Rate-dependent tipping points for forced dynamical systems

Rate-dependent tipping has recently emerged as an identifiable type of instability of attractors in non-autonomous dynamical systems. Other tipping mechanisms involve bifurcation and noise, but there have been several studies of rate-induced tipping in the presence of a parameter that shifts between asymptotically constant values. The objective of this work is to provide a mathematical framework for the rate-induced tipping for certain types of non-autonomous dynamic systems with more general parameter shifts, in particular between asymptotically periodic parameter variation, and to provide the necessary conditions for such behaviour to occur or not. For a specific model system, we examine rate-induced tipping near a saddle-node bifurcation that is subject to a parameter shift between constant and periodic forcing. We consider how the critical rates of rate-induced transitions can be defined and calculated using direct numerical simulation for such a system. We show that the critical rates can be characterized in terms of heteroclinic connections between a saddle equilibrium and a saddle periodic orbit for an extended autonomous system. We compute this using Lin’s method for an extended system and explore the dependence of critical rates on other parameters in the system. More generally, for parameter shifts between different types of periodic forcing we establish the criteria for the existence of a bounded solution to the general situation and the appearance of sudden changes in the system state (tipping points) for the non-autonomous system.