Chaos Theory Verifications!?

This is difficult to explain (particularly in simple text) because so many people misunderstand the simplest form of chaos theory, simple statistics. The first misunderstanding that people often have (and that you imply by how you worded your question) is that chaos theory should deal with certainties, it does not. It does exactly the opposite. Instead it produces quantifiable values of uncertainty. What quickly baffles people in chaos theory is the uncertainty is not about the data you've taken it is an uncertainty in which random distribution process probably produced the randomization in your data set.

In very simple terms chaos theory implies to an experimentalist where they should study to explain what is happening. Time and time again this mathematical analysis technique has lead to fascinating understandings by identifying the things required for understanding before they are understood. There are also times that a chaos theory analysis of clearly random process yields no understanding at all other than some unknown random factor is in play.

I shall try to add a bit to the explanation above.

Dynamic physical systems are described by differential equations of different kinds.

If systems are deterministic coefficients are constant. If coefficients are variable it can happen that for some values the results are unstable and even a minute value change can lead to a tremendous change in the result.

One of the best examples is the flow turbulence which starts at a threshold value of the fluid velocity but does not have a constant pattern. Such instabilities lead to the "butterfly effect" hypothesis which in fact is not anything else than an image how it could happen. In fact in chaotic systems losses of energy lead to a stabilization, after a while, if new energy is not supplied. For instance turbulence can decrease if fluid temperature goes down and viscosity increases and under circumstances even disappear.

For an example of where chaos theory produced a result, look at the Abelian sandpile model. This deceptively simple example of self organized criticality was not understood until chaos theory was applied.

Thank you for a link to that Model

Chaos happens when you have a non-linear system with feedback. In this case, the system behaves differently for large values than for small values and quickly becomes unpredictable.

A simple example is to take a scientific calculator with a cosine function (set to radians). Put in an initial value and repeatedly press <cos>, so you're calculating cos(cos(cos(cos...))). Depending on the initial value, the value will either converge, cycle between 2 values, cycle between 4 values, ..., or become completely chaotic.

Here is a physical example, where the force on a pendulum is non-linearly dependent on the position.

https://wiki.brown.edu/confluence/display/PhysicsLabs/3A95.50+Chaotic+Pendulum

This may help. Eureqa

Newton's *gravity* was never verified, but accepted as truth for centuries...until Einstein came along a "visualized" space-time warpage!

This is the universal chaos law Second Law of Thermodynamics

I once attempted to use the logistic difference equation to model the instability of a spectrographic spark from tungsten electrode to disc sample electrode (that slowly rotates, and is translated linearly (to work sort of like a record player). The instability was caused by injecting hydrogen partial pressure into the argon sheath gas, although other gases could also be used in introduce the instability.

is the logistic difference equation, where r is the critical parameter, and the index n represent successive (generational) values of X. One simply plugs in the previous answer to arrive at the next one in a sequence of events. The mapping of X in the critical parameter space is shown below:

http://www.stsci.edu/~lbradley/seminar/logdiffeqn.html

In my own experiment, this form of chaos was not followed, rather the probability of "jump" location of the spark was simply binomial probability, although the hydrogen gas pressure still did model as the critical parameter, it was merely the amplitude of the branched binomical probability density.

Tough guy, James.

A good starting point for Chaos theory is the book Chaos, Making a New Science by James Gleick 2011.(available for Kindle) Although the book does not give a formal mathematical treatment of the subject it does present a good overview of the topic and the history of its development from Edward Lorenz to present

Another point I would like to make, there is a difference between a random process and a chaotic process. In a random process, each time that you start from the same initial conditions, you will get a different outcome. A chaotic process, on the other hand is completely deterministic: if you start with the the same initial conditions, you will always have the same outcome. However, any infinitesimal change in the initial conditions can lead to a radically different outcome.

Come to my house any school day at 7:00 AM.

I'll show you chaos like you've never seen.

Lol.