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\centerline{\bf \Large Introduction to Dynamical Systems and Chaos}
\centerline{\bf \Large take home midterm execrises}

\centerline{Imre P\'eter T\'oth, fall 2005}
\bigskip
As a midterm, please choose and solve one of the following exercises.
The first two are theoretical, the second two are numerical, and the last 
one is mixed. Please work on 
your own -- of course, you are allowed (and even advised) to read the book 
and other literature. Also, please feel free to ask me for hints.

\begin{enumerate}

\item
State and prove Sharkovskii's Theorem. This is Challenge 3 from the book.

\item \footnote{This exercise is based on exercises in Robinson: 
Dynamical Systems -- CRC Press, 1995.}
Denote by $\Sigma_2$ the space of infinite sequences of $1$-s and $2$-s.
That is, $\Sigma_2=\{1,2\}^\IN$. We define the shift map $\sigma$ on 
$\Sigma_2$ by
$$\sigma((s_0,s_1,s_2,\dots)):=(s_1,s_2,s_3,\dots).$$

For any $\lambda>1$, define the metric $d_\lambda$ on $\Sigma_2$ by
$$d_\lambda(s,t)=\sum_{i=0}^\infty \frac{|s_j-t_j|}{\lambda^j},$$
for $s=(s_0,s_1,\dots), t=(t_0,t_1,\dots) \in \Sigma_2$.

\begin{enumerate}

\item
Prove that $d_\lambda$ is indeed a metric on $\Sigma_2$ for any 
$\lambda>1$.

\item
Prove that for any $\lambda,\mu>1$ the open sets defined by $d_\lambda$ 
and $d_\mu$ are the same, so the metric spaces $(\Sigma_2,d_\lambda)$ and 
$(\Sigma_2,d_\mu)$ are topologically the same. (Hint: show that for any 
$n\in \IN$ and $s\in \Sigma_2$, the set $\{t\in\Sigma_2 \mid t_j=s_j 
\text{ for $j\ge n$}\}$ is open in both metrics.)

\item
As a consequence, prove that the identity map $id$ is a homeomorphism from 
$(\Sigma_2,d_\lambda)$ to $(\Sigma_2,d_2)$ for any $\lambda>1$. Also, $id$ 
is a conjugacy from $(\Sigma_2,\sigma)$ to itself when the domain is 
considered with the metric $d_\lambda$, and the range with the metric 
$d_2$.

\item
A map $f:X \to X$ is called topologically mixing, if for any open subsets 
$U$ and $V$ of $X$, there exist a finite $n_0\in \IN$ such that for all 
$n\ge n_0$ the intersection $f^n(U) \cap V$ is nonempty.

Prove that $\sigma$ is topologically mixing on $\Sigma_2$ (with the 
metric $d_2$, say).

\item
Prove that the middle-third Cantor set $C$ (with the usual distance) is 
homeomorphic to $(\Sigma_2,d_2)$. (The usual coding will be a 
homeomorphism.)

\item
Let $f:C\to C$ be defined as $f(x)=3x\text{ (mod $1$)}$. Show that the 
range of $f$ is indeed in $C$ (actually, it is $C$ itself). Show that 
$(C,f)$ is conjugate to $(\Sigma_2,\sigma)$.

\item
Let $T:[0,1]\to [0,1]$,
$$T(x)=\begin{cases} 2x & \text{ if $x\le 1/2$} \\
                     2-2x & \text{ if $x> 1/2$} 
       \end{cases} $$
be the tent map. Let $I_1=[0,1/2]$ and $I_2=[1/2,1]$. Define the map
$h:\Sigma_2 \to [0,1]$ by
$$h(s)=\bigcap_{i=0}^\infty T^{-i}(I_{s(i)}),$$
where $s=(s(0),s(1),\dots)\in \Sigma_2$. (Here the notation $s(k)$ for the 
$k$-th element of the sequence $s$ is only used to avoid double 
subscripts.)

Prove that $h$ is well defined, that is, the intersection is a single 
point.

\item
Prove that $h$ is continuous, onto, and $T(h(s))=h(\sigma(s))$ for any 
$s\in\Sigma_2$. (That is, $h$ is a semi-conjugacy from $(\Sigma_2,\sigma)$ 
to $([0,1],T)$.

\item
Find the sets $h^{-1}(0)$, $h^{-1}(1)$, $h^{-1}(1/2)$.

\item
Prove that $h$ is one-to-one on most points, and at most two-to-one. What 
are the points in [0,1] that are the images of more than one sequence in 
$\Sigma_2$?

\item
Prove that $T$ is topologically transitive on [0,1].

\item
Show that $(\Sigma_2,\sigma)$ and $([0,1],T)$ cannot be conjugate, since 
there is no homeomorphism between $[0,1]$ and $\Sigma_2$.

\end{enumerate}

\item
The standard map is the family of maps $f_K:[0,2\pi]^2 \to [0,2\pi]^2$ 
defined by
$$f(x,y):=(x+y+K\sin(x),y+K\sin(x)),$$
where all values are taken mod($2\pi$), and $K\in \IR$ is some parameter.

If $K=0$, the sets $\{y=const\}$ are obviously invariant (they are called 
``invariant tori''). The system is not chaotic in any sense.
If $K>0$, but $K$ is small, most of these invariant tori survive, although 
they are deformed to some curves. However, as we take greater values of 
$K$, more and more of these invariant tori break up, and some kind of 
chaotic behaviour appears on a bigger and bigger part of the phase space.

Examine and demonstrate this transition from predictable to chaotic motion 
by numerically calculating orbits and drawing phase portraits for 
different values of $K$.

\item
Consider the logistic map $f(x)=ax(1-x)$ for the parameter value $a=3.56$
Find the periodic orbits of period at most $8$ with a good precision, 
using numerical calculation  of the dinamics. Find the Lyapunov exponents 
of these orbits. Describe the short and long term behaviour of 
trajectories that start near the period $4$ orbit.

\item
The Gauss map $g:(0,1)\to [0,1)$ is defined by
$g(x)=1/x \text{ (mod $1$)}$. 

Find (numerically) the Lyapunov exponent of the typical and some 
non-typical orbits of $g$, as precisely as you can. Some warings and 
hints:
\begin{itemize}
\item
for certain points in $(0,1)$, not all iterates of $g$ are defined. Which 
are these points? (Denote the set of such points with $H$.)
\item
$g$ has infinitely many fixed points. Finding the Lyapunov exponent is 
easy for these, but these points are -- of course -- highly non-typical.
\item
To find a typical Lyapunov exponent, choose a point at random, and follow 
the trajectory for a long time. Notice however, that (due to the way the 
computer operates) your random numbers tend to be in $H$, so you 
may not be able to do infinitely many iterations. On the other hand, 
numerical error is involved in your calculation at every step. 
Try to guess how long 
you should follow a trajectory, so that the error in your estimate on the 
Lyapunov exponent is reasonably small.
Convince yourself -- if this is true -- that your trajectory still 
resembles a ``typical'' one. For example, it doesn't seem to approach some 
periodic orbit.
\end{itemize}

\end{enumerate}

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