The paper by Edward Ott and Thomas M. Antonsen explores the low-dimensional dynamics of large systems of globally coupled phase oscillators in the infinite size limit. They derive explicit nonlinear ordinary differential equations for the macroscopic evolution of these systems, providing a reduced description that captures the essential dynamics. The authors focus on the Kuramoto model, where oscillators have Lorentzian frequency distributions, and show that the time evolution of the order parameter can be described by a single nonlinear first-order ordinary differential equation. This reduction is achieved by considering a restricted class of states, which are shown to form an invariant manifold. The method is also extended to other scenarios, including external driving, communities of oscillators, and time-delayed coupling, demonstrating its broad applicability. The results are validated through numerical simulations and compared with known analytical solutions, confirming the accuracy of the low-dimensional description. The paper highlights the utility of this approach in understanding the complex dynamics of large coupled oscillator systems.The paper by Edward Ott and Thomas M. Antonsen explores the low-dimensional dynamics of large systems of globally coupled phase oscillators in the infinite size limit. They derive explicit nonlinear ordinary differential equations for the macroscopic evolution of these systems, providing a reduced description that captures the essential dynamics. The authors focus on the Kuramoto model, where oscillators have Lorentzian frequency distributions, and show that the time evolution of the order parameter can be described by a single nonlinear first-order ordinary differential equation. This reduction is achieved by considering a restricted class of states, which are shown to form an invariant manifold. The method is also extended to other scenarios, including external driving, communities of oscillators, and time-delayed coupling, demonstrating its broad applicability. The results are validated through numerical simulations and compared with known analytical solutions, confirming the accuracy of the low-dimensional description. The paper highlights the utility of this approach in understanding the complex dynamics of large coupled oscillator systems.