Uncertainty Transformation via Hopf Bifurcation in Fast-Slow Systems

TL;DR

This paper investigates how uncertainty propagates through fast-slow dynamical systems near Hopf bifurcations, revealing how initial randomness transforms into various distribution types during bifurcation passage.

Contribution

It provides a theoretical framework for understanding uncertainty transformation in multiscale systems near Hopf bifurcations, including conditions for different distribution outcomes.

Findings

Random initial conditions can transform into symmetric copies or deterministic outputs.

Certain classes of vector fields lead to specific distribution types after bifurcation.

Theoretical conditions for different uncertainty transformation scenarios are established.

Abstract

Propagation of uncertainty in dynamical systems is a significant challenge. Here we focus on random multiscale ordinary differential equation models. In particular, we study Hopf bifurcation in the fast subsystem for random initial conditions. We show that a random initial condition distribution can be transformed during the passage near a delayed/dynamic Hopf bifurcation: (I) to certain classes of symmetric copies, (II) to an almost deterministic output, (III) to a mixture distribution with differing moments, and (IV) to a very restricted class of general distributions. We prove under which conditions the cases (I)-(IV) occur in certain classes vector fields.