  Coupled oscillator networks provide mathematical models for interacting
periodic processes. If the coupling is weak, phase reduction -- the reduction
of the dynamics onto an invariant torus -- captures the emergence of collective
dynamical phenomena, such as synchronization. While a first-order approximation
of the dynamics on the torus may be appropriate in some situations,
higher-order phase reductions become necessary, for example, when the coupling
strength increases. However, these are generally hard to compute and thus they
have only been derived in special cases: This includes globally coupled
Stuart--Landau oscillators, where the limit cycle of the uncoupled nonlinear
oscillator is circular as the amplitude is independent of the phase. We go
beyond this restriction and derive second-order phase reductions for coupled
oscillators for arbitrary networks of coupled nonlinear oscillators with
phase-dependent amplitude, a scenario more reminiscent of real-world
oscillations. We analyze how the deformation of the limit cycle affects the
stability of important dynamical states, such as full synchrony and splay
states. By identifying higher-order phase interaction terms with hyperedges of
a hypergraph, we obtain natural classes of coupled phase oscillator dynamics on
hypergraphs that adequately capture the dynamics of coupled limit cycle
oscillators.
