Exact Dimensional Reduction for Quasi-Linear ODE Ensembles

Abstract

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