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\begin{document}

\title{Differential Equations}
\author{Homework 7}
\date{Due in class Thursday, May 3, 2018}
\maketitle

\begin{enumerate}

\item \label{1} Fill in the details of Verhulst, Example 13.9: Consider the
augmented system
\begin{gather*}
  \dot x = \mu \, x - x^3 + x \, y \,, \\
  \dot y = - y + y^2 - x^2 \,, \\
  \dot \mu = 0 \,.
\end{gather*}
Near the origin, the center manifold can be parameterized by
\[
  y = h(x,\mu) \,.
\]
\begin{enumerate}
\item Writing $h$ as a power series in $x$ and $\mu$, show that
\[
  \dot u = - x^2 + \mbox{higher order terms} \,.
\]
\item Show that the dynamics on the center manifold is given by
\begin{gather*}
  \dot x = \mu \, x - 2 \, x^3 + O(x^3) \,, \\
  \dot \mu = 0 \,.
\end{gather*}
\item
Determine the critical points of the reduced system and their
stability as a function of $\mu$.
\end{enumerate}

\item (Verhulst, Exercise 13.3.) Find an approximation to the center
manifold $W_{\mathrm c}$ near the origin for the system
\begin{gather*}
  \dot x = -y + x \, z - x^4 \,, \\
  \dot y = x + y \, z + x \, y \, z \,, \\
  \dot z = - z - (x^2+y^2) + z^2 + \sin x^3 \,.
\end{gather*}
Use the Center Manifold Theorem to determine whether the origin is a
stable equilibrium point.

\item Consider the second order equation
\[
  y '' + y' - \mu \, y + y^2 = 0
\]
with parameter $\mu$.

\begin{enumerate}
\item Write the equation as a first order system.
\item Find the eigenvalues and eigenvectors of the system linearized
about the origin.
\item Explicitly transform the system into new coordinates $u$ and $v$
such that the linear part is diagonal.

\emph{Hint:} You should find (with appropriate definitions of $u$ and
$v$), that
\begin{gather*}
  \dot u = \mu \, (u+v) - (u+v)^2 \,, \\
  \dot v = -v - \mu \, (u+v) + (u+v)^2 \,.
\end{gather*}

\item As in Problem~\ref{1} above, augment the system by the parameter
equation $\dot \mu$ and approximate the center manifold in the
$u$-$v$-$\mu$ phase space near the origin.

\item Show that the local dynamics on the center manifold is given by
\[
  \dot u = \mu \, u  - u^2 + O(u^3) \,.
\]

\item Find the critical points of the reduced dynamics and determine
their stability; draw a bifurcation diagram in the $\mu$-$u$ plane.

\end{enumerate}
\end{enumerate}

\end{document}
