  It is known that explosive synchronization (ES) in an isolated network of
Kuramoto oscillators with inertia is significantly enhanced by the presence of
time delay. Here we show that time delay in one layer of the multiplex network
governs the transition to synchronization and ES in the other layers. We found
that a single layer with time-delayed intra-layer coupling, depending on the
values of time delay, experiences a different type of transition to
synchronization, e.g., ES or continuous, and the same type of transition is
incorporated simultaneously in other layer(s) as well. Hence, a suitable choice
of time-delay in only one layer can lead to a desired (either ES or continuous)
transition simultaneously in the delayed and other undelayed layers of
multiplexed network. These results offer a platform for a better understanding
of the dynamics of those complex systems which are represented by multilayered
framework and contain time delays in the communication processes.
