**“The
Sharkovskii Theorem****”**

By

**OĞUZHAN YÖRÜK**

**(BİLKENT UNIVERSITY)**

**Abstract:**** ** This
is a Senior Project presentation. The Sharkovskii
Theorem provides a special order of all natural numbers such that, for a continuous function f
on the real line, once the function f has orbits of length n than it has orbits
of lengths all other natural numbers that are dominated by n in this special
order. At the beginning of this special order there is the number 3 and hence,
this theorem has as consequence the fact that, if the function f has orbits of
lengths 3 then it has orbits of any length, which is one of the definition of chaotic behavior
under iteration. A detailed proof of this theorem will be presented.

**Date: ****Tuesday, December 16, 2014**

**Time: ****15:40**

**Place: ****Mathematics Seminar Room,
SA-141**

Tea and cookies will be served before the
seminar.