  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.
