On the origin of self-oscillations in large systems

TL;DR

This paper demonstrates that large systems with phase coexistence can exhibit self-oscillations due to linear feedback, with a bifurcation mechanism explained through mean field theory and validated in models like the Ising system.

Contribution

It introduces a simple theoretical framework linking feedback to self-oscillations in large systems, applicable across diverse phenomena.

Findings

Self-oscillations arise from feedback in systems with phase coexistence.

The Van der Pol equation models feedback dynamics effectively.

The theory explains oscillations in markets, queues, and biological networks.

Abstract

In this article is shown that large systems endowing phase coexistence display self-oscillations in presence of linear feedback between the control and order parameters, where an Andronov-Hopf bifurcation takes over the phase transition. This is simply illustrated through the mean field Landau theory whose feedback dynamics turns out to be described by the Van der Pol equation and it is then validated for the fully connected Ising model following heat bath dynamics. Despite its simplicity, this theory accounts potentially for a rich range of phenomena: here it is applied to describe in a stylized way i) excess demand-price cycles due to strong herding in a simple agent-based market model; ii) congestion waves in queueing networks triggered by users feedback to delays in overloaded conditions; iii) metabolic network oscillations resulting from cell growth control in a bistable phenotypic landscape.