Chaos Theory Definition 

This is a theory that explains the complex and unpredictable results that will be produced in systems that are sensitive to its (their) initial conditions.

The Chaos Theory

To explain more clearly, a small change in a system can cause a catastrophic effect in somewhere in the world.

“Believe it or not, the flutter of butterfly’s wings in India can cause Tsunami in Indonesia.”

This is what the Chaos Theory states.

Tsunami in Indonesia

Chaos Theory and Tsunami

The Butterfly Effect

The butterfly effect Is stated as, “The small differences in initial condition of a system (like minor rounding off errors in maths) can cause a series of effects which cannot be determined. This is applicable to deterministic systems as well.

The Butterfly Effect

In deterministic systems we can predict the outcome of a operation. But due to the nature of chaos even the deterministic one turns unpredictable.

Chaos – Means “in a state of disorder”

Properties of A Chaotic System

1. Sensitive to initial conditions: This is nothing but popularly called  as the butterfly effect.

Each point in such a system is arbitrarily closely approximated by other points with significantly different future trajectories. Thus, an arbitrarily small perturbation of the current trajectory may lead to significantly different future behavior. However, it has been shown that the last two properties in fact imply sensitivity to initial conditions.

2.Topologically Mixing

The system will evolve over time so that any given region or open set of its phase space will eventually overlap with any other given region. This mathematical concept of “mixing” corresponds to the standard intuition, and the mixing of colored dyes or fluids is an example of a chaotic system.

3. Periodic orbits must be dense

This is more self explanatory. The orbits of the system undergoing must be densely populated. 

Properties of a Chaotic System

History of Chaos theory

Till now you are reading this article, surely you should have got a question how this chaos theory evolved? Who experimented and experienced. Yes, we have beautiful answers for it.

Lorenz’s  Experiment

The first true experimenter in chaos was a meteorologist, named Edward Lorenz. In 1960, he was working on the problem of weather prediction. He had a computer set up, with a set of twelve equations to model the weather. It didn’t predict the weather itself. However this computer program did theoretically predict what the weather might be.

Lorenz's Attractor

He left the experiment as such, but one day he wanted to see it again.  So instead of starting from the first he started to take the readings from the middle of the sequence. He entered the number off his printout and left to let it run. When he came back an hour later, the sequence had evolved differently. Instead of the same pattern as before, it diverged from the pattern, ending up wildly different from the original. Eventually he figured out what happened. The computer stored the numbers to six decimal places in its memory. To save paper, he only had it print out three decimal places. In the original sequence, the number was .506127, and he had only typed the first three digits, .506.

“The amount of difference in the starting points of the two curves is so small that it is comparable to a butterfly flapping its wings”

Applications of Chaos Theory

The chaos theory has so many applications in wide range of fields like

1. Mathematics
2. Biology
3. Computer Science
4. Economics
5. Physics and even in
6. Robotics

Lorenz’s Attractor

In certain energy states, 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.(attractor)

Lorenz modeled the location of a particle moving subject to atmospheric forces and obtained a certain system of ordinary differential equations.

Lorenz Equations

The particle appears to move randomly, and yet obeys a deeper order, since it never leaves the attractor.

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.

The most real time application of chaos is in ecology. Here this theory is used to show how population growth under density dependence can lead to chaotic dynamics.

FRACTALS

The fractals are one of the most important elements of chaos theory.

A Fractal is a geometric which is capable of reproducing itself at different measurements. A Fractal can be split into multiple parts; each part will look alike as a reduced size copy.

These fractals have many interesting properties. They are,

1. Lack of well-defined scale
2. Self-Similarity

Examples of Fractals

1. Sierpenski triangle
2. Clouds
3. Pulmonary alveolar structure

Sierpensk Triangle

Real-Time examples of Chaos Theory

CHAOS WASHING MACHINES

In 1993, Goldstar Corporation created a Chaotic Washing Machine. The washing machine works on the basis of  “Chaos Theory”. The washing machine uses the fact that there are identifiable and predictable facts in non-linear systems. The goal of this machine is to produce clean and less tangled clothes.

There is a Main Pulsator which periodically creates recurring phenomenon that alternately increases and decreases the stirring level of water. As  a result there happens a Chaos Motion. The amazing washing machine hit market and increased the annual share of Goldstar to 1.5 million in 1993.

CHAOS IN STOCK MARKET

Chaos analysis has determined that market prices are highly random, but with a trend. The amount of the trend varies from market to market and from time frame to time frame.

CHAOS THEORY IN WEATHER FORECASTING

As we saw in Lorenz’s Experiment, the chaos model is more helpful in long range weather forecasting.

There are many variables associated with the weather: temperature, air pressure, wind speed, wind direction, humidity and many more. The equations which control the weather involve all of these variables.

Cheers,
R.Gopinath

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