  Nonlinear dynamical systems subjected to a combination of noise and
time-varying forcing can exhibit sudden changes, critical transitions or
tipping points where large or rapid dynamic effects arise from changes in a
parameter that are small or slow. Noise-induced tipping can occur where
extremes of the forcing causes the system to leave one attractor and transition
to another. If this noise corresponds to unresolved chaotic forcing, there is a
limit such that this can be approximated by a stochastic differential equation
(SDE) and the statistics of large deviations determine the transitions. Away
from this limit it makes sense to consider tipping in the presence of chaotic
rather than stochastic forcing. In general we argue that close to a parameter
value where there is a bifurcation of the unforced system, there will be a
chaotic tipping window outside of which tipping cannot happen, in the limit of
asymptotically slow change of that parameter. This window is trivial for a
stochastically forced system. Entry into the chaotic tipping window can be seen
as a boundary crisis/non-autonomous saddle-node bifurcation and corresponds to
an exceptional case of the forcing, typically by an unstable periodic orbit. We
discuss an illustrative example of a chaotically forced bistable map that
highlight the richness of the geometry and bifurcation structure of the
dynamics in this case. If a parameter is changing slowly we note there is a
dynamic tipping window that can also be determined in terms of unstable
periodic orbits.
