  Many complex dynamical systems in the real world, including ecological,
climate, financial, and power-grid systems, often show critical transitions, or
tipping points, in which the system's dynamics suddenly transit into a
qualitatively different state. In mathematical models, tipping points happen as
a control parameter gradually changes and crosses a certain threshold. Tipping
elements in such systems may interact with each other as a network, and
understanding the behavior of interacting tipping elements is a challenge
because of the high dimensionality originating from the network. Here we
develop a degree-based mean-field theory for a prototypical double-well system
coupled on a network with the aim of understanding coupled tipping dynamics
with a low-dimensional description. The method approximates both the onset of
the tipping point and the position of equilibria with a reasonable accuracy.
Based on the developed theory and numerical simulations, we also provide
evidence for multistage tipping point transitions in networks of double-well
systems.
