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\def\d{{\mathrm d}}
\def\e{{\mathrm e}}
\def\R{{\mathbb R}}
\def\Re{\operatorname{Re}}

\begin{document}

\title{Differential Equations}
\author{Homework 4}
\date{Due in class Tuesday, March 13, 2018}
\maketitle

\begin{enumerate}


\item (From Verhulst, Exercise 4.8.)  Consider the system
\begin{gather*}
  \dot x = \frac{\partial E}{\partial y} 
         + \lambda \, E \, \frac{\partial E}{\partial x} \,, \\
  \dot y = -\frac{\partial E}{\partial x} 
         + \lambda \, E \, \frac{\partial E}{\partial y} 
\intertext{with $\lambda \in \R$ and}
  E(x,y) = y^2 - 2 \, x^2 + x^4 \,. 
\end{gather*}
\begin{enumerate}
\item Put $\lambda = 0$.  Determine the critical points and their
character by linear analysis.  What happens in the nonlinear system?
Sketch the phase plane.

\item What happens to the critical points when $\lambda \neq 0$?

\item Choose $\lambda < 0$.  Determine the $\omega$-limit sets of the
orbits starting at $(\frac12,0)$, $(-\frac12,0)$, and $(1,2)$.
\end{enumerate}

\item (From Verhulst, Exercise 6.4.) For the equation
\[
  \dot x = A x + B(t)x \,,
\]
where $A$ is a constant $n \times n$ matrix with all eigenvalues
having strictly negative real part, and $B(t)$ is $n \times n$ matrix
depending continuously on $t$.  Show that there exists $\delta>0$ such
that if $\lVert B(t) \rVert \leq \delta$ for $t \geq 0$, then $0$ is
asymptotically stable.

\item Let $J = \lambda \, I + N$ be a Jordan block matrix of dimension
$k$ and eigenvalue $\lambda$.  (Thus, $N$ is the nilpotent matrix with
$1$ on the superdiagonal and $0$ everywhere else.)

Show that if $f$ is an analytic function, $f(J)$ is well
defined and
\[
  f(J) = \sum_{i=0}^k \frac{f^{(i)}(\lambda)}{i!} \, N^i \,.
\]


\item Consider the differential equation
\[
  \dot x = A(t) x
\]
with
\[
  A(t) = S(t)^{-1} B S(t)
\]
where
\[
  B =
  \begin{pmatrix}
  -1 & 0 \\ 4 & -1
  \end{pmatrix}
  \qquad \text{and} \qquad
  S(t) = 
  \begin{pmatrix}
    \cos(at) & - \sin(at) \\ \sin(at) & \cos (at)
  \end{pmatrix} \,.
\]
\begin{enumerate}
\item Show that, for any $t$, all eigenvalues of $A(t)$ have negative
real part.

\item Show, that for suitable choice of $a$, the differential equation
has solutions with $\lVert x(t) \rVert \to \infty $ as $t \to \infty$.
\end{enumerate}


\end{enumerate}




\end{document}
