Singularity of information flow at the Hopf bifurcation point

TL;DR

This paper explores how information flow, measured by the learning rate, behaves near the Hopf bifurcation in stochastic systems, revealing non-smooth behavior and linking dynamics to information processing.

Contribution

It introduces a singular perturbation approach to analyze the learning rate near bifurcation points in stochastic systems, extending deterministic bifurcation theory methods.

Findings

Learning rate remains finite in the deterministic limit.

Linear analysis fails near the bifurcation point.

Singular perturbation method accurately captures non-smooth behavior.

Abstract

We investigate the singular behavior of information flow near the Hopf bifurcation point by analyzing the learning rate, a key quantity in stochastic thermodynamics. As a model system exhibiting the Hopf bifurcation, we study the Brusselator. We first numerically compute the learning rate in the stationary regime and find that it remains finite even in the deterministic limit, suggesting that information flow can be quantified in deterministic dynamics through probabilistic descriptions. Linear analysis accurately reproduces the numerical results in the stationary regime but fails near the bifurcation point. To overcome this limitation, we employ the singular perturbation method, well known in deterministic bifurcation theory, and carry out the corresponding calculation explicitly for a stochastic system described by a Langevin equation. This allows us to evaluate the learning rate near the bifurcation point. We then theoretically derive its non-smooth behavior in the deterministic limit. Our results demonstrate that changes in dynamical behavior are reflected in the information flow and provide a basis for analyzing information processing in biochamical oscillations.