  We show that a one-dimensional differential equation depending on a parameter
$\mu$ with a saddle-node bifurcation at $\mu =0$ can be modelled by an extended
normal form $\dot y = \nu (\mu )-y^2+a(\mu )y^3$, where the functions $\nu$ and
$a$ are solutions to equations that can be written down explicitly. The
equivalence to the original equations is a local differentiable conjugacy on
the basins of attraction and repulsion of stationary points in the parameter
region for which these exist, and is a differentiable conjugacy on the whole
local interval otherwise. (Recall that in standard approaches local equivalence
is topological rather than differentiable.) The value $a(0)$ is Takens'
coefficient from normal form theory.
  The results explain the sense in which normal forms extend away from the
bifurcation point and provide a new and more detailed characterisation of the
saddle-node bifurcation. The one-dimensional system can be derived from higher
dimensional equations using centre manifold theory. We illustrate this using
two examples from climate science and show how the functions $\nu$ and $a$ can
be determined analytically in some settings and numerically in others.
