Coupled catastrophes in systems with bidirectional feedback

Catastrophes are present across many disciplines, ranging from the extinction of populations in ecosystems to the collapse of prices in financial systems. Here, we focus on how interactions between systems influence these catastrophes. Specifically, we study two bidirectionally coupled sub-systems—each of which possesses an S-shaped bifurcation curve with saddle-node bifurcations—and explore how interactions between them can lead to simultaneous bifurcations. There are four types of such coupled catastrophes – synchronization, anti-synchronization, consensus, and anti-consensus. Which of these behaviors manifests depends both the intrinsic dynamics of the subsystems and ways in which they are coupled. In general, there are three possible coupling classes – cooperation, competition, and predation. We develop an analytic/graphical methodology to determine and visualize the locus of the coupled catastrophes in parameter space, and we show which classes of couplings support different types of coupled catastrophes. Finally, we discuss several potential applications areas from distinct domains.