The role of coupling and timescales for interacting tipping elements

Sudden and abrupt changes can occur in a nonlinear system within many fields of science when such a system crosses a tipping point and rapid changes of the system occur in response to slow changes in an external forcing. These can occur when time-varying inputs cross a bifurcation. If an upstream system loses stability in this way it may cause a downstream system influenced by it to tip, especially if the downstream system evolves on a much faster timescale, in what we call an accelerating cascade of tipping elements. In this paper, we identify the conditions on the coupling and timescales of the systems resulting in various types of tipping (cascade) responses. We also present a prototypical example of a unidirectionally coupled pair of simple tipping elements with hysteresis. This allows us to map out the various types of response as a function of system parameters and to link it to bifurcations of the underlying system that may have multiple timescales.