I was trying to bring the dynamical system
$\dot{x} = x + 4rxe^{-x^2}$ to the Normal form of Pitchfork bifurcation -
$\dot{X} = RX - X^3$ by the change of variables and using the series for $e^x$.
But i could arrive at the following using $e^x$ series around $x=0$
$\dot{x} = (4r+1)x + (-4r)x^3 + O(x^5)$.
Now I am thinking to divide by $4r$ both sides, but How does that transform to the normal form?
Please advise.