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

Relaxation time of the global order parameter on multiplex networks: the role of interlayer coupling in Kuramoto oscillators