  Networks with different levels of interactions, including multilayer and
multiplex networks, can display a rich diversity of dynamical behaviors and can
be used to model and study a wide range of systems. Despite numerous efforts to
investigate these networks, obtaining mathematical descriptions for the
dynamics of multilayer and multiplex systems is still an open problem. Here, we
combine ideas and concepts from linear algebra and graph theory with nonlinear
dynamics to offer a novel approach to study multiplex networks of Kuramoto
oscillators. Our approach allows us to study the dynamics of a large, multiplex
network by decomposing it into two smaller systems: one representing the
connection scheme within layers (intra-layer), and the other representing the
connections between layers (inter-layer). Particularly, we use this approach to
compose solutions for multiplex networks of Kuramoto oscillators. These
solutions are given by a combination of solutions for the smaller systems given
by the intra and inter-layer system and, in addition, our approach allows us to
study the linear stability of these solutions.
