  We discuss the nonlinear phenomena of irreversible tipping for non-autonomous
systems where time-varying inputs correspond to a smooth "parameter shift" from
one asymptotic value to another. We express tipping in terms of pullback
attraction and present some results on how nontrivial dynamics for
non-autonomous systems can be deduced from analysis of the bifurcation diagram
for an associated autonomous system where parameters are fixed. In particular,
we show that there is a unique local pullback point attractor associated with
each linearly stable equilibrium for the past limit. If there is a smooth
stable branch of equilibria over the range of values of the parameter shift,
the pullback attractor will remain close to (track) this branch for small
enough rates, though larger rates may lead to rate-induced tipping. More
generally, we show that one can track certain stable paths that go along
several stable branches by pseudo-orbits of the system, for small enough rates.
For these local pullback point attractors, we define notions of
bifurcation-induced and irreversible rate-induced tipping of the non-autonomous
system. In one-dimension, we give a number of sufficient conditions for the
presence or absence of rate-induced tipping, and we discuss some applications
of our results to give criteria for irreversible rate-induced tipping in a
conceptual climate model example.
