Predicting bifurcations of almost-invariant patterns: a set-oriented approach

TL;DR

This paper develops a set-oriented approach to predict bifurcations of almost-invariant flow patterns in dynamical systems, providing a theoretical framework for understanding critical transitions like vortex splitting in geophysical flows.

Contribution

It introduces a novel set-oriented method using Markov chains to analyze bifurcations of almost-invariant sets in parameter-dependent flows, linking spectral properties to pattern break-up.

Findings

Eigenvector sign structure indicates bifurcation points

Spectral analysis predicts pattern splitting

Application to Duffing oscillator confirms method effectiveness

Abstract

The understanding and prediction of sudden changes in flow patterns is of paramount importance in the analysis of geophysical flows as these rare events relate to critical phenomena such as atmospheric blocking, the weakening of the Gulf stream, or the splitting of the polar vortex. In this work our aim is to develop first steps towards a theoretical understanding of vortex splitting phenomena. To this end, we study bifurcations of global flow patterns in parameter-dependent two-dimensional incompressible flows, with the flow patterns of interest corresponding to specific invariant sets. Under small random perturbations these sets become almost-invariant and can be computed and studied by means of a set-oriented approach, where the underlying dynamics is described in terms of a reversible finite-state Markov chain. Almost-invariant sets are obtained from the sign structure of leading eigenvectors of the corresponding transition matrix. By a flow pattern bifurcation we mean a qualitative change in the form of a break-up of an almost-invariant set, when a critical external parameter of the underlying dynamical system is reached. For different examples and settings we follow the spectrum and the corresponding eigenvectors under continuous changes of the underlying system and yield indicators for different bifurcation scenarios for almost-invariant sets. In particular, we study a Duffing-type oscillator, which is known to undergo a classic pitchfork bifurcation. We find that the set-oriented analogue of this classical bifurcation includes a splitting of a rotating pattern, which has generic precursor signal that can be deduced from the behavior of the spectrum.