Math 6410 • ODE

Let \(E\) be an open subset of \(\R^n\) containing the origin and \(x_0\). Assume that
\(\mathbf{f}\) is a continuously differentiable function mapping \(E\) into \(\R^n\). Suppose
\(\mathbf{f}(0)=0\) and \(A=D\mathbf{f}(0)\). Let \(U\) and \(V\) be subsets of \(E\) containing
the origin.
\begin{enumerate}
\item Define the maximal interval of existence \(I(x_0)\).
\item Define the flow \(\phi_t(x_0)\).
\item Define a homeomorphism \(H\) of \(U\) onto \(V\).
\item Suppose \(x_0\in U\). State the Hartman-Grobman theorem in terms of \(\phi_t\), \(A\),
\(H\) and \(x_0\). Be sure to include all necessary conditions.
\item Consider \(x=(x_1,x_2)\) with
\[
\begin{aligned}
  \dot{x}_1&=-x_1,\\
  \dot{x}_2&=x_2-x_1^2,
\end{aligned}
\]
and \(x(0)=x_0\). Write down the explicit solution to this equation. Give the linearized
system \(\dot{y}=Ay\), \(y(0)=y_0\) and its explicit solution. Give the homeomorphism \(H\)
which relates the two solutions.
\end{enumerate}