  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.
