**
Chaos theory is defined as the study of complex nonlinear dynamic systems. Complex
implies just that, nonlinear implies recursion and higher mathematical algorithms, and dynamic implies
nonconstant and nonperiodic.
**

**
One can call it 'hyperspace physics', quantum physics.
**

**
Thus chaos theory is, very generally, the study of forever changing
complex systems based on mathematical concepts of recursion, whether in the form of a recursive
process or a set of differential equations modeling a physical system.
**

**
Chaos theory attempts to explain the fact that complex and unpredictable results can and will occur in systems that are sensitive to their initial conditions.
**

**
I have read many theories as to how Chaos Theory first started.
**

**
Once source credits Professor Edward Lorenz of M.I.T., a meteorologist, as the first to discover evidence supporting chaos theory in 1960: Under certain conditions, the motion of a particle described by certain systems will neither converge to a steady state nor diverge to infinity, but will stay in a bounded but chaotically defined region. By chaotic, we mean that the particle's location, while definitely in the attractor,
might as well be randomly placed there. That is, the particle appears to move randomly, and yet
obeys a deeper order, since it never leaves the attractor. Lorenz modeled the location of a
particle moving subject to atmospheric forces and obtained a certain system of ordinary
differential equations. When he solved the system numerically, he found that his particle moved
wildly and apparently randomly. After a while, though, he found that while the momentary behavior
of the particle was chaotic, the general pattern of an attractor appeared. In his case, the pattern
was the butterfly shaped attractor now known as the Lorenz attractor.
**

**
**

Reversing Time

Under certain conditions, the motion of a particle described by certain systems will neither converge to a steady state nor diverge to infinity, but will stay in a bounded but chaotically defined region. By chaotic, we mean that the particle's location, while definitely in the attractor, might as well be randomly placed there. That is, the particle appears to move randomly, and yet obeys a deeper order, since it never leaves the attractor. Lorenz modeled the location of a particle moving subject to atmospheric forces and obtained a certain system of ordinary differential equations. When he solved the system numerically, he found that his particle moved wildly and apparently randomly. After a while, though, he found that while the momentary behavior of the particle was chaotic, the general pattern of an attractor appeared. In his case, the pattern was the butterfly shaped attractor now known as the Lorenz attractor. Click to see animated versions ~ Site 2

__The 'Butterfly Effect'__: This is one of the catch phrases of chaos theory. It refers to
sensitive dependence on initial conditions. In nonlinear systems, making small changes in the initial
input values will have dramatic effects on the final outcome of the system. A butterfly flaps its wings
and the weather changes in China. This is one of the catch phrases of chaos theory, called the
butterfly effect. It refers to sensitive dependence on initial conditions. In nonlinear systems, making
small changes in the initial input values will have dramatic effects on the final outcome of the system.

Theories abound as to real-life examples of this phenomenon:

1. The weather: small changes in weather effect larger patterns.

2. The stock market: slight fluctuations in one market can effect many others.

3. Biology: A small change in a virus in monkeys in Africa creates a "thunderstorm" of an effect on the human population around the world with the appearance of the AIDs virus.

4. Evolution: small changes in the chemistry of the early Earth gives rise to life.

5. Psychology: Thought patterns and consciousness altered by small changes in brain chemistry or small changes in physical environmental stimuli.

The butterfly effect occurs under two conditions:

1. The system is nonlinear.

2. Each state of the sytem is determined by the previous state. In other words, the output at each moment is repeatedly entered back into the system for another cycle through the mathematical functions that determine the system.

The result of the butterfly effect is unpredictability. Small differences in initial input can have dramatically different results after several cycles throught the system. In the fractals pictured above, points that are very close together can be different colors. The results can tend toward infinity at different rates or toward zero, even though the initial points are very close together.

Finally, one of the trademarks of these sorts of chaotic systems is self-similarity on different scales. If we were to change the boundaries in the pictures above, similar patterns would be found, no matter what scale we chose. Likewise, we may speculate about the fractal nature of nature itself.

SOCIETY FOR CHAOS THEORY IN PSYCHOLOGY AND LIFE SCIENCES CHAOS WITHOUT THE MATH CRYSTALINKS FRACTALS Chaos links, math science, and computer science all together. CREATION INDEX ALPHABETICAL INDEX OF ALL FILES CRYSTALINKS MAIN PAGE