Sunday, January 28, 2007

Nonlinear Dynamics: what is it?

So I'm taking a course in nonlinear dynamics this term, and its really cool. Nonlinear dynamics is the study of chaotic systems in physics. So what does that mean?

Chaos is a technical term in physics meaning that something is deterministic (follows knowable laws), but unpredictable. Weather is chaotic, we understand a lot about how temperature and air pressure affect the weather in the short term, but long term prediction is almost impossible, very small errors in the measurements of the weather now leads to very large errors later on, and since there is always some error in any measurement we are stuck.

That doesn't mean that things are hopeless, however. There is quite a bit we can learn about chaotic systems, especially by studying simple mathematical chaotic systems. One of the simplest is the logistic map. Basically given some value x, you can get a new value x' by x' = λ * x (1-x). Given suitable values of λ x will stay between 0 and 1, no matter how many times you iterate it. It is used to model populations in biology. I you iterate this a whole bunch of times (like 10000) one of three things will happen depending on your value of λ: 1) x could diverge off somewhere, these are what we would consider unphysical solutions, these values of λ are not interesting, and we do not study them. 2) x could settle on one (or more) stable values, we call these orbits. 3) x could be chaotic, essentially be randomly distributed within a certain range.

One very cool thing about this, is that if you start with a stable orbit and increase λ by a certain amount, you will have 2 orbits, increase it by a smaller amount and you will get 4 and so on, we call this a period doubling cascade and it goes on until the number of orbits is essentially infinite and the system becomes chaotic (the picture from wikipedia is a graphical representation of this). The interesting part about this is that if you take the ratio of the amounts you have to increase λ by to go from the start of the orbit to the end of the orbit, and the same for the next one you get a number that is constant for lots of similar types of systems. This is called quantitative universality.

OK, that's enough for now, maybe I'll post some more on this in the future.

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