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\title{Models of Weather and Climate}
\author{Summer Semester 2025, Exercise 2}
\date{Due Thursday, May 29, 2025}
\maketitle


In class, we have derived a linear response formula for a
time-independent perturbation of a dynamical system with a physical
measure.  Namely, consider the perturbed system
\begin{equation*}
  \dot x = f(x) + \delta f
\end{equation*}
and suppose that the unperturbed system $\dot x = f(x)$ has a physical
measure $\rho$.  Provided that $\delta f$ is independent of both $t$
and $x$, the perturbed expectation of the state variable $x$ with
respect to the physical measure is given by
\begin{equation*}
  \delta \rho(x)
  = \int_0^\infty \int M(t,x) \, \d \rho(x) \, \d t \, \delta f \,,
\end{equation*}
where $M(t; x_0)$ is the fundamental matrix of the linear system
\[
  \delta \dot x = Df(x(t)) \, \delta x
\]
where $x(t)$ solves $\dot x = f(x)$ with $x(0) = x_0$.

\begin{enumerate}
\item Write a code that computes
\[
  \bar M(t) = \int M(t,x) \, \d \rho(x)
\]
where the values of $x$ are sampled from a long-time integration of
the Lorenz system (as a substitute for sampling from the physical
measure) for different values of $t$.

\item Write a code that computes
\[
  \delta \rho(x) \approx \int_0^T \bar M(t) \, \delta f \, \d t \,.
\]

\item In Exercise 1, Problem 3 you have already set up a code where
you can test, using the Lorenz system, the response of $\rho(x)$ to
different $\delta f$ by direct simulation.  Adapt the code to directly
compute $\delta \rho(x)$ for selected $\delta f$ and compare your
result with that in Problem 2.

\end{enumerate}
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