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\centerline{\bf \Large Introduction to Dynamical Systems and Chaos}
\centerline{\bf \Large take home final exam - corrected version}

\centerline{Imre P\'eter T\'oth, fall 2005}
\bigskip
As an exam, please solve the following exercises. They are meant to be 
easier than the midterm exercises were, because your time is limited, but 
they cover a bigger part of the material. Please work on your own. Of 
course, using the book is allowed.

\begin{enumerate}

\item
Consider the function $f_a(x)=ax(1-x)$ where $a$ is some real 
parameter. Show three different values of $a$ where the dynamical system 
defined by $f_a$ exhibits three greatly different kinds of behaviour.

\item
Consider the two-dimensional map $f(x;y)=(x^2-x+2y-1;2x-2x^3+y)$. Check 
that 
the point $(1;0)$ is periodic, and determine the stability of its orbit.

\item
Consider the Smale Horseshoe map as shown in Figure 5.16 of the book.
\begin{enumerate}
\item Is there an orbit, which never enters the region marked as $V_L$ in 
Figure 5.16 ?

\item Is there an orbit, which never enters the region marked as $V_L$ in
Figure 5.16, but which is dense in the rest of the invariant Cantor set?

\item Is there an orbit, which never enters the region marked by $1$ in
Figure 5.17, but which is dense in the rest of the invariant Cantor set?
\end{enumerate}

\item
Take a chess-board, and draw a smaller chess-board (of size $\frac{1}{8}$ 
of the original) instead of every black field. (White fields remain 
entirely white.) Than repeat this step infinitely many times with the 
small black squares you have after the previous step. What is the box 
dimension on the set of points which remain black forever?

\item
Solve Exercise 7.12 from the book.

\end{enumerate}
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