Principal axes for stochastic dynamics

TL;DR

This paper presents a method to identify independent stochastic sources in complex systems modeled by coupled Langevin equations using eigenanalysis of local diffusion matrices, enhancing understanding and predictability.

Contribution

It introduces a novel eigenvalue-based algorithm to determine stochastic sources and eigendirections in high-dimensional Langevin systems, applicable to systems with bifurcations.

Findings

Eigenvalues and eigenvectors reveal stochastic source directions.

Transforming coordinates along eigenvectors improves system predictability.

Method successfully applied to systems with Hopf-bifurcation.

Abstract

We introduce a general procedure for directly ascertaining how many independent stochastic sources exist in a complex system modeled through a set of coupled Langevin equations of arbitrary dimension. The procedure is based on the computation of the eigenvalues and the corresponding eigenvectors of local diffusion matrices. We demonstrate our algorithm by applying it to two examples of systems showing Hopf-bifurcation. We argue that computing the eigenvectors associated to the eigenvalues of the diffusion matrix at local mesh points in the phase space enables one to define vector fields of stochastic eigendirections. In particular, the eigenvector associated to the lowest eigenvalue defines the path of minimum stochastic forcing in phase space, and a transform to a new coordinate system aligned with the eigenvectors can increase the predictability of the system.