Characterizing resonant solitary states as relative equilibria via normal form reduction
We investigate solitary states in coupled oscillator networks, where a single oscillator detaches from a synchronized cluster and evolves at a distinct mean frequency. We focus on resonant solitary states, in which the mean frequency of the detached oscillator is shaped by resonance with a mode of the cluster. Here, we introduce a normal-form and symmetry-based approach to identify such states. We reduce the high-dimensional network dynamics to a two-dimensional second-order system that couples a rotator representing the solitary oscillator to a harmonic oscillator representing the dominant Laplacian mode of the cluster. We use normal-form transformations to introduce symmetry into this system. Using this symmetry, we reformulate the frequency selection problem as the search for relative equilibria of the normal form symmetry. We derive explicit algebraic conditions for the existence of these relative equilibria, and hence for resonant solitary states and their effective frequencies.