  This work considers the timescales associated with the global order parameter
and the interlayer synchronization of coupled Kuramoto oscillators on
multiplexes. For the two-layer multiplexes with initially high degree of
synchronization in each layer, the difference between the average phases in
each layer is analyzed from two different perspectives: the spectral analysis
and the non-linear Kuramoto model. Both viewpoints confirm that the prior
timescales are inversely proportional to the interlayer coupling strength.
Thus, increasing the interlayer coupling always shortens the transient regimes
of both the global order parameter and the interlayer synchronization.
Surprisingly, the analytical results show that the convergence of the global
order parameter is faster than the interlayer synchronization, and the latter
is generally faster than the global synchronization of the multiplex. The
formalism also outlines the effects of frequencies on the difference between
the average phases of each layer, and identifies the conditions for an
oscillatory behavior. Computer simulations are in fairly good agreement with
the analytical findings and reveal that the timescale of the global order
parameter is at least half times smaller than timescale of the multiplex.
