Collective dynamics of multiplex networks with distinct adaptation rules

We investigate the collective dynamics of multiplex networks of adaptively coupled Kuramoto oscillators, where distinct adaptation rules regulate the structural evolution of each layer. We uncover a wide variety of collective states, including incoherent states, antipodal-incoherent states, antipodal cluster states, antipodal-partially coherent states, mixed states, and antipodal states. These collective states are characterized using a set of intralayer and interlayer macroscopic order parameters. The former characterizes the self-organization within each layer, while the latter quantifies the dynamical correlation between the layers. Our results highlight the interplay of different adaptation mechanisms mediated via multiplexing that leads to rich collective states, which are not otherwise observed in a single-layer counterpart. Further, we provide an analytical estimate of the critical interlayer coupling strength for interlayer frequency synchronization using the collective coordinate approach, which agrees well with the simulation results.