creek of the Ark
Conor Flynn32158Big Thompson Gnar
Dave Stentiford11050 Gore Canyon
Brent Wilkins02117 Green Truss
J. Cody Howard100100 Norrish Creek, BC
Martin No-mullet200100 South fork Salmon
Nathan Werner10175 Clear creek of the Ark


History:

May 23, 2009: Natalie loses to Nathan and Leif on multiple laps of the Clear creek of the Ark. The scores were N:2, L:0, N:0

May 21, 2009: Leif and Natalie tie 1-1 on the First Gorge of Lime creek

May 20, 2009: Leif and Natalie tie 1-1 on South Mineral creek

May 19, 2009: Leif loses to Natalie on a couple high water runs of Oh Be Joyful. The score was L:2, N:1

May 17, 2009: Leif loses to Natalie on the Black Rock section of Clear Creek. The score was L:1, N:0

May 16, 2009: Natalie, Leif, and Nathan tie 0-0-0 on multiple laps of Spencer Heights.

May 15, 2009: Natalie loses to Leif on a multi-lap expedition to the Poudre Narrows. L:1, N:2

May 11, 2009: Leif and Natalie Tie 0-0 on the Rio Embudo.

May 10, 2009: Natalie Loses to Leif on the Upper Taos Box. The scores are L:0, N:1

May 8, 2009: Leif and Natalie Tie 1-1 on Eldorado Canyon.

April 27, 2009: Natalie loses to Leif on Upper and Middle Narrows of the Poudre, during some slalom training. The score was L:0, N:1

April 25, 2009: Leif loses to Natalie on the Barrel Springs section above Shoshone on the Colorado river. The score was N:2, L:well... sort of like... half of a swim. (Leif says:"I never left the boat, but I did pull my skirt to escape a certain hole.")

January 2, 2009: Melissa loses to Martin, Leif and Natalie on the Limestone Bluffs section of the south fork Salmon. The score is M:1+S, N:0, L:0, M:0

January 1, 2009: Natalie loses to Leif, Damon, Martin, and Chris (from Willow Creek) on the Carhood section of Willow creek. The score is N:1, everyone else:0

December 31, 2008: Leif and Natalie tie on lower Willow creek. The score is slightly dubious due to a collision.

November 28, 2008: Brent loses to Natalie and Leif on the Farmlands into the Green Truss section of the White Salmon. The score is B:4, N:3, L:1 (+1 extra flip on BZ falls, which nobody else ran)

November 23, 2008: Brent ties with Natalie, losing to Leif and Cody Howard on Norrish creek in BC. The score is B:2, N:2, L:0, CH:0

October 30, 2008: Brent loses to Conor, Natalie and Leif on the Big Thompson Gnar. The score is B:4, N:1, C:0, L:0

October 12, 2008: Natalie and Leif tie 0-0 on the Big T gnar section.

October 5, 2008: Dave loses to Conor and Leif on the Poudre Narrows and upper Poudre Falls. D:1, C:0, L:0

October 4, 2008: Natalie loses to Conor, Leif and Dave on Gore Canyon. The scores are N:2, L:1, C:1, D:1

September 28, 2008: Conor loses to Leif on Gore canyon. The score is C:2 (counting a rope extraction from tunnel falls as 1 roll), L:0

September 27, 2008: Conor and Leif tie 0-0 on the South Saint Vrain Narrows, but agree that since it was too shallow to flip over even on purpose, they won't put it on the scoreboard.

September 20, 2008: Conor and Leif tie 0-0 after 2 laps on the Gnar Section of the Big Thompson.

August 26, 2008: Conor Loses to Leif and Natalie on Cheeseman Canyon. Score: C:1, L:0, N:0.

Let the smack-talking begin!