The role of multiple time scales for rate-induced tipping phenomena

Many systems in nature are characterized by the coexistence of different stable states for a given set of environmental parameters and external forcing. Examples for such behavior can be found in different fields of Earth system, e.g. ecosystems and climate dynamics. As a consequence of the coexistence of a multitude of stable states, the final state of the system depends strongly on the initial condition.  The set of initial conditions which all converge to the same stable state is called the basin of attraction. In addition, the dynamics of complex systems is often characterized by the different time scales on which certain processes act. We show that the interplay of these different time scales is important particularly for the case of rate-induced tipping. This tipping phenomenon occurs when the rate of change of an internal parameter or an external forcing is varying on a different timescale as the intrinsic timescale of the system.  The system can track its original stable state under such time-dependent forcing as long as the rate of environmental change is slow. If this rate is larger than a critical rate the system will tip and obey a rather different dynamical behavior, either by approaching a different stable state or by visiting temporarily different parts of the state space.  We study the role of the relative size of the basins of attraction and the location of their boundaries in rate-induced tipping and demonstrate that the decision whether a trajectory tips or tracks the original stable state depends crucially on the changes in the basins of attraction, in particular their size and, more importantly on their boundaries, that also “move” in state space under a time-dependent variation of intrinsic parameters/external forcing.  This dependence is discussed for the two cases of smooth basin boundaries made up by the stable manifolds of saddle points and fractal basin boundaries where chaotic saddles embedded in the boundary influence the tipping of trajectories.