Analysis and Predictability for Tipping Points with Leading-Order Nonlinear Terms

TL;DR

This paper extends the analysis of tipping points in stochastic systems by incorporating leading-order nonlinear terms, deriving variance scaling laws, and evaluating early-warning signs beyond linear approximations.

Contribution

It generalizes the understanding of early-warning signs for tipping points to systems with nonlinear deterministic terms, providing new theoretical and numerical insights.

Findings

Derived variance scaling laws near bifurcations with nonlinear terms

Proved existence of stationary distribution for fast variables with nonlinearities

Validated scaling laws and predictability measures through numerical simulations

Abstract

Tipping points have been actively studied in various applications as well as from a mathematical viewpoint. A main technique to theoretically understand early-warning signs for tipping points is to use the framework of fast-slow stochastic differential equations. A key assumption in many arguments for the existence of variance and auto-correlation growth before a tipping point is to use a linearization argument, i.e., the leading-order term governing the deterministic (or drift) part of stochastic differential equation is linear. This assumption guarantees a local approximation via an Ornstein-Uhlenbeck process in the normally hyperbolic regime before, but sufficiently bounded away from, a bifurcation. In this paper, we generalize the situation to leading-order nonlinear terms for the setting of one fast variable. We work in the quasi-steady regime and prove that the fast variable has a well-defined stationary distribution and we calculate the scaling law for the variance as a bifurcation-induced tipping point is approached. We cross-validate the scaling law numerically. Furthermore, we provide a computational study for the predictability using early-warning signs for leading-order nonlinear terms based upon receiver-operator characteristic curves.