Double stochastic resonance in the mean-field $q$-state clock models

A magnetic system with a phase transition at temperature $T_c$ may exhibit double resonance peaks under a periodic external magnetic field because the time scale matches the external frequency at two different temperatures, one above $T_c$ and the other below $T_c$. We study the double resonance phenomena for the mean-field $q$-state clock model based on the heat-bath-typed master equation. We find double peaks as observed in the kinetic Ising case ($q=2$) for all $q\ge 4$, but for the three-state clock model ($q=3$), the existence of double peaks is possible only above a certain external frequency since it undergoes a discontinuous phase transition.