Some scaling behaviors in a circle map with two inflection points

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.