Abstract
An important challenge in several disciplines is to understand how sudden changes can propagate among coupled systems. Examples include the synchronization of business cycles, population collapse in patchy ecosystems, markets shifting to a new technology platform, collapses in prices and in confidence in financial markets, and protests erupting in multiple countries. A number of mathematical models of these phenomena have multiple equilibria separated by saddle-node bifurcations. We study this behaviour in its normal form as fast–slow ordinary differential equations. In our model, a system consists of multiple subsystems, such as countries in the global economy or patches of an ecosystem. Each subsystem is described by a scalar quantity, such as economic output or population, that undergoes sudden changes via saddle-node bifurcations. The subsystems are coupled via their scalar quantity (e.g. trade couples economic output; diffusion couples populations); that coupling moves the locations of their bifurcations. The model demonstrates two ways in which sudden changes can propagate: they can cascade (one causing the next), or they can hop over subsystems. The latter is absent from classic models of cascades. For an application, we study the Arab Spring protests. After connecting the model to sociological theories that have bistability, we use socioeconomic data to estimate relative proximities to tipping points and Facebook data to estimate couplings among countries. We find that although protests tend to spread locally, they also seem to ‘hop' over countries, like in the stylized model; this result highlights a new class of temporal motifs in longitudinal network datasets.
Highlights
Sudden changes propagating among coupled systems pose a significant scientific challenge in many disciplines, yet we lack an adequate mathematical understanding of how local sudden changes spread [1]
Our findings suggest that small changes in these central nodes can suffice to trigger a regime shift in a peripheral node close to its tipping point
Livelihoods could improve if sudden adoption of technologies in coupled markets were facilitated [13,14], or if coupled recessions and booms in economies were better managed [5,6,7,8,13], or if social uprisings spreading among countries were better forecast [11,12,32]
Summary
Sudden changes propagating among coupled systems pose a significant scientific challenge in many disciplines, yet we lack an adequate mathematical understanding of how local sudden changes spread [1]. One (such as compact discs replacing cassettes or electric cars replacing fuel cars) can change abruptly [13], and movement of people between distinct markets can facilitate adoption of the new technology [14] In each of these examples, a system consists of distinct subsystems that (i) change suddenly between equilibria and (ii) are coupled. These couplings move the locations of the latter subsystems’ bifurcations This model allows us to explore how regime shifts can synchronize and spread. If the driven subsystem Y drives a third subsystem Z (i.e. if XQYQZ ), one possible behaviour is a cascade of regime shifts, one triggering another like falling dominoes. A sequence of regime shifts can ‘hop’ over intermediate subsystems This phenomenon is not observed in classic models of cascades (e.g. percolation, epidemic spreading and sandpile models). Our findings suggest that small changes in these central nodes ( potentially triggered by a large change in a small node adjacent to it) can suffice to trigger a regime shift in a peripheral node close to its tipping point
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