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 ...
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.
