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Coupled oscillator networks provide mathematical models for interactingperiodic processes. If the coupling is weak, phase reduction -- the reductionof the dynamics onto an invariant torus -- captures the emergence of collectivedynamical phenomena, such as synchronization. While a first-order approximationof the dynamics on the torus may be appropriate in some situations,higher-order phase reductions become necessary, for example, when the couplingstrength increases. However, these are generally hard to compute and thus theyhave only been derived in special cases: This includes globally coupledStuart--Landau oscillators, where the limit cycle of the uncoupled nonlinearoscillator is circular as the amplitude is independent of the phase. We gobeyond this restriction and derive second-order phase reductions for coupledoscillators for arbitrary networks of coupled nonlinear oscillators withphase-dependent amplitude, a scenario more reminiscent of real-worldoscillations. We analyze how the deformation of the limit cycle affects thestability of important dynamical states, such as full synchrony and splaystates. By identifying higher-order phase interaction terms with hyperedges ofa hypergraph, we obtain natural classes of coupled phase oscillator dynamics onhypergraphs that adequately capture the dynamics of coupled limit cycleoscillators.

Higher-Order Interactions in Phase Oscillator Networks through Phase Reductions of Oscillators with Phase Dependent Amplitude