Math 6410 • ODE

\begin{enumerate}
\item[(a)] State the Center Manifold Theorem for rest points. Briefly explain its
importance in bifurcation theory.
\item[(b)] Using an approximate center manifold, determine the nature of the bifurcation
at the origin when \(\epsilon=0\).
\[
\begin{aligned}
  \dot{x} &= \epsilon x-x^3+y^2,\\
  \dot{y} &= -y+x^2.
\end{aligned}
\]
\end{enumerate}