Math 6410 • ODE

Consider the nonlinear system in \(\mathbb{R}^3\):
\[
\begin{cases}
\dot{x}=y-x^2,\\
\dot{y}=-z+xy,\\
\dot{z}=-\lambda z+x^2,
\end{cases}
\]

where \(\lambda > 0\) is a parameter.
\begin{enumerate}

\item
Find all critical points of the system and classify them when \(\lambda > 0\).


\item
Analyze the stability of the origin \((0,0,0)\) using the center manifold theorem. Compute the
reduced system on the center manifold up to quadratic terms.


\item
Discuss how the stability of the origin changes as \(\lambda\to 0^+\). What happens if \(\lambda=0\)?

\end{enumerate}