Early warning signals for primary and secondary bifurcation to oscillatory instabilities

TL;DR

This paper introduces a spectral visibility graph method to provide early warnings for both primary and secondary bifurcations leading to oscillatory instabilities in engineering systems, capturing harmonic evolution and enabling adaptive risk assessment.

Contribution

The study presents a novel spectral visibility graph approach that detects bifurcations in signals, adaptable to different bifurcation sequences and sensitive to harmonic content changes.

Findings

Effective early warning for primary bifurcations.

Detection of secondary bifurcations in complex systems.

Applicable to thermo-acoustic and aero-acoustic instabilities.

Abstract

In several natural and engineering systems, changes in control parameters can trigger bifurcations that lead to sustained or growing periodic oscillations, indicating the onset of oscillatory instabilities. Such emergent behaviour often results from positive feedback between interacting subsystems, resulting in large-amplitude oscillations that can be detrimental. Several precursors are available to provide early warning of an impending oscillatory instability. In reality, practical systems may exhibit different sequences of bifurcations, including a primary bifurcation to an oscillatory state that may be either continuous or abrupt, followed by an abrupt secondary bifurcation, and further transitions beyond the secondary bifurcation. Existing precursors for oscillatory instabilities typically forewarn the onset of the primary bifurcation to an oscillatory state and tend to saturate once the system enters the oscillatory regime. Notably, primary bifurcations often involve lower amplitudes compared to the more severe states after secondary bifurcation. In this study, we propose a methodology based on spectral visibility graphs to get forewarning for both primary and secondary bifurcations. The method inherently captures the evolution of the harmonic content of the signal relative to other frequency components. The approach employs tuning of a single sensitivity parameter to detect different sequences of bifurcation. We demonstrate the usefulness of our method for thermo-acoustic and aero-acoustic instabilities in multiple engineering systems involving turbulent flow and reactions. Our methodology can help systems prepare in advance or even avoid undesirable transitions. Tuning the sensitivity parameter allows adaptive, risk-based warnings, ensuring high performance without tipping into undesirable regimes.