Controlling systems that drift through a tipping point

TL;DR

This paper explores how small, timely interventions can steer systems through tipping points with multiple attractors, especially under noise, by identifying a window of opportunity for effective control.

Contribution

It introduces a method for controlling noisy systems near bifurcations using minimal perturbations, highlighting a temporal control window after the bifurcation.

Findings

Small perturbations can reliably steer the system toward a desired attractor.

Control is most effective within a specific time window after the bifurcation.

The required perturbation size is smaller when applied during the control window.

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.