Contrasting chaotic and stochastic forcing: tipping windows and attractor crises

Abstract

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 cause the system to leave one attractor and transition to another. We consider tipping in the presence of bounded chaotic forcing; previous work has primarily modelled noise forcing via a stochastic differential equation, but this is only a good model of chaotic forcing in the “fast chaos” limit. We argue that for low amplitude chaotic forcing 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 via a boundary crisis/non-autonomous saddle-node bifurcation and corresponds to an exceptional trajectory of the forcing, typically by an unstable periodic orbit. We discuss an illustrative example of a chaotically forced bistable map that highlights the richness of the geometry and bifurcation structure of the dynamics in this case. If there is also a ramping of a parameter we note there is a dynamic tipping window that can also be determined in terms of unstable periodic orbits.