Normal form of subcritical pitchfork bifurcation. I'm working the a dynamical system $\dot{x} = r x - \frac{x}{1+x^2}$. I have already worked out that it is a subcritical pitchfork bifurcation. At least, that what my bifurcation diagram shows. Since the bifurcation point is $(r, x) = (1, 0)$, I was able to do a change of variables where $\alpha = r-1$ and get $\dot{x} = \alpha x + \frac{x^3}{1+x^2}$. I'm unclear on what is required to satisfy normal form. Normal for a subcritical pitchfork bifurcation is of the form $\dot{y} = \beta y + y^3$. What I have $\dot{x}$ looks sufficient except for the $x^2+1$ in the denominator. Is $\dot{x} = \alpha x + \frac{x^3}{1+x^2}$ already normal? Do I need to do more?

$$ \frac{1}{1+x^2}=1-x^2+x^4-\ldots $$