Generalization of complex Riccati-based dimension reduction to heterogeneous complexified Kuramoto ensembles

Determine how the dimension-reduction results based on the complex Riccati equation, which have been shown to apply to ensembles of identical complexified oscillators, generalize to complexified Kuramoto systems with heterogeneous natural frequency distributions.

Background

In the Conclusions and Open Questions section, the authors discuss extending analytical understanding from finite-size and identical-oscillator settings to broader regimes. They note that for identical complexified oscillators, recent work shows that a dimension-reduction method using the complex Riccati equation is applicable, offering a powerful analytic handle on the dynamics.

They emphasize that it is not yet established how these reductions translate to systems with heterogeneity in natural frequencies. Clarifying this would bridge results for identical complexes to more realistic settings where oscillators are non-identical, and it may inform connections to classical reductions like the Ott–Antonsen or Watanabe–Strogatz approaches.