State Space Partitioning

This paper develops a way to represent a chaotic system through transitions between regions of its state space. Time-series data define a finite-volume approximation of the equation governing probability, producing a continuous-time Markov model. The method accommodates flexible partitions, noisy observations, and uncertainty caused by finite data. The Lorenz equations provide a worked example of recovering statistical behavior without trying to track an individual chaotic trajectory indefinitely. Readers get the mathematical construction and practical methodology behind a statistical reduced-order model, with the companion paper extending it to atmospheric flow.

This paper applies state-space partitioning to an atmospheric system governed by the compressible Euler equations with gravity and rotation on a thin spherical shell. Representative flow states and their observed transitions form a finite-state approximation of the system’s probability dynamics. The resulting model recovers selected stationary distributions and temporal correlations and identifies global Koopman modes. Two partitioning strategies reveal how choices about grouping high-dimensional states affect the statistics that are retained. Readers get a worked example of taking the theory from the first paper into a large fluid simulation, including its successes and limits as a statistical approximation.