# Chaos Theory & The Butterfly Effect

## How Chaos Theory Began

It all started when Henri Poincare failed to find a solution to the 3-body problem, which is the study of the effect of gravity of three bodies on each other. However, he made an impressive realization which is that the slightest impossibly-minor change in the initial state of one of the bodies can cause a drastic change in its predicted path. This is called sensitive dependence on initial conditions, the main principle of Chaos Theory.

## How The Interest In Choas Theory Arised

Many years later, MIT meteorologist and mathematician, Edward Lorenz created 12 differential equations that predicted the weather. One day, he discovered that if he ran the computer calculations starting from two points with a 0.000127 difference between them, the outcomes will be completely different.

That was when his attention was drawn to chaos theory; he wrote a paper about it and about the butterfly effect. Since then, the public's interest in this theory started building up.

The picture to the right illustrates the outcome of Edward Lorenz's calculations.

+ The idea of a butterfly originated from the shape of the Lorenz attractor which looks like a butterfly. However, the question "does the flap of a butterfly's wings in Brazil set of a tornado in Texas?" was just a figure of speech said by Edward to clarify the extent of sensitivity on initial conditions.

## The Logistic Equation

Chaotic equations are non-linear. The logistic equation is one of the most famous and studied chaotic equations. It looks much simpler than it really is.
In biology, it is used to predict animal populations.

r is the driving parameter: rate of maximum population growth

Xn (as n tends to infinity): the predicted population

Xn converges to one limit when r is less than 3, then Xn experiences period doubling until r= 3.57 after which is experiences Chaos.

Many Principles of chaos theory can be observed with in the logistic equation, such as:

1. Unpredictability: since we can't know the initial population with absolute accuracy, it is impossible to predict future population since the slightest difference will lead to a different outcome.
2. Order and Disorder: the white sectors within the chaotic range represent order within disorder. These often occur in surprising unpredictable manners.
3. Feedback: a system tends to be chaotic when there is feedback, which means that the result affects the next result. Here, the change in population in a certain manner will lead to another change of population in another manner.

Above are some values of Xn over successive generations for r=3.8 and Xi=0.4 for blue curve and Xi=0.41 for red curve. We can observe how the outcomes are completely different.

## How is Chaos Theory Useful?

Chaos theory might seem useless to you at first since it's so "random" and unpredictable. But in truth, chaos theory has many application in various fields:
• Medicine: studies of epilepsy (seemingly random seizures)...
• Electrical systems: encryption systems, random number generators, communications...
• Engineering: damage monitoring and control, fluid turbulence...
• Economics: stock markets
• Social studies...

Below are covers of some books about the applications of chaos theory in several fields

## For example, chaos theory is used in microwaves

Using chaos theory, the microwave is made to produce blasts of random microwave radiations instead of organized and constant ones. Over time, the random blasting defrosts the food more evenly and therefore quicker whereas organized and constant blasts of microwave radiation tended to defrost only specific portions of the food. The use of chaos defrost has led to 60% reduction in time for defrosting food.

By SherineZaatari GS 2013