### Abstract

## Chaos Theory

Niclas Wohlleben, Klaus Greven

The topic of our project is the iteration of real functions within the unit interval. We use a parabolic function f(x)=4λx(1-x),
where λ which controls the height of the parabola can take values between 0 and 1.
Iteration means choosing a point x_{0} and inserting it in the above function.
The result is the next x-value that will be entered in the function. This is repeated many times and visualised graphically.
A home PC was available to assist with these calculations.
Looking at the sequence of values, it becomes apparent that the sequence approaches some value and remains constant thereafter for some values of λ.
This can be easily seen graphically and such points are called fixed points.
For larger values of λ, there are two of these fixed points, for even larger λ 4, then 8 and so forth.
These bifurcations lead to the chaotic regime where there are so many fixed points that they cannot be identified any more.
Our project focuses on the regime that is not chaotic and the transition to the chaotic regime.
In this regime we can calculate the appearance and properties of fixed points based on the function and its derivative.
Further directions that we will consider include the use of different functions or exponentiation of the above function.