Controlling a tipping point

The concept of a tipping point is familiar: A generally stable system evolves to where the stability disappears and the system suddenly and irreversibly switches to a dramatically different, perhaps catastrophic state. The system may be a power grid that experiences a slow rise in demand, leading to a large-scale power outage, or a climate system, subject to increasing greenhouse gases, that has an abrupt shift in mean temperature. For a system with multiple potential outcomes, Takashi Nishikawa (Northwestern University) and Edward Ott (University of Maryland) report that it may be possible to steer the system to a desirable outcome by giving it a small, carefully chosen push. They theoretically consider the evolution of a noisy one-dimensional system: At each iteration, the current position is mapped to a new position, where the mapping depends on a slowly varying parameter plus some random noise. When the parameter reaches a threshold value—a so-called bifurcation—the system will shift to one of two states, but which it is can depend very sensitively on the noise fluctuations or the particular details of how the parameter varies. The researchers show with simulations and analytically that if the noise amplitude is low, one can get a 90% chance of reaching the desired outcome through a one-time shift in the system’s position by an amount just a few times larger than the noise level. Surprisingly, the required nudge is smallest during a window of opportunity that occurs a finite time after the bifurcation. Although the method can be generalized to higher dimensions, the authors caution that it requires having an accurate system model. (T. Nishikawa, E. Ott, Chaos 24, 033107, 2014, doi:10.1063/1.4887275 .)