Parameter shifts for nonautonomous systems in low dimension: Bifurcation- and Rate-induced tipping

We discuss the nonlinear phenomena of irreversible tipping for nonautonomous systems where time-varying inputs correspond to a smooth "parameter shift" from one asymptotic value to another. We define notions of bifurcation-induced and rate-induced irreversible tipping of the nonautonomous system in terms of local pullback point attractors and present several results on how nontrivial dynamics for nonautonomous systems can be deduced from analysis of the structure of the bifurcation diagram for an associated autonomous system where parameters are fixed at intermediate values. In one-dimension, we give a number of sufficient conditions for the presence or absence of the less understood rate-induced tipping, and we discuss consequences of our results in a conceptual climate model example.