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\centerline{{\sc Problem set 6}}
\centerline{Math 207A, Fall 2018}
\centerline{Due: Fri., Nov. 16}


\bigskip\bigskip\noindent
{\bf 1.} Plot the bifurcation diagram and representative phases lines for the scalar ODE
\[
x_t = \mu x + x^3 - x^5.
\]
Identify the bifurcation points and classify them. How would the system behave if $\mu$ is increased quasi-statically from $-\infty$ to $+\infty$?
When is the equilibrium $x=0$ linearly stable but unstable to sufficiently large perturbations?

\bigskip\noindent
{\bf 2.} Consider the scalar ODE
\[
x_t = \lambda + \mu x - x^2,
\]
wheer $\lambda, \mu\in \Rl$ are parameters. Sketch the bifurcation diagram for the equilibria as a function of $\mu$ for fixed $\lambda$
in the cases $\lambda < 0$, $\lambda = 0$, and $\lambda > 0$. Identify the bifurcation points and classify them in each case.

\bigskip\noindent
{\bf 3.} Two rigid rods of length $L$ are connected by a torsional spring with spring constant $k$ and are subject to a compressive force of strength $F$. Explain why a reasonable model for the potential energy of the system is
\[
V(x) = \frac{1}{2} k x^2 - 2 F L(1-\cos x),
\]
where $x$ is the angle of the rods to the horizontal. If the rod is strongly damped with damping constant $\beta > 0$, then the ODE for its motion is
$\beta x_t + V'(x) = 0$. Show that a nondimensionalized form of the ODE is
\[
x_t + x - \mu \sin x = 0,\qquad \mu = \frac{2FL}{k}.
\]
Sketch a bifurcation diagram for the ODE and classify the bifurcation that occurs.

\bigskip\noindent
{\bf 4.}  Consider the system
\begin{align*}
&x_t = 1 - x - \beta xy,
\qquad
y_t
=\beta xy - (1+\gamma) y,
\end{align*}
where $\beta, \gamma >0$ are positive parameters. Sketch the bifurcation diagram for the equilibria as a function of $\beta$
and show that a bifurcation occurs at some $\beta = \beta_*(\gamma)$. What kind of bifurcation is it? Sketch typical phase planes
on $\Rl^2$ (using numerical solutions if you prefer) for $\beta$ close to $\beta_*$ when $\beta < \beta_*$,  $\beta = \beta_*$, and $\beta > \beta_*$.
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