Threshold for Kendall’s τ to infer imminent bifurcation

Determine a statistically justified threshold or decision rule for Kendall’s τ—the rank correlation between the control parameter u and an early warning signal s(u)—that reliably indicates the approach of a bifurcation in stochastic nonlinear dynamical systems. The criterion should specify which values of τ are sufficiently large to infer an upcoming bifurcation and under what assumptions on data generation and noise this inference is valid.

Background

Kendall’s τ is widely used to assess the performance of early warning signals (EWSs) that are expected to increase as a control parameter u approaches a bifurcation. However, the paper highlights a core limitation: despite broad use, there is no established τ threshold that demarcates when a bifurcation is imminent. This uncertainty undermines practical decision-making based on τ alone.

The authors point out further limitations of τ, such as high values occurring in settings without true bifurcations and the inability of τ to quantify the remaining distance to the bifurcation. These issues motivate the need for more robust criteria and methods beyond τ or, at minimum, for principled rules on interpreting τ in bifurcation detection.