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\begin{document}

\title{Ordinary Differential Equations}
\author{Midterm Exam}
\date{April 5, 2018}
\maketitle

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\begin{enumerate}

\item Consider the system
\begin{gather*}
  \dot x = - x + y^2 \,, \\
  \dot y = -y^3 + x^2 \,.
\end{gather*}

\begin{enumerate}
\item Find all critical points and determine their linear stability. 

\item Clearly, the origin is a critical point where linear stability
analysis is inconclusive.  Show that the origin is an asymptotically
stable equilibrium point.

\emph{Hint:} Show that $V (x,y) = x^2 \, (1+y) + y^2$ is a Lyapunov
function in some neighborhood of the origin.

You may find the following elementary inequality useful:
\[
  a \, b \leq \frac{\delta}2 \, a^2 + \frac1{2 \delta} \, b^2
\]
for all $a,b \in \R$ and $\delta >0$.

\item Sketch the phase portrait.

\end{enumerate}
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\item Consider the $n$-dimensional linear system
\[
  \dot x = A(t)x
\]
where $A(t)$ is $T$-periodic.  Recall that the Floquet theorem states
that a fundamental matrix solution to this equation can be written as
\[
  \Phi(t) = P(t) \, \e^{Bt}
\]
where $P(t)$ is a $T$-periodic and $B$ a constant $n \times n$ matrix.

\begin{enumerate}
\item Consider the case $n=1$.  Determine $B$ and give necessary and
sufficient conditions so that every solution $x(t)$ remains bounded as
$t \to \infty$.

\emph{Hint:} Introduce the average
\[
  \overline A = \frac1T \int_0^T A(s) \, \d s \,.
\]

\item Consider the case $n=2$ with
\[
  A(t) = g(t) \,
  \begin{pmatrix}
    \lambda & 1 \\ 0 & \lambda
  \end{pmatrix}
\]
where $g(t)$ is a $T$-periodic real-valued function.  Determine $B$
and give necessary and sufficient conditions so that every solution
$x(t)$ remains bounded as $t \to \infty$.

\end{enumerate}
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\item Consider the system
\begin{gather*}
  \dot x = y \,, \\
  \dot y = -x + y \, (1 - x^2 - 2 \, y^2 ) \,.
\end{gather*}
Show that the system has at least one periodic orbit.
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\item Consider an autonomous planar differential equation.
\begin{enumerate}
\item Suppose a point $x$ is not on a periodic orbit.  Show that a
period orbit cannot be at the same time the $\omega$-limit set and the
$\alpha$-limit set of the orbit through $x$.

\item Show that the region bounded by a periodic orbit must contain an
equilibrium point.

\emph{Hint:} Argue by contradiction and use part (a).
\end{enumerate}
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\item Suppose $x(t)$ is a solution to the $n$-dimensional
differential equation
\[
  \dot x = f(x) + \varepsilon \, g(x) 
\]
where $f,g \colon \R^n \to \R^n$ are smooth vector fields.  Further,
let $y(t)$ solve the equation
\[
  \dot y = f(y)
\]
with $y(0) = x(0)$.

\begin{enumerate}
\item Show that $x(t) - y(t) = O(\varepsilon)$ on the time scale $1$,
assuming that both solutions exist on this time scale.

\item Suppose you know that the solution $y(t)$ is bounded (therefore
exists) on some time interval $[0,T]$.  What can you say about the
time interval of existence for $x(t)$ as $\varepsilon \to 0$?
\end{enumerate}
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\end{enumerate}

\end{document}
