The Chaos Theory

[Edge]

The name "chaos theory" comes from the fact that the systems that the theory describes are apparently disordered, but chaos theory is really about finding the underlying order in apparently random data.

 

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, pg. 141)

 

http://www.imho.com/grae/chaos/chaos.html

 

Well, we have all heard of the "Chaos Theory". I have read things about this and I find it interesting. It seems to be true... kinda relates with Entropy. Anyway, I wonder, is this theory more complex than "one slight change on a variable may lead to very different results over a large period of time" or it is not that "chaotic"?

 

Thoughts on this theory?--

[Tormod]

I just finished the book "Deep Simplicity: Chaos Complexity and the Emergence of Life" by John Gribbin and it was a very interesting (although a bit heavy) read. The basics of chaos theory really isn't that hard to grasp, but the implications are mind-boggling.

 

Generally chaos theory can be used to do a lot of things, but most of all it helps to identify the lack of equilibrium in dynamic systems. Basically, a system in equilibrium will be a "dead" system. An example used here is life itself - if the atmosphere is in equilibrium, it will not be life-sustaining. Atmospheres which contain life (as we know it) will affect the atmosphere it lives in, by creating cycles which brings the system out of equilibrium. The system in itself will always attempt to reach equilibrium (or least energy demanding path - this is related to the laws of conservation of energy, and thus entropy) but life disturbs it.

 

So a living system is chaotic, and is always "on the edge" - small changes can have drastic effects, but changes on all levels are also unavoidable. This is most easily observed in weather patterns, for example - you can't have a never-ending wind going from one point to another. Winds move from high pressure areas to low pressure areas, thus balancing out the difference between the two. The problem is that chaotic systems are also extremely complex, which makes it impossible to make good predictions - just like it is in meteorology.

 

Chaos theory is very interesting stuff!

[sanctus]

 

 

Anyway, I wonder, is this theory more complex than "one slight change on a variable may lead to very different results over a large period of time" or it is not that "chaotic"?

 

Thoughts on this theory?--

 

The supposition of seeing the results only in a large period of time isn't always true. Sometimes it happens after a few iterations.

A way used to study chaos is consider a box with hard spheres inside. At the start you know exactly all the positions and the impulsions of every sphere. Then you look at its evolution and then you change slightly something and you look again at the evolution. See for example: http://cherrypit.princeton.edu/papers/paper-153.pdf

 

One of the most interesting application of chaos theors is the KAM theorem (if I remember right there is a thread about that on the forum) with which under certain suppositions you can prove the stability of the solar system.

[sanctus]

By the way, what is your actual question?

[ixa]

Chaos theory has some interesting applications in , for example, physics, economics and medical sciences. Many body systems, like the solar system, can exhibit chaotic behaviour. Other systems that can be chaotic include fusion plasmas and the atmosphere. Also, it has been shown that a healthy human heart behaves chaotically, whereas a strictly periodic behaviour is a sign of a heart condition.

 

Several methods to detect chaos can be used, one of the most straightforward being the calculation of the rate of separation of different solutions; in a chaotic system, the solutions with different initial solutions diverge at an exponential rate. The rate of separation can be identified with the (largest) Lyapunov exponent of the system. These exponents can be calculated, for example, from measured numeric data, e.g. time series. There are also many other methods that can be used to detect chaos, for example non-linear prediction methods. A good reference on various detection methods and chaos in general is a book by J.C. Sprott, "Chaos and time-series analysis". Sprott also has a nice website on chaos, "http://sprott.physics.wisc.edu/".

 

Chaos is in a sense also related to entropy. If one defines entropy in the usual way that is used in information sciences (C.E. Shannon, The Bell systems technical journal vol 27, 379) a chaotic system produces data that has higher entropy than a non-chaotic (and non-random) one. This fact has been used to detect heart diseases and chaos in fusion plasmas (I can give references if someone is interested).

 

And to answer the question, chaos theory is much more complicated than just a theory that states "one slight change on a variable may lead to very different results over a large period of time". Mathematically it is basically a theory on non-linear dynamics, and in the real world it has many significant and important applications.

[Boerseun]

Every morning I leave my house in a chaotic state. Every evening, when I return, my house is ordered. I do not understand this, neither do I attempt to explain it.

 

I have either stumbled upon the solution to the world energy crisis, or I don't pay my housekeeper enough.

[Edge]

By the way, what is your actual question?

Well, it's mainly this.

 

Can order be obtained from disorder?

 

I guess... this thread is mostly for looking a explanation of the Chaos Theory. As it seems very interesting... and real.

[Tarantism]

eh how can order be derived from chaos???? that makes no sense at all, seeing as how chaos is the exact opposite of order. hm.

[paultrr]

eh how can order be derived from chaos???? that makes no sense at all, seeing as how chaos is the exact opposite of order. hm.

 

I think it would pay for you to check out: http://www.imho.com/grae/chaos/chaos.html One can get order out of chaos. Also, quantum randomness tied in with chaos theory has its own interesting ability to generate order out of what appears to be pure randomness.

[Turtle]

I think it would pay for you to check out: http://www.imho.com/grae/chaos/chaos.html One can get order out of chaos. Also, quantum randomness tied in with chaos theory has its own interesting ability to generate order out of what appears to be pure randomness.

 

___I wish I could have said so much in so little so elegantly as Paul. Chaos is a poor name for a rich subject; it unabashedly redefines the very ideas of 'order', randomness', 'pattern'.

 

 

 

 

 

:evil:

[sanctus]

Well, it's mainly this.

 

Can order be obtained from disorder?

 

I guess... this thread is mostly for looking a explanation of the Chaos Theory. As it seems very interesting... and real.

 

 

There is a very simple example where you obtain order from disorder: take a magnet in a magnetic field at high temperatures. If the temperature is high enough the will be no overall magnetization that means the spins are disoredered now you cool it down and eventually you get a permanent magnetization that means the spins are aligned with the field i.e you have order.