Multiple Devil's staircase in a discontinuous circle map

The multiple Devil's staircase, which describes phase-locking behavior, is observed in a discontinuous nonlinear circle map. Phase-locked steps form many towers with similar structure in winding number(W)-parameter(k) space. Each step belongs to a certain period-adding sequence that exists in a smooth curve. The Collision modes that determine steps and the sequence of mode transformations create a variety of tower structures and their particular characteristics. Numerical results suggest a scaling law for the width of phase-locked steps in the period-adding (W=n/(n+i), n,i∈int) sequences, that is, Δk(n)∝n^-τ (τ>0). And the study indicates that the multiple Devil's staircase may be common in a class of discontinuous circle maps.