Decoupled synchronized states in networks of linearly coupled limit cycle oscillators

Networks of limit cycle oscillators can show intricate patterns of synchronization such as splay states and cluster synchronization. Here we analyze dynamical states that display a continuum of seemingly independent splay clusters. Each splay cluster is a block splay state consisting of sub-clusters of fully synchronized nodes with uniform amplitudes. Phases of nodes within a splay cluster are equally spaced, but nodes in different splay clusters have an arbitrary phase difference that can be fixed or evolve linearly in time. Such coexisting splay clusters form a decoupled state in that the dynamical equations become effectively decoupled between oscillators that can be physically coupled. We provide the conditions that allow the existence of particular decoupled states by using the eigendecomposition of the coupling matrix. Additionally, we provide an algorithm to search for admissible decoupled states using the external equitable partition and orbital partition considerations combined with symmetry groupoid formalism. Unlike previous studies, our approach is applicable when existence does not follow from symmetries alone and also illustrates the differences between adjacency and Laplacian coupling. We show that the decoupled state can be linearly stable for a substantial range of parameters using a simple eight-node cube network and its modifications as an example. We also demonstrate how the linear stability analysis of decoupled states can be simplified by taking into account the symmetries of the Jacobian matrix. Some network structures can support multiple decoupled patterns. To illustrate that, we show the variety of qualitatively different decoupled states that can arise on two-dimensional square and hexagonal lattices.