Approximating the bifurcation diagram of weakly and strongly coupled leading-following systems

The potential of dynamical systems to undergo bifurcation-induced tipping has received much attention recently in climate and ecology research. In particular, such systems can form an intricate interacting network, creating the possibility of cascading critical transitions in which tipping of one element results in the tipping of another. In this paper, we focus on unidirectionally coupled scalar subsystems in which one component is driven by a polynomial equation. We investigate such interacting systems beyond the so-far used setting of linearly interacting bistable subsystems. In these cases, we show how the bifurcation diagram of the coupled system can be approximated using asymptotic methods, starting from the simpler bifurcation diagram of the decoupled problems. We study the limits in which the coupling is weak or strong, yielding approximations of the equilibrium branches and their stability. Those results are illustrated using conceptual models for the ocean circulation driven by wind and density and for the interacting ocean circulation and Amazon rainforest.