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\title{Ordinary Differential Equations}
\author{Final Exam}
\date{May 22, 2018}
\maketitle

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\begin{enumerate}

\item Consider the system
\begin{gather*}
  \dot x = \mu + x^2 + x \, y + y^2 \,, \\
  \dot y = x^2 + x \, y - y \,, \\
  \dot \mu = 0 \,.
\end{gather*}

\begin{enumerate}
\item Consider the critical point $(0,0,0)$ and compute the linearly
stable, linearly unstable, and linearly neutral subspaces.

\item Note that the system has a two-dimensional center manifold.
Compute an expression for the center manifold up to terms of second
order.

\item Show that the motion on the center manifold is given by
\begin{gather*}
  \dot u = \mu + u^2 + \text{h.o.t.} \,, \\
  \dot \mu = 0 \,.
\end{gather*}

\item Now treat $\mu$ as a parameter and show that the system in
center manifold coordinates has a bifurcation at $\mu=0$.  Name that
bifurcation and sketch a bifurcation diagram in center manifold
coordinates.  (Extra credit if you manage to sketch the bifurcation
in the original $2+1$-dimensional phase space.)

\end{enumerate}
\hfill(5+5+5+5)


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\item Consider the system
\begin{gather*}
  \dot x = \mu \, x + y - x \, (x^2 + y^2) \,, \\
  \dot y = -x + \mu \, y - y \, (x^2 + y^2) \,.
\end{gather*}
\begin{enumerate}
\item Show that the origin is the only critical point and determine
the linear stability of the origin as a function of the real parameter
$\mu$.

\item Write the system in polar coordinates.

\item Show that the system has a periodic orbit for $\mu>0$.  Conclude
that the system has an Andronov--Hopf-bifurcation at $\mu=0$.

\item Sketch the bifurcation diagram.

\end{enumerate}
\hfill(5+5+5+5)
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\item \label{1} Prove the following simple averaging theorem.

Consider the equation
\[
  \dot x = \varepsilon \, f(x,t) + \varepsilon^2 \, g(x,t;\varepsilon) \,,
\]
where $f \colon \R^n \times [0,\infty) \to \R^n$ is $T$-periodic for
fixed $x \in \R^n$.  Assume that $f$ and $g$ are smooth.  Suppose that
there exists a ball of radius $R>0$ on which $f$, $g$, and all their
derivatives are bounded independent of $\varepsilon$.

Define
\[
  \overline f (y) = \frac1T \int_0^T f(y,t) \, \d t
\]
and suppose that the solution of
\[
  \dot y = \varepsilon \, \overline f(y) \, \qquad
  y(0) = x(0)
\]
remains in a ball of radius $R/2$.  Then
\[
  x(t) - y(t) = O(\varepsilon)
\]
on the time scale $1/\varepsilon$.

\emph{Hint:}  Define
\[
  u(x,t) = \int_0^t (f(x,s) - \overline f (x)) \, \d s \,,
\]
set
\[
  x = z + \varepsilon \, u
\]
and derive a differential equation for $z$. 
\hfill(10)

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\item Consider \emph{Mathieu's equation}
\[
  \ddot x + (1+2 \varepsilon \, \cos(2t)) \, x = 0 \,.
\]
\begin{enumerate}
\item Use the ansatz
\begin{gather*}
  x = z_1 \, \cos t + z_2 \, \sin t \,, \\
  \dot x = -z_1 \, \sin t + z_2 \, \cos t 
\end{gather*}
to show that
\begin{gather*}
  \dot z_1 = 2 \varepsilon \, \sin t \, \cos (2t) \, (z_1 \, \cos t +
  z_2 \, \sin t) \,, \\
  \dot z_2 = -2 \varepsilon \, \cos t \, \cos (2t) \, (z_1 \, \cos t +
  z_2 \, \sin t) \,.
\end{gather*}

\item Use averaging (see attached table for useful trigonometric
identities!) to conclude that
\[
  x(t) = \tfrac12 \, (x(0) + \dot x(0)) \, \e^{-\tfrac12 \varepsilon
  t} \, (\cos t + \sin t) +
  \tfrac12 \, (x(0) - \dot x(0)) \, \e^{\tfrac12 \varepsilon
  t} \, (\cos t - \sin t) + O(\varepsilon)
\]
on the time scale $1/\varepsilon$.  (Check the assumptions of the
statement in Problem~\ref{1}!)
\end{enumerate}
\hfill(10+10)

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\item Consider the system
\begin{gather*}
  \varepsilon \, \dot y = z + g(\phi) \,, \\
  \varepsilon \, \dot z = - y \,, \\
  \dot \phi = I \,, \\
  \dot I = 0 \,,
\end{gather*}
where $\phi \in \mathbb T$, where $\mathbb T = [0,2\pi]$ with
identification of the endpoints.

Show that the $y$-$z$ dynamics can be parameterized via $z=(\phi,I)$
provided the resonance condition
\[
  I \neq \frac1{\varepsilon k} \,, \quad k \in \mathbb Z
\]
is satisfied.

\emph{Hint:} Fix $I$, expand $g(\phi)$ and $z(\phi, I)$ as a Fourier
series, and match coefficients.
\hfill(10)

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\end{enumerate}
\end{document}
