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An important challenge in several disciplines is to understand how suddenchanges can propagate among coupled systems. Examples include thesynchronization of business cycles, population collapse in patchy ecosystems,markets shifting to a new technology platform, collapses in prices and inconfidence in financial markets, and protests erupting in multiple countries. Anumber of mathematical models of these phenomena have multiple equilibriaseparated by saddle-node bifurcations. We study this behavior in its normalform as fast--slow ordinary differential equations. In our model, a systemconsists of multiple subsystems, such as countries in the global economy orpatches of an ecosystem. Each subsystem is described by a scalar quantity, suchas economic output or population, that undergoes sudden changes via saddle-nodebifurcations. The subsystems are coupled via their scalar quantity (e.g., tradecouples economic output; diffusion couples populations); that coupling movesthe locations of their bifurcations. The model demonstrates two ways in whichsudden changes can propagate: they can cascade (one causing the next), or theycan hop over subsystems. The latter is absent from classic models of cascades.For an application, we study the Arab Spring protests. After connecting themodel to sociological theories that have bistability, we use socioeconomic datato estimate relative proximities to tipping points and Facebook data toestimate couplings among countries. We find that although protests tend tospread locally, they also seem to "hop" over countries, like in the stylizedmodel; this result highlights a new class of temporal motifs in longitudinalnetwork datasets.

Coupled catastrophes: sudden shifts cascade and hop among interdependent systems