Comments, discussion?
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The Paradigm Revolution 
Dude I think you are the guy to go to for some di-rection here. I read an article the other day about the chaos theory in relation to economy…it was talking about how the same three principles apply to all elements in the circle of the chaos theory (mathematics, weather, economy, ecology etc) The principles are: must be sensitive to initial conditions, must be topologically mixing, and it’s periodic orbits must be dense. I looked up more on this and it appears that ‘the sensitivity to initial conditions’ is more commonly called the butterfly effect and topological mixing is actually a facet of topology…which i didn’t know, but has something to do with the constant change and manipulation of an element (that’s probably the wrong term) without it’s tear or break. wikipedia topology if you don’t know what i’m talking about haha. my question is about the orbit thing, I don’t understand it, can you send me in a direction to find some more information on this in real terms rather than mathematical jargon? I’m curious about how these principles can apply to things like weather and economy simultaneously…even more relative perhaps, it is showing up in digital art (the butterfly effect thing) on a random basis apparently and I’m trying to wrap my mind around the whole thing. To see if I buy into it or if its nonsense hippie shit.
MFC said this on April 3, 2010 at 2:21 pm |
Hey MFC. Damn chaos mathematicians. They make everything more complicated than it needs to be.
Chaos Theory, as you know, just describes the interaction of seemingly random events that are actually related. So chaos theory should predict the splatter of yoke when you drop an egg, or the exact makeup of water splashing during a cannonball.
Sensitivity to initial conditions, as you pointed out, refers to the makeup of things before an event in which they are involved. The mass and composition of the egg, the density of the floor, speed and density of the air, and all of the other “initial conditions” are obviously relevant
when trying to figure out where the yoke is going to go when you drop the egg.
Topological Mixing refers to tendancy for things to get messy. Specifically, it refers to the extent of overlap between all possible outcomes of an event. The more interaction between all possible outcomes (called phase space) the more chaotic an event. Painting a dot is less chaotic than slapping the canvas with a wet brush, even though they both have similar initial conditions. The number of possibilities of paint-on-canvas is more finite with a dot than with a splash. An unbroken egg is less chaotic than a broken one, even though their initial conditions regarding the make-up of the egg are identical. The more topological mixing, the less likely that the initial conditions alone will be capable of predicting the outcome of an event. These are oversimplified examples, math does a better job of explaining this phenomenon. Imagine factoring really big numbers: the intial conditions are complex, but there is very little, if any, topological mixing, as the same rule applies to all possible states of a number. This makes predicting the outcome a relatively easy task.
Density of periodic orbits refers to the time dimension of all possible outcomes of an event. So it essentially identifies when topoligical mixing occurs within a phase space. Knowing when the mixing occurs is necessary to establish the magnitude of the mixing, just like knowing the distance traveled over a given amount of time is necessary to calculate the velocity of an object.
So, a simple mathematical model for determining the outcome of a chaotic event would be:
Sensitivity Init. Cond. X Top. Mixing
_______________________________ = chance of a given outcome
Density of Periodic Orbits
The way that chaotic systems interact is almost infinitely reduceable. One could argue that changes in the weather cause changes in the stock market, but the extent of that is entirely dependent on the equation above. Since there is relatively little toplogical mixing between these two systems at any given point in time, they could loosely be called independent systems, but chaos theory demonstrates how that is untrue. Even a small amount of topological mixing within a given periodic orbit is enough to permit the possibility that the two dynamic systems could effect the outcome of one another, regardless of how unlikely that may be. There is way more topological mixing between stock traders (or the banking sector, corporate activity, Mardi Gras, etc) and the stock market in any given orbit, so the sensitivity to those initial conditions is more likely to influence the outcome of an event (like the stock market crashing) than is sensitivity to the initial conditions of the weather.
The conventional butterfly effect analogy is similar to this logical reduction. A butterfly flapping its wings does result in topological mixing with a weather system on the other side of the world, but it is extremely minute compared to factors like air temperature or humidity. Therefore a butterfly’s wings are very unlikely to affect the outcome of a weather system, but they do interact with one another. Chaos mathematics seeks to quantify that interaction.
Hope that answers your question.
Wil Finley said this on April 3, 2010 at 3:37 pm |
Yes, that def. answers the majority of the question. I had a pretty good wrap on it through the arts perspective but I needed clarification on the theory side to make it work in my head. It turns out that art historians have made a relationship between the death of an artist and the increase in use of that particular artists function and aesthetic. He was saying that things that are seemingly chaotic (in this case death of an artist) are, in fact, dependent upon and are a factor in things like art movements and genre classification throughout the past 60 years. ….however small…like the butterfly flapping his wings, a death of someone not important in the arts community leads to an increase in understanding and use of that particular style. ie. Warhol dies and the eighties spin into a supermarket color schemed, large production pop icon style of art (called post-modern) These artists were the generation subsequent to the ones giving Warhol the strongest criticism on the lack of importance and theory in his work….his death, while only mourned and honored in the pop culture NY scene, over the course of 10 years changed art into a different format, medium, Idea, direction and plight. This all made sense but the person who wrote the article was basically saying that the most uncoordinated events (chaos) generally equal a paving of the road for things that seem unavoidable in retrospect but would have never have been possible without the lack of order to begin with…he summed it up with saying that the Chaos theory is an all encompassing theory that can explain and predict the outcome of events. It just seems to me that it would make sense in the stock market or anything where numbers played a role…. I mean, math is infinite but definable none-the less…it seems a bit impossible to predict the future of art…I guess you could in predicting the use of new mediums and different social situations. But it’s funny to think that we are trying to put an equation to chaos. I like it.
Thanks homie, you made some sense out of the variables and that’s what I was going for.
Holler.
MFC said this on April 4, 2010 at 7:15 pm |