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{\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
$x=0, \pm \frac{1}{2}$.

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


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

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\epsfig{file=314-3-6.eps, width=50mm}
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Therefore saddle at (0,0), sinks at $(\pm\frac{1}{2},0)$.

Two basins of attraction:

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