Math 6410 • ODE

A nonconstant periodic orbit near the origin is created or destroyed as the parameter
\(\mu\) passes through zero in the one parameter family of equations
\[
\begin{aligned}
  \dot{x} &= y+\mu x,\\
  \dot{y} &= -x+\mu y-x^2y.
\end{aligned}
\]
\begin{enumerate}
\item[(a)] Determine whether the origin is stable or unstable for the system when
\(\mu=0\).
\item[(b)] For which values of \(\mu\) near zero does the system have a nonconstant
periodic orbit near the origin? Is this orbit attracting or repelling? Why?
\end{enumerate}