  We consider how breakdown of the quasistatic approximation for attractors can
lead to rate-induced tipping, where a qualitative change in tracking/tipping
behaviour of trajectories can be characterised in terms of a critical rate.
Associated with rate-induced tipping (where tracking of a branch of quasistatic
attractors breaks down) we find a new phenomenon for attractors that are not
simply equilibria: partial tipping of the pullback attractor where certain
phases of the periodic attractor tip and others track the quasistatic
attractor. For a specific model system with a parameter shift between two
asymptotically autonomous systems with periodic attractors we characterise
thresholds of rate-induced tipping to partial and total tipping. We show these
thresholds can be found in terms of certain periodic-to-periodic (PtoP) and
periodic-to-equilibrium (PtoE) connections that we determine using Lin's method
for an augmented system.
