Projective Embedding of Dynamical Systems: uniform mean field equations

TL;DR

This paper introduces PEDS, a method for embedding dynamical systems into higher dimensions using projector operators, and shows that a specific uniform mean field projector preserves fixed points and stability properties.

Contribution

The paper defines PEDS and proves that the uniform mean field projector provides a mean field approximation while preserving key stability features of the original system.

Findings

Uniform mean field projector simplifies the equations of motion.

Stable fixed points are preserved under the embedding.

Unstable fixed points become saddle points in the embedding.

Abstract

We study embeddings of continuous dynamical systems in larger dimensions via projector operators. We call this technique PEDS, projective embedding of dynamical systems, as the stable fixed point of the dynamics are recovered via projection from the higher dimensional space. In this paper we provide a general definition and prove that for a particular type of projector operator of rank-1, the uniform mean field projector, the equations of motion become a mean field approximation of the dynamical system. While in general the embedding depends on a specified variable ordering, the same is not true for the uniform mean field projector. In addition, we prove that the original stable fixed points remain stable fixed points of the dynamics, saddle points remain saddle, but unstable fixed points become saddles.