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\begin{document}

\title{Models of Weather and Climate}
\author{Summer Semester 2025, Exercise 1}
\date{Due Thursday, May 8, 2025}
\maketitle

\begin{enumerate}

\item (From Kaper/Engler, Chapter 7, Exercises 2, 3.)
\begin{enumerate}
\item Let $A \colon t \mapsto A(t)$ be a differentiable function
defined on a neighborhood of $t=0$ whose values are real $n\times n$
matrices, and suppose that $A(0) = I$, the $n \times n$ identity
matrix.  Prove that
\[
  \Bigl( \frac{\d}{\d t} \det A(t) \Bigr)(0)
  = \Tr \frac{\d A}{\d t}(0) \,.
\]
\item Let $\phi_t$ be the flow of the initial value problem
\[
  \dot x = f(x) \,,
\]
i.e., $x(t) = \phi_t(x_0)$ is a solution to the differential equation
with $x(0)=x_0$.  Assume, for simplicity, that $f \in C^2(\R^n, \R^n)$, and that
the flow is defined on all of $\R^n$ and for all positive times.
Prove that, for all $x \in \R^n$,
\[
  \Bigl( \frac{\d}{\d t} \det D \phi_t \Bigr)(0)
  = \nabla \cdot f \,.
\]
\item Let $U \in \R^3$ be a measurable set.  Apply the result from (b)
to the Lorenz equations
\begin{align*}
  \dot x & = - \sigma \, (x-y) \,, \\
  \dot y & = \rho \, x - y - x \, z \,, \\
  \dot z & = -\beta \, z + x \, y 
\end{align*}
to show that, for all $t$,
\[
  \frac{\d}{\d t} \Vol (\phi_t (U))
  = - (\sigma + \beta + 1) \,  \Vol (\phi_t (U)) \,.
\]
Use this result to explain why the attractor $\mathcal A$ for the
Lorenz equations cannot have a subset of positive volume.
\end{enumerate}


\item Recall from class that the definition of the (leading) Lyapunov
exponent $\lambda$ implies the following.  Suppose $x(t)$ and $y(t)$
are two solutions to the differential equation $\dot x = f(x)$.  Write
$\xi(t) \equiv x(t) - y(t)$ to denote their difference.  Then
\[
  \lVert \xi(t) \rVert \approx \e^{\lambda t} \, \lVert \xi(0) \rVert
\]
so long as the initial perturbation $\xi(0)$ is small and $t$ is not
too large.
\begin{enumerate}
\item Use this property to estimate the leading Lyapunov exponent for
the Lorenz system with standard parameters experimentally.

\item Suppose you are able to specify the initial condition with an
error of $10^{-3}$.  For how long do you expect you can predict the
evolution of the system until the error in the solution exceeds $10\%$
of its maximal value on the attractor?  All errors are measured in the
Euclidean norm in $\R^3$.

\end{enumerate}


\item Suppose you add the forcing term $\gamma \, \sin t$ to the
$x$-component equation and $\gamma \, \cos t$ to the $y$-component
equation.  Determine experimentally whether $\langle x \rangle$,
$\langle z \rangle$, and/or $\langle x^2 \rangle$ respond linearly to
small changes in $\gamma$.  How large can $\gamma$ become before the
response of the system is not smooth in $\gamma$?


\end{enumerate}
\end{document}
