Comment on Low-dimensional behavior of generalized Kuramoto model by S. Ameli and K. A. Samani

In the paper Ameli and Samani (Nonlinear Dyn , 2022), the authors formulate low-dimensional evolutions of the macroscopic order parameters in the generalized Kuramoto model, in which the quenched disorder of the heterogeneous natural frequencies and coupling strength are correlated via a weighted absolute value function. The authors state that the collective dynamics, as well as the global bifurcation of various attractors, can be delineated in the framework of the low-dimensional manifold. We argue that such low-dimensional descriptions for the frequency-weighted coupling are not correct in general. This contradiction is explained from several aspects, including the Ott-Antonsen reduction and the forward (backward) critical points corresponding to the onset (vanishing) of synchrony. Remarkably, we uncover that the singularity of the frequency-weighted coupling forbids the analytical continuation, but can vastly simplify the coherent behaviors of the system. Importantly, we justify that our analysis can be extended to a wide class of systems involving the frequency-weighted coupling scheme.

Automated summary

In the paper by Ameli and Samani (Nonlinear Dyn, 2022), the authors propose a low-dimensional evolution of macroscopic order parameters in the generalized Kuramoto model with correlated quenched disorder of natural frequencies and coupling strength. They claim that the collective dynamics and global bifurcation of attractors can be described using the low-dimensional manifold. However, we argue that these low-dimensional descriptions for frequency-weighted coupling are generally incorrect. Our analysis reveals that the singularity of the frequency-weighted coupling prohibits analytical continuation but significantly simplifies the system's coherent behaviors. We demonstrate that our findings can be applied to a wide range of systems involving the frequency-weighted coupling scheme.