Exact dimensional reduction for quasi-linear ODE ensembles

We present an exact dimensional reduction for ensembles of N identical dynamical units governed by ordinary differential equations of order M with quasi-linear structure. In these systems, each unit follows a linear differential equation whose coefficients depend nonlinearly on the ensemble of variables, such as a mean field, giving rise to a large class of network dynamical systems. We derive M+1 closed-form macroscopic equations of order M with variables that exactly capture the full microscopic dynamics and that allow for the exact reconstruction of individual trajectories from the reduced system. This dimensional reduction facilitates a simplified analysis of collective behavior in a new class of coupled oscillator networks and provides computationally efficient exact representations of large-scale dynamics. We illustrate our approach on two examples, highlighting new families of solvable models relevant to physics, biology, and engineering that are now amenable to simplified analysis.