Symmetry-adapted Koopman operator for networks of phase oscillators

Koopman operators are linear, infinite-dimensional operators that can potentially capture the behavior of nonlinear dynamical systems, offering a unified description for prediction and control. For equivariant systems, estimating the operators while respecting the symmetry leads to expensive computations that scale with a power of the order of the symmetry group. Here, following the work of Salova et al. (2019) on radial basis functions, we construct a dictionary of complex polynomials that leverages the system’s symmetry to yield a low-dimensional representation, thereby overcoming radial basis functions’ computational limitations. We apply this method to a network of phase oscillators with global phase invariance, obtaining compressed descriptions, verified spectra, and good forecasting power.