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\begin{document}
\begin{center}
{\LARGE {Mathematics 118A, Winter 1998}}
\end{center}
\begin{center}
\large Final Exam
\end{center}
\bigskip

\begin{enumerate}

\item Recall that the trace of a matrix $A$, denoted by $\tr A$, is
the sum of the diagonal elements of $A$.  Set
\[
  \trip A \trip = |\tr A|.
\]

\begin{enumerate}
\item Prove that $\trip A+B \trip \leq \trip A \trip + \trip B \trip$.
\item Is $\trip \cdot \trip$ a norm?  Prove the statement or give a
counter example.
\hfill (2+3)
\end{enumerate}

\item Recall that a Hamiltonian system in the plane is defined through
its Hamiltonian function $H=H(p,q)$.  The equations of motion are
\[  \dot p = - \frac{\partial H}{\partial q}, \qquad
    \dot q = \frac{\partial H}{\partial p}.
\]
Show that the phase flow corresponding to this system is volume
preserving.  \hfill (3)

\item Consider the second order equation
\[
  \ddot x + x - \tfrac12 x^2 = 0.
\]

\begin{enumerate}
\item Find and analyze the equilibrium points by linearization.

\item Find a first integral.

\item Use the first integral as a Lyapunov function to show that the
origin of the phase plane is a Lyapunov stable critical point.

\item Is the origin asymptotically stable?  Explain your reasoning.

\item Sketch the phase portrait of the system.

Hint: Recall that orbits are level curves of the first integral.

\item Are there periodic orbits?  How many?

(Justify your answer.  The hint to part (e) may be useful here, too.)
\end{enumerate}
\hfill(4+2+2+2+2+2)

\item  Consider the system
\begin{align*}
  \dot x & = (x-y) \sqrt{x^2+y^2} - x \\
  \dot y & = (x+y) \sqrt{x^2+y^2} - y .
\end{align*}  

\begin{enumerate}
\item Use Bendixon's criterion to show that if a periodic orbit
exists, part or all of it must lie outside a circle of radius
$\tfrac23$ centered at the origin.

\item Find the periodic orbit.  (There is only one---why?)

\item Is the periodic orbit stable?  Why or why not?

\item Linearize the system about the periodic solution.

\item State at least one solution to the linearized equation.
(Explain your answer!)

\item Recall that the Floquet exponents $\lambda_1, \dots, \lambda_n$
of the linear equation
\[
  \dot z = A(t) \, z,
\]
where $A(t)$ is a $T$-periodic $n \times n$ matrix, satisfy
\[
  \sum_{j=0}^n \lambda_j = \frac1T \int_0^T \tr A(t) \, dt
  \pmod{\tfrac{2\pi i}{T}}.
\]
Use this result to give an alternative proof of your answer to part
(c).

\item Sketch the phase portrait.
\hfill(2+2+2+2+2+2+2)
\end{enumerate}

\item The cuckoo is a bird that doesn't breed its own eggs, but places
  them into the nests of other birds.  When a young Cuckoo grows, it
  kills all of its step-siblings.

A very simple model of a cuckoo population $x$ and one host species
$y$ (the robin, say) is the system
\begin{align*}
  \dot x & = x y - k x \\
  \dot y & = y - y^2 - x y,
\end{align*}  
where the positive parameter $k$ is related to the natural death rate
of the cuckoo.  (For simplicity, other biologically relevant
parameters have been set to one.)
\begin{enumerate}
\item Explain briefly why this system models the situation described
above.   What is the role of each of the terms?

\item Determine the critical points of the system.  (There are three.)
Are they all relevant, i.e., do they correspond to nonnegative bird
populations?

\item For which values of $k$ can the two species co-exist?  Can the
cuckoo or the robin ever become extinct?

To answer this question, analyze the stability of the critical points
and sketch the phase portrait for different values of $k$.

Hints:
\begin{itemize}
\item One of the critical points is associated with the eigenvalues
\[
  \lambda = \frac{-k \pm \sqrt{5 k^2 - 4 k}}{2}.
\]
\item Look very carefully at your answer to part (b) again!

\item You may not need a full linear analysis of all the critical
points as some information can be gathered directly from the
equations. 
\end{itemize}
\end{enumerate}
\hfill (3+3+8)



\end{enumerate}
\end{document}
