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\centerline{{\sc Problem set 4}}
\centerline{Math 207A, Fall 2011}
\centerline{Due: Wed., Oct. 26}


\bigskip\noindent
{\bf 1.} Newton's method for the iterative solution of the scalar equation $f(x) =0$ is
\[
x_{n+1} = x_n - \frac{f(x_n)}{f^\prime(x_n)}.
\]
If $f(x) = x^2 -2$, show that this equation becomes
\[
x_{n+1} = \frac{x_n}{2} + \frac{1}{x_n}.
\]
What are the fixed points of this system? Determine their stability. Compute $x_4$ numerically if $x_0 = 3$.

\bigskip\noindent
{\bf 2.} Find the fixed points of the system
\[
x_{n+1} = -\frac{\mu}{2} \tan^{-1} x_n
\]
and determine their stability.
Show that a period-doubling bifurcation occurs at $\mu = 2$. Is the resulting period-two
orbit stable or unstable?

\bigskip\noindent
{\bf 3.} Consider the discrete dynamical system on the circle for $x_n \in \Ts$
\[
x_{n+1} = x_n + \mu \qquad (\mathrm{mod}\ 2\pi)
\]
corresponding to rotation by an angle $\mu \in \Ts$. Describe the structure
of the orbits and how they depend on $\mu$.

\bigskip\noindent
{\bf 4.} Carry out numerical experiments for iterations of the logistic map
\[
x_{n+1} = \mu x_n (1-x_n)
\]
where $1\le \mu \le 4$ and $0\le x_0 \le 1$. (You can write
your own program or use the MATLAB script provided on the course website.)
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