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\title{Differential Equations}
\author{Summer Semester 2024, Exercise 6}
\date{Due Wednesday, June 19, 2024}
\maketitle

\begin{enumerate}

\item (From Teschl, Problem 6.18.)   Show that $L(x) = x_1^2 + x_2^2$ is a Lyapunov function for the
system
\begin{gather*}
  \dot x_1 = x_2 \,, \\
  \dot x_2 = - \eta \, x_2 - x_1
\end{gather*}
where $\eta \geq 0$ in a neighborhood of the equilibrium point
$x^*=(0,0)$.  What can you say about the stability of the equilibrium
point?

\item (From Teschl, Problem 6.27.) Consider a general system with
friction of the form
\[
  \ddot q = - \eta(q) \, \dot q - U'(q)
\]
with $\eta(q) > 0$ and $U(q)$ some differentiable function with
Lipschitz continuous derivative $U'(q)$.
\begin{enumerate}
\item Show that there are no non-trivial periodic solutions.

\emph{Hint:} Use the energy.

\item Show that minima of $U$ correspond to asymptotically stable
equilibria in phase space.
\end{enumerate}


\item Study the stability of the equilibria of
\[
  \dot x = 1 - 2 \mu x + x^2
\]
as a function of $\mu$.  Which bifurcations occur?

\item A very simple global energy balance model for planet Earth reads
\[
  C \, \dot T = (1-\alpha(T)) \, Q - \varepsilon \, \sigma \, T^4
\]
where $C$ is the heat capacity of the planet, the precise value does
not matter here,
$\sigma = 5.67 \cdot 10^{-8} \, \mathrm{W} \, \mathrm{m}^{-2} \,
\mathrm{K}^{-4}$ is the so-called Stefan constant (the corresponding
term in the equation describes the energy emitted from Earth via
black-body radiation per area), $\varepsilon = 0.6$ a ``Greenhouse
effect'' constant that models that parts of the black-body spectrum
are reflected back to the surface, $Q$ is the solar energy reaching
Earth per area with a current value of
$Q_0 = 342 \, \mathrm{W} \, \mathrm{m}^{-2}$, and $\alpha$ is the
albedo, i.e., the fraction of solar radiation reflected back into
space.  It is temperature dependent and assumed to follow the profile
\[
  \alpha(T) = 0.5 - 0.2 \,
    \tanh \frac{T - 265 \, \mathrm{K}}{10 \, \mathrm{K}}
\]
which models a smooth transition from full ice cover when
$T \ll 273\, \mathrm{K}$ so that the albedo is high to a surface
without any ice when $T \gg 273 \, \mathrm{K}$ so that albedo is
low.\footnote{Details about this and other models can, for example, be
found in H. Kaper, H. Engler, \emph{Mathematics and Climate}, SIAM,
2013.}  In the following, we shall use $Q/Q_0$ as the bifurcation
parameter.

\begin{enumerate}
\item Use Matplotlib to plot the location of equilibria in a diagram
with $Q/Q_0$ on the horizontal and $T$ on the vertical axis.

\item Describe the approximate location and type of bifurcations.

\item Which branches of equilibria are stable, which are unstable?
\end{enumerate}





\end{enumerate}
\end{document}
