What's a simple explanation for the chaos theory?

Question

I'm just looking for a simple, easy to understand, paragraph or two on the chaos theory.

Answer 1

Since all the initial conditions cannot be known, in complex systems complete predictability is impossible to attain. The result is apparent "chaos" (unpredictability) even though the system in question is fully complying with the known rules of physics. The obvious example is weather, which man has struggled to predict throughout history, but since we cannot know every fluctuation in surface temperature, every eddy in the wind cycle, and every localized disturbance (a large bonfire, for example) we can only predict weather to a limited degree for a relatively large land area (not at all for, say, your yard).

Answer 2

A system exhibits chaotic behavior if the behavior appears random but is in fact governed mostly by a deterministic process.

Anyone with a spreadsheet can construct a simple system that exhibits chaotic behavior. Do the following:
1. In cell A1, enter a random number between -2 and 2.
2. In cell A2, enter the formula: =A1^2 - 2.
3. Fill A2 down a hundred rows or so.

Examine the list. Look at the numbers, but also look at a graph of the numbers. If you do this, you will notice that the list you obtain has the following properties:
1. The numbers appear to be without order, at first glance.
2. The numbers stay between -2 and 2.

Even though the list is generated very simply, and the only randomness is what we put into it at the beginning, the list wanders all over the place and never settles down.

Other respondents have discussed some of the terms associated with chaos. You can use this spreadsheet to examine a few of those.

Change the number you put in A1 slightly. If it was 0.12345, change it to 0.12346. You should notice that the bottom of the new list looks completely different from the original, if you examine a few cells at a time. This is the phenomenon called sensitive dependence on initial conditions: when the initial conditions are altered, the long-term behavior of the system is totally altered.

Now enter the number -1 into A1. The list consists entirely of -1. Now enter -1.61803398874989 (it's the golden mean). The list bounces back and forth between two numbers. If you enter -1.80193773580484, it will circulate between three numbers. You can find special numbers like these, called periodic points, throughout the interval [-2,2] -- they fill it up, in the same way that the rational numbers do. This phenomenon is called recurrence, and is another characteristic of a chaotic system.

Answer 3

First, a quick touch on Quantum Theory. There is a certain amount of uncertainty in any measurement, so that we can not know both the exact location and exact momentum of an object. Keep that idea on the back burner.

Now, imagine making a formula to describe the path of a bunch of gas molecules in a jar. These molecules will all be hitting (interacting with) one another and the imperfect edge of the jar. it is easy to imagine that if you had an absolutely perfect knowledge of every variable in the jar that you would be able to calculate the exact path of all of these particles. What if, however, your velocity was off by just a tiny amount with one of those particles. In its first interaction it would hit the other particle at a different point and they would both go zinging off in the wrong directions and pretty soon none of the gas molecules would be traveling according to your calculated routes. This can be summed up to small errors in initial condition leads to large errors after a given amount of time in a chaotic system.

Now because we know that there is a limit to how precise we can take a measurement we know that in large interacting systems precise details about the future cannot be directly calculated. However, there are still many ways analyze the these systems using probabilistic mathematics.

Answer 4

I think one important component is "sensitive dependence on initial conditions". This is also known as the "butterfly effect", in which the breeze created by a butterflys wings in Brazil causes a hurricane to rage across the US Eastern Seaboard a few months later. In a nutshell, this just means that in chaotic systems, a very small change in the initial conditions can give wildly different results. No butterfly, no hurricane.

Another important concept is self-similarity, especially with a class of mathematical relations called fractals. This is seen in nature all the time. For example, the branches of a tree have branches themselves, which in turn have smaller branches, and eventually twigs. In fact, the trunk of the tree is really just a special branch.

If you want to play around with chaos and fractals, do a google search for a program that lets you explore the Mandelbrot set. You'll get to make some very cool pictures, and it'll demonstrate both of the concepts I mentioned.

Answer 5

In multivariate systems, the results of a computation are often a function of the initial state system, such systems are chaotic.
The system is deterministic if we know the initial conditions to the proper significant figures. However if we do not know the initial conditions of a chaotic system to the proper precision, statistically expected fluctuations in the initial system can lead to completely different end states (a.k.a. the butterfly effect). However for a system to be truly chaotic, it depends on other factors.

Answer 6

First Theory of Chaos:
The Universe was originated by a Big Bang
Therefore, there was explosions everywhere
And this gave origin to the chaos in all the universe

Second Theory of Chaos
People who spoke the same language try to be like God
God confuses them by giving each one of them a different language

Answer 7

out of chaos theory has it the the universe developed and perhaps the way things are going it may soon go back to chaos, or perhaps this earth we live on is part of chaos.

Answer 8

order arises from chaos. it might not be the order that you want, but eventually all energy from an event will have been expended and a new order arises.

throw a deck of cards in the air-----------total chaos in action

eventually they land all over the floor-------------new order.

Answer 9

Expect the unexpected.

Answer 10

S**t happens.