Chaos theory (I need a crash course)

I would very much like (following someones advice) to learn about Chaos Theory in a basic and then perhaps more advanced sense...

If someone can provide a brief crahs course, or a website that could get me started, in terms normal people can understand (with an AS in Physics, lol)... I would be highly grateful!

Thanks all.

-Rob

Reply 1

Squishy

Strange advice, eh...

This is a pretty good page, and so is this. The standard book most people recommend for Chaos Theory is "Chaos" by James Gleick, but it's a bit waffly.

Reply 2

jpowell

Chaotic functions are ones that are incredibly sensitive to initial condtions. So even a very small change in the initial condtions will cause massive changes in the results.

Also check out Chaos by James Gleick.

I had a few lectures on chaos theory last year as part of the wave phenomena course. There are certain types of equations called non-linear partial differential equations, whereas linear equations can be solved "nicely" non-linear equations can get very complicated and are chaotic.

The textbook example of a chaotic function is:

f(n+1) = f(n)*(1-f(n))*k

This is a recursive function, to find the next value sequentially you multiply the previous value by 1 minus the previous value, and multiply by a constant k.

The idea is to take a value of k, and plug in values of f(1) = x and see if the function converges on a specific value as n increases.

Take k = 2.1 and any value as your f(1), i'm going to start with f(1) = 0.2. On my calculator I type in 3, hit enter, and then type in Ans*(1-Ans)*2.1 and because Ans is the previous value, hitting return emulates the recursive function. The function is only well behaved when you use numbers less than 1, convince yourself that a number greater than 1 will cause the function to go to -infinity, and a number less than 1 will have the same effect, so only start with numbers between 0 and 1

so lets see what happens

0.2
0.336
0.4685184
0.5229187068
0.5238969389
...
and after a few more iterations it tends to a number
0.5238095238

That number is called an attractor.

Now try any other value of f(1) between 1 and 0 and you will fall into this 'attractor'.

Lets change k, try k = 2.6, this takes a lot longer to converge, but it does converge to 0.6153846154

Those two examples were not that interesting, but what about k = 3.2? This doesn't converge to a single number, but instead oscillates between 0.5130445095 and 0.7994554905.

But we still have not reached chaos.

Let us now try k = 3.8 and f(1) = 0.5, after 50 iterations we are at 0.5371401901 with no sign of the number converging, now try 50 iterations with f(1)=0.5001, we get 0.6305575200.

That is chaos, a tiny change in the initial conditions, just 1 ten thousandth and after 50 iterations the values were completely different.

If you really want to learn more then do check out the James Gleick book, it is excellent.

Reply 3

RobbieC

I am considering popping to my library tomorrow and checking out that book... as this idea is really starting to get my mind working, and interest me... I have spoken to my AS Physics teacher about it before, but she doesn't really like talking about stuff and says im stupid... well not really, but kinda.. Lol

Yeah, thanks for the suggested reading and topic, Squishy, and the highly mathematical explanation AMM, as we all know how much I love maths! :s-smilie:

Lol...

ty

Reply 4

vortex_199

You can find most of the definitions on this sitehttp://www.s-cool.co.uk/default.asp just click aslevel then physics. You'll see something at the side saying physics definitions.

As for crash coursing get the book 'AS Physics in a week', though you could do it in a couple days.