Math 6410 • ODE

Let \(A\) be an \(n\times n\) real matrix and let \(\varphi(t,x_0)\) denote the solution
of the initial value problem
\[
  \begin{cases}
    \dfrac{dx}{dt}=Ax,\\
    x(0)=x_0.
  \end{cases}
\]
Suppose that \(k\) not necessarily distinct eigenvalues of \(A\) have non-positive real
part and the rest have positive real part
\[
  \Re\lambda_1\le\cdots\le\Re\lambda_k\le0
  <\Re\lambda_{k+1}\le\cdots\le\Re\lambda_n.
\]
Show that there is a \(k\)-dimensional subspace \(E\subset\R^n\) such that
\begin{enumerate}
\item[(a)] \(E\) is forward invariant: for every \(x_0\in E\) we have
\(\varphi(t,x_0)\in E\) for all \(t\ge0\);
\item[(b)] If \(x_0\notin E\) then the distance from \(\varphi(t,x_0)\) to \(E\) tends
to infinity as \(t\to\infty\).
\end{enumerate}