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The understanding and prediction of sudden changes in flow patterns is ofparamount importance in the analysis of geophysical flows as these rare eventsrelate to critical phenomena such as atmospheric blocking, the weakening of theGulf stream, or the splitting of the polar vortex. In this work our aim is todevelop first steps towards a theoretical understanding of vortex splittingphenomena. To this end, we study bifurcations of global flow patterns inparameter-dependent two-dimensional incompressible flows, with the flowpatterns of interest corresponding to specific invariant sets. Under smallrandom perturbations these sets become almost-invariant and can be computed andstudied by means of a set-oriented approach, where the underlying dynamics isdescribed in terms of a reversible finite-state Markov chain. Almost-invariantsets are obtained from the sign structure of leading eigenvectors of thecorresponding transition matrix. By a flow pattern bifurcation we mean aqualitative change in the form of a break-up of an almost-invariant set, when acritical external parameter of the underlying dynamical system is reached. Fordifferent examples and settings we follow the spectrum and the correspondingeigenvectors under continuous changes of the underlying system and yieldindicators for different bifurcation scenarios for almost-invariant sets. Inparticular, we study a Duffing-type oscillator, which is known to undergo aclassic pitchfork bifurcation. We find that the set-oriented analogue of thisclassical bifurcation includes a splitting of a rotating pattern, which hasgeneric precursor signal that can be deduced from the behavior of the spectrum.

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