Question:
Consider the system
\begin{align} \frac{dx}{dt} & = y \\ \frac{dy}{dt} & = -(x^2+\mu)y - (x^2+\nu)x \end{align}
Conduct Hopf analysis at the parameter values where Hopf bifurcations occur.
Attempt:
When we only have one parameter, I understand that a bifurcation occurs at the parameter value where the critical point(s) change stability. However, what is the definition of a bifurcation point when there are two or more parameter values?
In this particular example, I see that the critical points are at $(0,0)$ and $(\pm \sqrt{-\nu},0)$, where the latter only exists when $\nu<0$.
Going from $\nu>0$ to $\nu<0$, the critical point $(0,0)$ goes from being stable to unstable, and we also gain two extra critical points - so is it right to say that we have a bifurcation at $\nu=0$?