<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\documentclass[a4paper,12pt]{article}
\usepackage{epsfig}
\newcommand{\ds}{\displaystyle}
\parindent=0pt
\begin{document}

{\bf Question}

Let $f(x,y)=\left(\sin \frac{\pi}{3}x,\ds\frac{y}{2}\right)$. Find
all the fixed points and their stability.  Give a sketch
indicating basins of attraction and stable/unstable manifolds.

\medskip

{\bf Answer}

Fixed points: $\sin \frac{\pi}{3}x=x,\ \frac{y}{2}=y$ so $y=0$ and
$x=0, \pm \frac{1}{2}$.

$DF(x,y)=\left(\begin{array} {rr} \frac{\pi}{3}\cos \frac{\pi}{3}x
&amp; 0\\ 0 &amp; \frac{1}{2} \end{array} \right):$


$DF(0,0)=\left(\begin{array} {rr} \frac{\pi}{3} &amp; 0\\ 0 &amp;
\frac{1}{2} \end{array} \right),\
DF(\pm\frac{1}{2},0)=\left(\begin{array} {rr} \frac{\pi}{2\sqrt3}
&amp; 0\\ 0 &amp; \frac{1}{2} \end{array} \right)$.

\begin{center}
\epsfig{file=314-3-6.eps, width=50mm}
\end{center}

Therefore saddle at (0,0), sinks at $(\pm\frac{1}{2},0)$.

Two basins of attraction:

\begin{center}
$\begin{array}{l}
x&gt;0 (\textrm{attracted to }(\frac{1}{2},0))\\
x&lt;0 (\textrm{attracted to }(-\frac{1}{2},0))
\end{array}
\ \ \ 
\begin{array}{c}
\epsfig{file=314-3-7.eps, width=50mm}
\end{array}$
\end{center}
\end{document}
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