# Chaos Theory & The Butterfly Effect

### How a Fluttering Butterfly Wing Can Cause A Tornado

## How Chaos Theory Began

*main principle*of Chaos Theory.

## How The Interest In Choas Theory Arised

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

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*.

__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.__Order and Disorder__: the white sectors within the chaotic range represent order within disorder. These often occur in surprising unpredictable manners.__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?

*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