Dynamical equivalence between Kuramoto models with first- and higher-order coupling

The Kuramoto model with high-order coupling has recently attracted some attention in the field of coupled oscillators in order, for instance, to describe clustering phenomena in sets of coupled agents. Instead of considering interactions given directly by the sine of oscillators' angle differences, the interaction is given by the sum of sines of integer multiples of these angle differences. This can be interpreted as a Fourier decomposition of a general $2{\pi}$-periodic interaction function. We show that in the case where only one multiple of the angle differences is considered, which we refer to as the "Kuramoto model with simple $q$th-order coupling," the system is dynamically equivalent to the original Kuramoto model. In other words, any property of the Kuramoto model with simple higher-order coupling can be recovered from the standard Kuramoto model.