I did a search on “chaos theory” and nothing came up (seems odd). Anyway, here’s my questions:
Is this really a “new” kind of math, or just another strain in “older” math?
Is it controversial or largely accepted?
Does it have another name? Is complexity the same as chaos?
What are the best books for non-math types on this?
Finally, does it have any applications to social sciences?
Thanks! 
“Chaos theory” is more or less an umbrella term for those types of systems and mathematical equations which have certain characteristics, so it really isn’t a new “branch” of mathematics in and of itself.
It is pretty much accepted.
I’m sure it does but I’m drawing a blank for some reason. Oh yes, Hi Opal!
Chaos: The Making of a New Science by James Gliek
Lots.
James Gleick.
Chaos theory isn’t a different kind of math at all (I think?) so much as a realization of the role played by equations that don’t have a simple linear solution. Or, in less math-centric terms, of the fact that things can be deterministic without being deterministic in a simple straightforward “you put in x you get out x’” way at all – even to the point that the description of the deterministic cause requires such exactitude (due to sensitive dependence on initial conditions) that it is virtually as complicated as describing the entire environment and its entire history, at which point you’ve come damn close to reconciling determinism with non-determinism.
It has applications everywhere, not to mention mis-applications and attempted applications and so forth.
Hmmm…I’m no expert but I think I can take a fair shot at this.
I think it is hard to state what is ‘new’ about Chaos Theory. I think what is new is the idea that people can still get meaningful information from systems that were previously considered too chaotic to model accurately. This comes down more to probability analysis and I suppose some new techniques have been applied but at the bottom line its math isn’t all that new.
I believe it is largely accepted. Anyone who is a meteorologist understands that the weather is just too complex a beast to accurately model. With some use of chaos theory they can now get fairly good predictions out to a few days but as chaos theory suggests the predictions go to crap very fast beyond a certain point. Faster computers have been applied to the problem to consider more information but the gains in forecasting further out are marginal at best.
I think complexity is the same as chaos in this regard. Theoretically there is nothing chaotic about a system such as, say, the weather. In theory if you had every piece of relevant information about the weather you could accurately forecast the weather very far out. Unfortunately it is the complexity that gets you hence the notion of the Butterfly Effect. Miss the effect of a butterfly flapping its wings in Brazil and your forecast will eventually fall apart and you’ll miss predicting a tonado in Kansas. It is simply too complex a task to get ALL the information you need for a really good prediction. Still, underlying all that ‘chaos’ particles and the like are following very well defined physical parameters.
Dunno.
Maybe…not sure. Isaac Asimov in his Foundation Series postualtes the creation of something called Psychohistory whereby you can predict large scale movements in society (but individual effects are unpredictable). Kind of a neat idea. Possible? Maybe. Possible today…don’t think so. So far it remains in the realm of sci-fi.
Society for Chaos Theory
Complexity, Complex Systems & Chaos Theory
Chaos Theory
Chaos Without the Math
What is Chaos? An Interactive Course
Chaos Theory: A Brief Introduction
Thanks.
So how accurately CAN they predict the weather?
Chaos theory is more commonly called nonlinear dynamics by those who work in the field. When it was first introduced, it spawned a whole new framework for studying certain types of systems, so in that sense it was a new kind of math. The field itself is not controversial, although it remains to be seen how useful it will be in solving complex problems. (I’m sure there are also controversies within the field about specific ideas, as in all fields.)
As far as the details go, the basic idea of nonlinear dynamics is that for a given set of parameters (e.g. position of a dust particle in a weather system), the values at a later time can be very sensitive to the initial conditions. Nonlinear systems have what are called fixed points and basins of attraction, sort of high-probability regions where a given starting point is likely to end up. However, unlike in conventional classical mechanics, starting two identical systems at nearly identical points is no guarantee that the solutions will be within a predictable range of each other. They may result in identical trajectories, or wildly different ones. It’s an interesting field.
For the record, I think the James Gleick book is just dreadful. (I tried reading it both as a layperson when it came out, and later having taken some physics, and didn’t like it either time.) I don’t have a non-technical recommendation right now, although I’ll try to remember to look for one the next time I go to Borders.
Yes and no. It’s really just the study of systems of non-linear differential equations, but it’s pretty different from other ways of studying the same objects. You might say that it’s a new subdivision of an old branch of mathematics.
Mathematically speaking, very little is ever controversial. Particular applications may or may not be controversial–but most aren’t. Chaos theory works.
Complexity theory means something very different to me, but even in the realm of systems theory, I think that complexity theory is something different. Chaos theory is one of the branches of complexity theory, I think.
Gleick’s is good. Strangely enough, it’s hard to find decent books on chaos theory geared at the mathematician. 
Probably. But IME, social scientists tend to study systems that are best modeled by discrete structures. If you use continuous approximations, then chaos theory may be quite helpful. Applications to economics, and the stock market in particular, have been studied quite a bit.