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{\LARGE {Mathematics 118A, Winter 1998}}
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\large Homework Assignment 1, due Friday, January 16
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\begin{enumerate}
\item Recall the definition of a one-parameter group of
diffeomorphisms $\{g^t\}$ on an open subset $M \subset \mathbb{R}^n$:

The mapping
\begin{gather*}
  g \colon \R \times M \to M, \\
  g (t,x) = g^t x \quad \mbox{for } t \in \R, x \in M,
\end{gather*}
must satisfy
\begin{enumerate}
\item $g \in C^1$,
\item $g^t$ is a diffeomorphism for every $t \in \R$,
\item the family $\{g^t \colon t \in \R \}$ is a one-parameter group
of transformations on $M$.
\end{enumerate}
Show that requirement (b) is redundant.  I.e., show that (a) and (c)
imply (b). (Hint: What is the inverse of $g^t$?)

\item Let $\{g^t\}$ be a one-parameter group of diffeomorphisms on an
open subset $M \subset \R^n$, $x \in M$, and $v(x)$ the phase velocity
\begin{equation*}
  v(x) := \frac{d}{dt} g^t x \bigg|_{t=0} .
\end{equation*}
Prove that if $x$ is a fixed point of the phase flow, then $v(x)=0$.

\item Consider the equation
\begin{equation*}
  \dot x = x (1-x).
\end{equation*}
\begin{enumerate}
\item Find and characterize the critical points.  Sketch the flow in
the (one-dimensional) phase space.
\item Solve the equation.
\item Sketch the solution in the extended phase space.
\end{enumerate}

\item Consider the equation
\begin{equation*}
  \ddot x - \lambda \dot x - (\lambda - 1) (\lambda - 2) x = 0
\end{equation*}
with $\lambda$ a real parameter.

Find the critical points and characterize these points.  Sketch the
flow in the phase plane and indicate the direction of the flow.

\end{enumerate}
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