  Let $\Phi$ be a quasi-periodically forced quadratic map, where the rotation
constant $\omega$ is a Diophantine irrational. A strange non-chaotic attractor
(SNA) is an invariant (under $\Phi$) attracting graph of a nowhere continuous
measurable function $\psi$ from the circle $\mathbb{T}$ to $[0,1]$.
  This paper investigates how a smooth attractor degenerates into a strange
one, as a parameter $\beta$ approaches a critical value $\beta_0$, and the
asymptotics behind the bifurcation of the attractor from smooth to strange. In
our model, the cause of the strange attractor is a so-called torus collision,
whereby an attractor collides with a repeller.
  Our results show that the asymptotic minimum distance between the two
colliding invariant curves decreases linearly in the parameter $\beta$, as
$\beta$ approaches the critical parameter value $\beta_0$ from below.
  Furthermore, we have been able to show that the asymptotic growth of the
supremum of the derivative of the attracting graph is asymptotically bounded
from both sides by a constant times the reciprocal of the square root of the
minimum distance above.
