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\centerline{{\sc Problem set 8}}
\centerline{Math 207A, Fall 2018}
\centerline{Due: Fri., Dec. 7}


\bigskip\bigskip\bigskip\noindent
{\bf 1.} Sketch the phase plane of the system
\[
x_t = x^2, \qquad y_t = -y.
\]
Linearize the system about the equilibrium $(0,0)$ and determine
the unstable, stable, and center subspaces of the equilibrium.
What is the stable manifold $W^s(0,0)$?
Show that there are many possible choices of an invariant ($C^1$) center manifold
$W^c(0,0)$ that is tangent to the center subspace at $(0,0)$.


\bigskip\noindent
{\bf 2.}  The following Selkov system for $x(t), y(t) \ge 0$, depending on parameters $a,\mu > 0$,  provides a simple model of glycolysis:
\[
x_t = -x + ay + x^2 y,\qquad y_t = \mu -a y -x^2y.
\]
(a) Find the fixed point and classify it as a function of the parameters. Show that if $0<a<1/8$, then
there are possible Hopf bifurcations as $\mu$ increases from $0$ to $\infty$. What are the possible
Hopf bifurcation points $(x_0,y_0;\mu_0)$?

\smallskip\noindent
(b) By plotting the phase planes numerically, show that Hopf bifurcations occur at the points in (a) and determine
whether they are subcritical or supercritical. 


\bigskip\noindent
{\bf 3.} Consider the following discrete dynamical system for $x_n \in \Rl$ depending on a parameter $\mu \in \Rl$:
\[
x_{n+1} = -\frac{\mu}{2} \tan^{-1} x_n.
\]
(a) Describe the fixed point(s) of the system and determine their stability.
What bifurcations of fixed points occur as $\mu$ increases from $-\infty$ to $\infty$?

\smallskip\noindent
(b) Show that a period-doubling bifurcation occurs at $\mu = 2$. Is the resulting period-two
orbit stable or unstable?


\newpage\noindent
{\bf 4.} The H\'enon map on $\Rl^2$ is given by
\begin{align*}
x_{n+1} &= a - b y_n - x_n^2,
\\
y_{n+1} &= x_n.
\end{align*}
(a) Find the fixed point and determine its stability.

\smallskip\noindent
(b) Carry out a numerical exploration of this map for various values of the parameters $a, b \in \Rl$.
It's up to you how much you want to explore, especially at the end of the quarter, but you should provide a plot of
the forward orbit of the point $(x_0,y_0) = (0,0)$ for $a=1.4$ and $b=-0.3$  in the region $-2 \le x \le 2$, $-2 \le y \le 2$
and briefly discuss the result.


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