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\def\classtitle{Dynamical Systems}
\def\classnumber{Math A4500}
\def\assignment{Homework 7}

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\newcommand{\R}{{\mathbb R}}% The real numbers

\begin{document}
\begin{center}
{\Large \bf \classnumber: \classtitle: \assignment}\\
\end{center}

\noindent
{\bf Name:} INSERT YOUR NAME HERE

\begin{questions}
\question {\bf Devaney 3e: I.12 \#3:}
Let $E_\lambda: \R \to \R$ be given by $E_\lambda(x)=\lambda e^x$.
Prove that $E_\lambda (x)$ has a unique ﬁxed point when $\lambda<0$. Show that this fixed point is attracting if $\lambda > -e$ and repelling if $\lambda < -e$.
\begin{solution}
Answer here!
\end{solution}


\question {\bf Devaney 3e: III.27 \#2:}
Describe the dynamics of the linear maps whose matrix representation is
given below. Identify precisely the stable and unstable sets.
\begin{parts}
\part
$\left(\begin{array}{rr}
2 & 1 \\
0 & \frac{1}{2}
\end{array}\right)$
\begin{solution}
Answer part 1 here!
\end{solution}

\part
$\left(\begin{array}{rr}
1 & 2 \\
3 & 4
\end{array}\right)$
\begin{solution}
Answer part 2 here!
\end{solution}

\part
$\left(\begin{array}{rr}
2 & 1 \\
1 & 1
\end{array}\right)$
\begin{solution}
Answer part 3 here!
\end{solution}

\part
$\left(\begin{array}{rrr}
\frac{1}{2} & 0 & 0 \\
1 & 2 & 0 \\
0 & 0 & 3
\end{array}\right)$
\begin{solution}
Answer part 4 here!
\end{solution}

\part
$\left(\begin{array}{rrr}
1 & 1 & 0 \\
1 & 2 & 0 \\
0 & 0 & 2
\end{array}\right)$
\begin{solution}
Answer part 5 here!
\end{solution}

\end{parts}

\question {\bf Devaney 3e: III.27 \#4:}
Consider the linear map
$$L({\mathbf x})=\begin{pmatrix}
\frac{1}{2} & 0 \\
0 & \frac{1}{3}
\end{pmatrix}.$$
Prove that $L^n(\mathbf x) \to 0$ for all $\mathbf x \in \R^2$. Prove that, if $\mathbf x$ does not lie on the y-axis, then the orbit of $\mathbf x$ tends to $\mathbf 0$ tangentially to the x-axis.
\begin{solution}
Answer here!
\end{solution}


\question {\bf Devaney 3e: III.28 \#2:}
Let $\Sigma_2$ be the $2$-sided shift space on the alphabet ${\mathcal A}=\{0,1\}$.
Prove that the shift map $\sigma:\Sigma_2 \to \Sigma_2$ is a homeomorphism.
\begin{solution}
Answer here!
\end{solution}


\question {\bf Devaney 3e: III.28 \#6:}
Let $\mathbf s^* \in \Sigma_2$. Prove that $W^s({\mathbf s^*})$ consists of precisely those sequences whose entries agree with those of $\mathbf s^*$ to the right of some entry of $\mathbf s^*$.
\begin{solution}
Answer here!
\end{solution}


\question {\bf Devaney 3e: III.28 \#8:}
Let $\mathbf 0 = (\ldots 00.000 \ldots)$ and
$\mathbf 1 = (\ldots 11.111 \ldots)$ be the two constant sequences in $\Sigma_2$. A sequence $\mathbf s$ is a heteroclinic sequence
if $s \in W^s(\mathbf 0) \cap W^u(\mathbf 1)$.
Describe the entries of such a heteroclinic sequence.
Prove that such sequences are dense in $\Sigma_2$.
\begin{solution}
Answer here!
\end{solution}


\end{questions}
\end{document}
