Controlling systems that drift through a tipping point

Abstract

Slow parameter drift is common in many systems (e.g., the amount of greenhouse gases in the terrestrial atmosphere is increasing). In such situations, the attractor on which the system trajectory lies can be destroyed, and the trajectory will then go to another attractor of the system. We consider the case where there are more than one of these possible final attractors, and we ask whether we can control the outcome (i.e., the attractor that ultimately captures the trajectory) using only small controlling perturbations. Specifically, we consider the problem of controlling a noisy system whose parameter slowly drifts through a saddle-node bifurcation taking place on a fractal boundary between the basins of multiple attractors. We show that, when the noise level is low, a small perturbation of size comparable to the noise amplitude applied at a single point in time can ensure that the system will evolve toward a target attracting state with high probability. For a range of noise levels, we find that the minimum size of perturbation required for control is much smaller within a time period that starts some time after the bifurcation, providing a "window of opportunity" for driving the system toward a desirable state. We refer to this procedure as tipping point control.

Parameter shifts for nonautonomous systems in low dimension: bifurcation- and rate-induced tipping

“…However, close tracking of a continuous path that traverses several stable branches of (1) cannot always be guaranteed, even if r is small enough. If the variation of Λ along a branch of attractors brings the system to a bifurcation point where no stable branches are nearby, we have the ingredients of a bifurcation-induced or B-tipping point; no matter how slowly we change the parameter, there will be a sudden and irreversible change in the state of the system x on, or near where the parameter passed through a dynamic bifurcation [4,8,23].…”

“…There have been a range of papers in the applied sciences that use the notion of a tipping point in applications such as climate systems [9,18,34] or ecosystems [30,11]. Further work has attempted to find predictors or early warning signals in terms of changes in noise properties near a tipping point [9,10,33,31,28], other recent work on tipping includes [22,23]. Indeed there is a large literature on catastrophe theory (for example [2] and references therein) that addresses related questions in the setting of gradient systems.…”

“…Consider the system(23) with b(λ) and d(λ) depending smoothly on a single real parameter λ ∈ [0, 1]. Suppose that λ varies with time by a parameter shift Λ ∈ P(0, 1).…”

Enhancing noise-induced switching times in systems with distributed delays

“…Furthermore, noise can force the system out of the stable steady state, or it can lead to switching to another stable steady state. This phenomenon is similar to the tipping process in dynamical systems, where relatively small changes in input can lead to sudden and disproportional changes in output, for example, due to a slow variation in parameter (B-tipping) or inclusion of noise (N-tipping) [23][24][25]. For systems not in thermal equilibrium, as in the case of noise-induced switching between states, the probability distribution is no longer of the Boltzmann form.…”

Quantifying nonergodicity in nonautonomous dissipative dynamical systems: An application to climate change

“…This is the case with Lagrangian coherent structures [7], or with power grids experiencing an increasing demand [8]. For other examples see [9]. Since there is a growing concern that the observed increase of greenhouse gases may lead to dramatic changes in the Earth System, perhaps the most striking example for continuous parameter shifts occurs in the dynamics of climate change [10,11].…”