Phase transition in higher-order Kuramoto model with multiple δ functions distribution

We study the thresholds of different synchronization phases in the higher-order Kuramoto model,where the oscillators’ frequency distribution is composed of multiple [Formula: see text] functions. We show that as the coupling strength increases, oscillators in different frequency groups will enter synchronization clusters from low to high frequencies, leading to multiple phase transitions. Thanks to the simplification of [Formula: see text] function, the thresholds of phase transitions of general higher-order Kuramoto model up to arbitrary order are obtained analytically. The thresholds are determined by the characters of the [Formula: see text] functions, which are confirmed through numerical simulations. Our method can be extended to higher-order Kuramoto model with arbitrary number of harmonic terms. We find that the thresholds will decrease with the increase in the number of harmonic terms. By the frequency distribution with multiple [Formula: see text] functions, we provide a delicate way to study the behavior of the thresholds in higher-order Kuramoto model.