Equilibrium point for this system

Question

$\dot x=y(y^2-\lambda)$

$\dot y=x+\lambda$

I have found a Equilibrium point so far $(x^*,y^*)=(-\lambda,0)$ ‚can you tell if this system has another equilibrium point?

Answer 1

From the first equation, you forgot that we could have

$$y(y^2-\lambda) = 0$$

This leads to $y = 0$ or $y = \pm~\sqrt{\lambda}$.

From the second equation, we have

$$x = -\lambda$$

This leads to three critical points

$$(x^*, y^*) = (-\lambda, 0),(-\lambda,-\sqrt{\lambda}),(-\lambda, \sqrt{\lambda})$$

Please note that from the comments that I forgot to mention that there are three critical points when $\lambda \ge 0$, and one critical point otherwise.