Resonant solitary states in complex networks

Partially synchronized solitary states occur frequently when a synchronized system of networked oscillators with inertia is perturbed locally. Several asymptotic states of different frequencies can coexist at the same node. Here, we reveal the mechanism behind this multistability: additional solitary frequencies arise from the coupling between network modes and the solitary oscillator’s frequency, leading to significant energy transfer. This can cause the solitary node’s frequency to resonate with a Laplacian eigenvalue. We analyze which network structures enable this resonance and explain longstanding numerical observations. Another solitary state that is known in the literature is characterized by the effective decoupling of the synchronized network and the solitary node at the natural frequency. Our framework unifies the description of solitary states near and far from resonance, allowing to predict the behavior of complex networks from their topology.