Chaos Theory

[dark angel]

In mathematics and physics, chaos theory describes the behavior of certain nonlinear dynamical systems that may exhibit dynamics that are highly sensitive to initial conditions (popularly referred to as the butterfly effect). As a result of this sensitivity, which manifests itself as an exponential growth of perturbations in the initial conditions, the behavior of chaotic systems appears to be random. This happens even though these systems are deterministic, meaning that their future dynamics are fully defined by their initial conditions, with no random elements involved. This behavior is known as deterministic chaos, or simply chaos.

The ButterFly Effect:

"It has been said that somthing as small as a flutter of a butterfly's wing can ultimatly cause a typhoon half the way around the world,"-Term meaning the more technical notion of sensitive dependence on inital conditions in the Chaos Theory.

I was wondering wether anyone has any veiws on this topic?
lol Um I was watching the Film "The Butterfly Effect" and thought this might be a good topic to discuss...

[Zotar]

I heard of o something called chaos theory Mathematics. If it has anything to do with the above, it is probably very advanced. What is it?

[Firesnake]

Aah I saw the butterfly affect
The theory that everything you do.. even the smallest thing can re-write your total life is an interesting one. my opinion: there is no proof for and against it so I pass

[Revenant]

As much as I belive in magic I want to say that it is more then possible. There are lots of things in the world, to whitch science can't give an explenation. Actualy I like this theory

[JET73L]

Yeah, small mistakes giving rise to more and more exponentially greater irregularities. It works great as a theory, or when explaining problems in experiments, or with the limits of things that require an inherent accuracy (such as clockwork mechanisms), but in the chaotic realm of uncontrolled nature, the entirety of the world is composed of butterfly wings, and if chaos is let to roam free, it will eventually develop patterns within.

[dark angel]

Dynamic Geomag: Chaos Theory explained

^ Umm yeah somthing i found..its a simple pendulum demonstrating the Chaos theory. The pendulum ends in a south magnetic pole, attracted by the four coloured north poles below...

Mathematical theory
Sarkovskii's theorem is the basis of the Li and Yorke (1975) proof that any one-dimensional system which exhibits a regular cycle of period three will also display regular cycles of every other length as well as completely chaotic orbits.

Mathematicians have devised many additional ways to make quantitative statements about chaotic systems. These include: fractal dimension of the attractor, Lyapunov exponents, recurrence plots, Poincaré maps, bifurcation diagrams, and transfer operator.

Sharkovskii's Theorem
In mathematics, Sharkovskii's theorem is a result about discrete dynamical systems. One of the implications of the theorem is that if a continuous discrete dynamical system on the real line has a periodic point of period 3, then it must have periodic points of every other period.

Suppose

f : R → R
is a continuous function. We say that the number x is a periodic point of period m if f m(x) = x (where f m denotes the composition of m copies of f) and having least period m if furthermore f k(x) ≠ x for all 0 < k < m. We are interested in the possible periods of periodic points of f. Consider the following ordering of the positive integers:

3, 5, 7, 9, ... ,2·3, 2·5, 2·7, ... , 22·3, 22·5, ..... , 24, 23, 22, 2, 1.
We start, that is, with the odd numbers in increasing order, then 2 times the odds, 4 times the odds, etc., and at the end we put the powers of two in decreasing order. Sarkovskii's theorem states that if f has a periodic point of period m and m ≤ n in the above ordering, then f has also a periodic point of period n.

As a consequence, we see that if f has only finitely many periodic points, then they must all have periods which are powers of two. Furthermore, if there is a periodic point of period three, then there are periodic points of all other periods.

Sharkovskii's theorem does not state that there are stable cycles of those periods, just that there are cycles of those periods. For systems such as the logistic map, the bifurcation diagram shows a range of parameter values for which apparently the only cycle has period 3. In fact, there must be cycles of all periods there, but they are not stable and therefore not visible on the computer generated picture.

Interestingly, the above "Sharkovskii ordering" of the positive integers also occurs in a slightly different context in connection with the logistic map: the stable cycles appear in this order in the bifurcation diagram, starting with 1 and ending with 3, as the parameter is increased. (Here we ignore a stable cycle if a stable cycle of the same order has occurred earlier.)

Veiws
I think.....really.....i dont know what to think...Theres scarse evidence to support the theorys, even though models and demo's have been constructed...I would like to think that maybe if this process is out there then maybe we could use it for our advantage...though as its been shown with the pendulum its terribly hard to predict...

[bond of flames 2.0]

The scientist on Jurassic Park had this whole cult following on the Chaos theory...the one who breaks his leg and notices the water shake in the foot print.
It totally makes sense cuz you can't predict what is going to happen with every action that you do or occurs. By sitting here right now we could be going down another path of life...but we don't know cuz we never went down any of the other paths.
The way the scientist explained it was hold your hand flat in the air or on a table or something just as long as its steady and flat...Drop a little bit of water on your hand and watch the way it falls off your hand. Do it again...will it go the same way or another..we don't know.
Brings me to an episode of stargate where they have this magic rock or mirror or some shit that allows you to see into multiple dimensions that have all occured based on actions that they have done. In one dimension they might be dead or in the other they may be wealthy or dead...who knows.
The whole thing makes sense but it doesn't at the same time. Its also like an escape theory that you can use to put a blame to something that occurs.
The bad part is is that we can't go back and to see what other paths we could have taken.

[Zotar]

hhhmmm.... tHis topic is like the total opposite of "Our life a story?" But it seems to have everything to do with it. Do little events influence us as individuals as well? Do we have any choice? What is in Controll? Chaos? Or Predestination? I think that Predestination effects Chaos which in turn effects those things that would happen. (That comes from my religious perspective, don't let it offend you.)

[Firesnake]

http://en.wikipedia.org/wiki/Chaos_theory

You gotta love them wikipedia

[Zotar]

Danget Firesnake! Now we have nothing to debate about. You just killed the thread. (JK)