  We propose a concept to generate and stabilize diverse partial
synchronization patterns (phase clusters) in adaptive networks which are
widespread in neuro- and social sciences, as well as biology, engineering, and
other disciplines. We show by theoretical analysis and computer simulations
that multiplexing in a multi-layer network with symmetry can induce various
stable phase cluster states in a situation where they are not stable or do not
even exist in the single layer. Further, we develop a method for the analysis
of Laplacian matrices of multiplex networks which allows for insight into the
spectral structure of these networks enabling a reduction to the stability
problem of single layers. We employ the multiplex decomposition to provide
analytic results for the stability of the multilayer patterns. As local
dynamics we use the paradigmatic Kuramoto phase oscillator, which is a simple
generic model and has been successfully applied in the modeling of
synchronization phenomena in a wide range of natural and technological systems.
