Kuramoto

Kuramoto refers to Yoshiki Kuramoto, a Japanese physicist who introduced the Kuramoto model in 1975 to study synchronization phenomena in populations of coupled oscillators. The model has become a standard framework in nonlinear dynamics and complex systems.

The Kuramoto model describes N phase oscillators with phases theta_i and natural frequencies omega_i drawn from a distribution g(omega). The evolution equation is d theta_i/dt = omega_i + (K/N) sum over j of sin(theta_j - theta_i). The order parameter r e^{i psi} = (1/N) sum over j of e^{i theta_j} quantifies synchronization; r near 0 indicates incoherence, while r near 1 indicates strong synchronization. As coupling K increases above a critical value K_c depending on g, a transition to partial synchronization occurs, with a subset of oscillators locking to a common frequency.

Extensions and impact: The Kuramoto-Sakaguchi model adds a phase lag parameter alpha, affecting the synchronization transition. The Ott-Antonsen reduction provides a low-dimensional description for certain frequency distributions. The model has wide applications in neuroscience, power grids, circadian rhythms, laser arrays, and engineered synchronization, and is frequently cited as a paradigmatic model for studying collective behavior in complex systems.