We consider the uniparametric differential equation $$\frac{dx}{dt} = \left(\lambda - x^2 + 2x\right)\left(\lambda - x^2\right).$$ I want to determine the equilibrium points as well as their stability depending on $\lambda$.
I know that (correct me if I'm wrong) if $\lambda < -1$ then there don't exist equilibrium points; if $-1 < \lambda < 0$ then the equilibrium points are $x_1 = 1 + \sqrt{1 + \lambda}$ (unstable) and $x_2 = 1 - \sqrt{1 + \lambda}$ (stable); if $\lambda > 0$ then the equilibrium points are $x_1$, $x_2$, $x_3 = \sqrt{\lambda}$ and $x_4 = -\sqrt{\lambda}$; if $\lambda = -1$ then it is $x_1 = x_2 = 1$ (unstable), and if $\lambda = 0$ then they are $x_1 = 2$ (unstable) and $x_2 = x_3 = x_4 = 0$ (stable).
How could I determine the stability of $x_1, x_2, x_3, x_4$ if $\lambda > 0$? Which cases should I consider and how can I realize? Moreover, I know that an equilibrium point is stable if it is asymptotically stable considering a one-dimensional system, but how could I notice that an equilibrium point is asymptotically stable, as well as stable?