Chaos Theory?

Question

I've read the definition on Wikipedia and know the total basics of the concept but can anyone dumb it down a bit for me, as I don't have a thesaurus and a mathematician handy to explain the wikipedia article.

Thanks.

Answer 1

Best Answer - Chosen by Asker

A very basic into to Chaos Theory would be to imagine this scenario:

Imagine a wildlife park in Africa.
The population of animals consist of lions and gazelles.
Lions eat gazelles.
If, in any year, there are lots of lions, they eat lots of gazelles, so the following year, there are not very many gazelles left. This causes a food shortage for the lions, so their numbers fall. Fewer lions allow the population of gazelles to recover.

Let A be the population of lions, and G be the population of Gazelles, expressed as a proportion so that A+G = 1.

Let's introduce a constant, S, which indicates how sexually active the Lions are.

So in any year, the population proportion of Lions next year (A) would be dependent on sexual activity (S), proportion of Lions this year (a), and how much food they have (G, which, if you remember, is 1-a)

How we have an equation of all 3 things:
A = S*a*(1-a)

Now open Excel or some other spreadsheet.
Enter some starting value between 0 and 1 (exclusive) in cell A1.
Enter this in to A2 : =4*A1*(1-A1) and hit return.
(the 4 is my value of S. You can experiment with other values, but 4 works well)
Click on A2 and drag down as far as you like, and the cells below will show the population of lions year by year. If you've done it correctly, you'll see what looks like random numbers, but it's merely the result of iterating a very simple equation!

That's Chaos!

Asker's Rating:

Asker's Comment:

Nice analogy :)

Answer 2

wow thats a lot to slim down;

Ok to start with;

Basicaly there is suposed to be a pattern even in the most chaotic of sinareos, (smoke, explosions, weather, storms etc)

Even though these seem to be random events (looking at the particles) if everthing was repeated exactly as it happed in the first occurance it would move exactly the same way.

But because in real life getting everything exactly the same down to the subatomic level is so difficult, repeated results are very few. (thats why its a thoery because you carn't prove it.)

Answer 3

There's no simple explanation of chaos theory. When things get small, all the rules go wrong. You need to start with infinitessimal calculus, then work on to quantum mechanics, then (when you come off the medication) you're ready to contemplate chaos.

Answer 4

The flapping of a single butterfly's wing today produces a tiny change in the state of the atmosphere. Over a period of time, what the atmosphere actually does diverges from what it would have done. So, in a month's time, a tornado that would have devastated the Indonesian coast doesn't happen. Or maybe one that wasn't going to happen, does.
(Ian Stewart, Does God Play Dice? The Mathematics of Chaos).

Answer 5

even in what seems like total, random chaos, there truly is an order. the fun is in deciphering.

Answer 6

1 small things diverge

2. x`= (1-x)^b this is the logistics equation, and is how states evolve,

3 for some states the evolution is quite smooth for others it is not.

4. you can't predict which is which

Answer 7

In a complex system, eg weather, planetary orbits etc. then the feedback & intereractions between the parts make what happens very sensitive to changes of the starting conditions.

So, although the system will behave predicatbly to known inputs, because it is so sensitive it is practically impossible to know the starting conditions with sufficient precission to acurately predict what will happen in the real system.

Answer 8

A couple of fundamental ideas:

* chaos = unpredictible - physical systems whose future behaviour cannot be predicted despite knowledge of the theoritical laws controlling that behaviour.

Reasons for difficulty/impossibility of future prediction:

* too many variables - to make accurate prediction of future behaviour of a complicated system (like weather) requires knowledge of more variables than may be realistically possible to measure

* complexity on continuum of scales - systems that exhibit 'fractal' nature have behaviour that depends on the state of the system as observed at many different size scales. To make accurate predictions the conditions on all those scales needs to be considered.

* non-linear math underpinning - mathematical descriptions of some complicated systems are given with equations that are very difficult to disentangle and use in simple ways. Incredibly powerful computing systems are needed to perform the calculations for predicting future behaviour & even then the predictions can become uncertain quite quickly.

Answer 9

Chaos theory was first suggested by scientists trying to make very long-range predictions of the weather. It was discovered (in computer data) that even very small changes in the assumed initial conditions could send the predictions way off the mark. It was suggested that even the beating of a butterfly's wings could upset the initial conditions and the butterfly has been selected as the symbol for chaos theory.

Source(s):

http://whatis.techtarget.com/definition/…

Answer 10

When you are holding a garden hose and you move your hand slightly, the place where the stream of water lands changes slightly. Therefore this is not a chaotic system.

When you change your aim point slightly on the cue ball when you break in a game of eight ball pool, the place where the cue ball finally comes to rest can vary greatly. Therefore this is a chaotic system.

Answer 11

If you have a self-contained system in which there are no random events, then in theory you should be able to predict what the system will do in the future based on what its current state is, or if it's more complex from what it's current and immediate past state is.

For example, if I stand at the top of a building with an apple and let it go, then simply from knowing that one position I could predict where the apple would be 1 second later, and how long it would take to hit the ground.

With more complex systems, you start needing to know more about past history. For example let's say a town has a one way system which means that for the postman to deliver all the letters he has to cycle down the same small street twice during his round. The first time he turns right at the bottom and the second time he turns left (he's a very unimaginative postman and always follows precisely the same route ;-) ). If I see the postman passing I can't tell how much further he has to cycle to complete his round, but if I watch until he gets to the end of the road, then I will have enough information to be able to calculate it. In other words a single observation wasn't enough to predict his behaviour but a small series of observations was.

Chaos Theory says that there are some systems whose behaviour is so complex, that in order to predict their future behaviour you would have to know exactly what they had been doing not just now or for the last few hours or even days, but from when they first started.

The most common example is the weather. In theory we understand all of the factors which cause it to rain, but the interaction of them is so complex that we can't actually predict with certainty where/when it will happen - the best we can do is a probability.

Answer 12

Generally speaking, "chaos theory" is the study of certain non-linear systems. Linear systems are far easier to understand but non-linear ones are often extremely complex.

In a "chaotic" system you have what is called sensitive dependence on initial conditions. Suppose your initial conditions are some set of numbers X0 and after a period of many iterations, the result is P(X0). Initially, your Ps are reasonably predictable but no matter how small a change in X0 you make, eventually the difference in the resulting Ps will become very different.

A very simple system to try is x=λx(1-x) where you feed the result back into the system. In other words, choose some x0 then x1=λx0(1-x0) then x2=λx1(1-x1) and so on.

Use a spreadsheet to iterate this equation, say, 1000 times starting with λ=1. For any initial 0<x<1 the system converges rapidly but as λ increases towards λ=3, it takes longer and longer for the system to converge for most starting points. And then when λ>=3 it becomes periodic where xn+k = xn means the system has period k (in other words iterating the system k times will give you the number you started with). But in all these situations, if x0=a and p(x0)=b then p(x0+e)~p(x0) for small e.

But as your λ increases roughly so that λ>3.57 you start to see a dramatic change in it's behavior so that if p(x0) and p(x0+e) are very different. This shows what chaos really is. A a small change in your initial condition leads to a large and unpredictable divergence at some point in the future.

This general principle applies even to the space program which is why NASA always needs to provide course corrections to its interplanetary spacecraft.