Friday, November 11, 2005

Dynamical Instability

Dynamical Instability

Dynamical Instability (2005)

Too much chaos in your life? Sorry. It can't be helped.

From a panel on "What is Chaos?" at the University of Texas:

You can now learn about dynamical instability, which to most physicists is the same in meaning as chaos. Dynamical instability refers to a special kind of behavior in time found in certain physical systems and discovered around the year 1900, by the physicist Henri Poincaré. Poincaré was a physicist interested in the mathematical equations which describe the motion of planets around the sun. The equations of motion for planets are an application of Newton's laws, and therefore completely deterministic. That these mathematical orbit equations are deterministic means, of course, that by knowing the initial conditions -- in this case, the positions and velocities of the planets at a given starting time -- you find out the positions and speeds of the planets at any time in the future or past.

Of course, it is impossible to actually measure the initial positions and speeds of the planets to infinite precision, even using perfect measuring instruments, since it is impossible to record any measurement to infinite precision. Thus there always exists an imprecision, however small, in all astronomical predictions made by the equation forms of Newton's laws. Up until the time of Poincaré, the lack of infinite precision in astronomical predictions was considered a minor problem, however, because of a tacit assumption made by almost all physicists at that time. The assumption was that if you could shrink the uncertainty in the initial conditions -- perhaps by using finer measuring instruments -- then any imprecision in the prediction would shrink in the same way. In other words, by putting more precise information into Newton's laws, you got more precise output for any later or earlier time. Thus it was assumed that it was theoretically possible to obtain nearly-perfect predictions for the behavior of any physical system.

But Poincaré noticed that certain astronomical systems did not seem to obey the rule that shrinking the initial conditions always shrank the final prediction in a corresponding way. By examining the mathematical equations, he found that although certain simple astronomical systems did indeed obey the "shrink-shrink" rule for initial conditions and final predictions, other systems did not. The astronomical systems which did not obey the rule typically consisted of three or more astronomical bodies with interaction between all three. For these types of systems, Poincaré showed that a very tiny imprecision in the initial conditions would grow in time at an enormous rate. Thus two nearly-indistinguishable sets of initial conditions for the same system would result in two final predictions which differed vastly from each other.

Poincaré mathematically proved that this "blowing up" of tiny uncertainties in the initial conditions into enormous uncertainties in the final predictions remained even if the initial uncertainties were shrunk to smallest imaginable size. That is, for these systems, even if you could specify the initial measurements to a hundred times or a million times the precision, etc., the uncertainty for later or earlier times would not shrink, but remain huge.

The gist of Poincaré's mathematical analysis was a proof that for these "complex systems," the only way to obtain predictions with any degree of accuracy at all would entail specifying the initial conditions to absolutely infinite precision. For these astronomical systems, any imprecision at all, no matter how small, would result after a short period of time in an uncertainty in the deterministic prediction which was hardly any smaller than if the prediction had been made by random chance. The extreme "sensitivity to initial conditions" mathematically present in the systems studied by Poincaré has come to be called dynamical instability, or simply chaos.

Because long-term mathematical predictions made for chaotic systems are no more accurate that random chance, the equations of motion can yield only short-term predictions with any degree of accuracy. Although Poincaré's work was considered important by some other foresighted physicists of the time, many decades would pass before the implications of his discoveries were realized by the science community as a whole. One reason was that much of the community of physicists was involved in making new discoveries in the new branch of physics called quantum mechanics, which is physics extended to the atomic realm.

Why might it be groovy to merge dynamical instability with quantum mechanics? Because, according to Chris Clarke at Green Spirit:

The combination of dynamical instability with quantum theory demonstrates that, however one interprets quantum theory, the universe is fundamentally unpredictable on a large scale as well as a small scale. Dynamical instability is sometimes called "the butterfly effect": a butterfly flapping its wings in Brazil can cause a hurricane in Bengal.

Uh-oh. The Butterfly Effect? Be like a jury and disavow any mental pictures of this guy

You've just been chaotically punk'd, dude...

Like wow. One sec I'm hanging with Demi, and the next sec I'm zapped into this blog. Go fig.

because we aren't going there, although I'm surprised to find myself here. My background is in literature and creative writing -- not science. And you can certainly read in-depth articles about chaos theory with a click of a quick Google search. But as someone who uses fractals as a base to make digital art, I find all of this pandemonium more than just cool. It's a means to get at art.

Let's see if I can summarize. Dynamical instability (chaos) describes complex motion and the dynamics of systems -- systems that are sensitive and not necessarily stable. Chaotic systems can be determined mathematically but are nearly impossible to predict. Chaos concerns itself with whether it is possible to make precise, long-range predictions of any system if the initial conditions are known and generally accurate.

So where do fractals fit in? Fractals are geometric shapes that are extremely complex, self-similar, and infinitely detailed. Fractals are tied to chaos because they have definite and defined properties and because they are extremely complex systems. Chaos and quantum mechanics reveal the universe (vast and minute) is inherently unstable, and, not coincidentally, Nature is filled with fractal patterns.

That's my nutshell grasp, but, as I admitted earlier, I'm out of my league here. I'd certainly welcome comments or links from mathematicians or fractal artists who can clarify the big picture better than I.

What intrigues me as an artist is to take this snapshot of a complex system and radically modify (some would say destroy) its established definition. My fractals become isotopic -- incredibly unstable. They mutate and are quickly less predictable. In fact, often, they are no longer fractal at all -- although they usually retain shards of fractal forms. However, that doesn't mean that my methods or the progressive stages of a given image are serendipitous. My process is discovery-oriented, but I also am familiar enough with graphics programs to sense in advance what choices produce what effects.

I'm probably sinking deeper into a tar pit of viscous abstraction today with each word I write. I lack any real understanding of complex mathematics, and I feel awkward talking (even in vague terms) about my artistic process. In fact, now that I reflect back to last spring, much of my blogging is probably dynamically unstable. I had ideas-parameters when I started this post. So how in the name of Poincaré did I end up here?

~/~

I've never posted a random (chaotic?) music shuffle, so why not. Spin the wheel:

1. "SUS" (Peel Session) -- The Ruts
2. "Broken Ship" -- Immaculate Machine
3. "Danger Bird (Live) -- Neil Young and Crazy Horse
4. "Strode Rode" -- Sonny Rollins
5. "Fill Me with Your Light" -- Clem Snide
6. "Night Light" -- Sleater-Kinney
7. "Trinkle Tinkle" -- Thelonious Monk with John Coltrane
8. "Get Out of This" -- Dinosaur Jr.
9. "Pull of the Moon" -- The Mermen
10. "Tangerine" -- Big Head Todd and the Monsters

Hmmm. This looks more like a list of what I've been listening to lately. Note to self: infuse additional tunes.

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Friday Faves
Blogs and sites I've enjoyed this week:

Fractal Art:
Reward your senses by visiting Mindy Sommers at Peapod Design. Lush, gorgeously saturated, post-processed fractals that allow mathematics to uncover connective patterns found in nature. She is fractaldom's finest watercolorist and landscape artist.

Political Blog:
Go hang with Attaturk and friends over at Rising Hegemon. Spot-on analysis of current political theatrics mixed with hilarious and satirical uppercuts that always land and do damage. I especially dig their sharp-edged, narrative photoblogging. Also, captioning fun galore.

Cultural Blog:
Feast your eyes on the stills of Square America -- self-described as "a gallery of vintage snapshots and vernacular photography." The site is truly expansive -- packed with picture windows revealing countless moments of lived lives. I find the gallery of photobooth shots pleasureable, and (if you're feeling more Type A) a series exploring "America's love affair with high-caliber weaponry" will have you locked and loaded.

3 comments:

Anonymous said...

Whatever passes
Over the fields
Before the harvest

May be arbitrary
But the human mind
Wants to make sense of it all.

Whatever cannot be
Translated and understood
Must not exist

And instead must be
Outside causality,
Useless as poetry.

Dr. Mike

Tim said...

The quotation and your summary are actually one of the few good descriptions of chaos theory I've seen. The problem is that being a mathematical or physical topic it's usually described in those terms: Ones which most artists or generally creative people aren't, how should I say, "familiar" with. I mean, we can't be experts on everything.

The only thing I would add, and I don't have a math or science background either, is probability. In weather forecasting, like hurricane track prediction, the uncertainty of the system is approached by assigning probablities to outcomes.

So while a system may be chaotic, like the weather, and we can't perfectly predict it, some results are more probable than others.

This is what I like about algorithmic art: although it's mechanical, at least in its origins, it's as unpredictable and creative as the weather. I like to gradient map old bw photos in my graphics program. And like you said, while the process is partly predictable, new things keep happening. That is, it's creative and "discoverential" (new word).

Anonymous said...

Word Shavings

The word splinters
And goes unheard
Among whisperers
Hidden in the wood

Tensile flakes shear
Fragile textiles whose
Arthritic aches wear
Down roseate hues

Greening stalks shred
Where torsos twist free
Nothing more need be said
Because anything can be

Dr. Mike

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