By investigating numerically a circle map with two cubic inflection points,
we find that the fractal dimension D of the set of quasiperiodic windings
at the onset of chaos has a variety of values, instead of a unique value li
ke 0.87. This fact strongly suggests that a family of universality classes
of D appears as the map has two various inflection points. On the other han
d, at the quasiperiodic transition with the golden mean winding number, the
ratios delta(n) of the width of the mode lockings when going from one Fibo
nacci level to the next do not converge to a fixed value or a limit cycle i
n most cases. In this sense, local scaling is broken due to the interaction
of the two inflection points of the map. Based on the above observations,
it seems that the global scaling is more robust than the local one, at leas
t for the maps we considered.