Math 6410 • ODE

Consider
\[
\begin{aligned}
  \frac{dx}{dt} &= Ax+f(x),\\
  x(0) &= x_0
\end{aligned}
\tag{2.1}
\]
where \(x\in\R^n\) and \(A\) is a constant \(n\times n\) matrix. Suppose that \(A\) is
semisimple, that \(f\in C^2\) and that \(f(0)=0\), \(D_xf(0)=0\).
\begin{enumerate}
\item Define what is meant by the stable manifold, the center manifold and the unstable
manifold at the origin for (2.1).
\item Give a representation of the solution of (2.1) when \(f=0\), in terms of the
eigenvalues of \(A\) and the associated projections and nilpotents. Indicate how to
compute the projections.
\item Discuss the asymptotic behavior of the solution in (b) for \(t\to\infty\).
Characterize the manifolds defined in (a) as precisely as possible.
\item Give a representation of the solution of (2.1) for \(f\ne0\). Under what
conditions, if any, on \(A\) and the initial data can one conclude that this solution
exists for all \(t\in\R^+\)? You need not prove all the assertions for the last part, but
should at least state your reasons.
\item State the Hartman-Grobman theorem for
\[
  x'=F(x)
\]
and state clearly what further conditions, if any, are needed on (2.1) to apply the
Hartman-Grobman theorem to (2.1).
\end{enumerate}