Projective Embedding of Dynamical Systems: uniform mean field equations
This work provides a theoretical framework for analyzing dynamical systems through projective embeddings, which could be useful for researchers in applied mathematics and physics, but it appears incremental as it builds on existing embedding techniques.
The authors tackled the problem of embedding continuous dynamical systems into higher dimensions using projector operators, introducing PEDS, and proved that with a uniform mean field projector, the equations simplify to a mean field approximation while preserving stability properties of fixed points.
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.