Rate-induced tipping (R-tipping) occurs when time-variation of input parameters of a dynamical
system interacts with system timescales to give genuine nonautonomous instabilities. Such instabilities appear as the input varies at some critical rates and cannot, in general, be understood
in terms of autonomous bifurcations in the frozen ...
Rate-induced tipping (R-tipping) occurs when time-variation of input parameters of a dynamical
system interacts with system timescales to give genuine nonautonomous instabilities. Such instabilities appear as the input varies at some critical rates and cannot, in general, be understood
in terms of autonomous bifurcations in the frozen system with a fixed-in-time input.
This paper develops an accessible mathematical framework for R-tipping in multidimensional
nonautonomous dynamical systems with an autonomous future limit. We focus on R-tipping via
loss of tracking of base attractors that are equilibria in the frozen system, due to crossing what we
call regular R-tipping thresholds. These thresholds are anchored at infinity by regular R-tipping
edge states: compact normally hyperbolic invariant sets of the autonomous future limit system
that have one unstable direction, orientable stable manifold, and lie on a basin boundary. We
define R-tipping and critical rates for the nonautonomous system in terms of special solutions
that limit to a compact invariant set of the autonomous future limit system that is not an
attractor. We focus on the case when the limit set is a regular edge state, introduce the concept
of edge tails, and rigorously classify R-tipping into reversible, irreversible, and degenerate cases.
The central idea is to use the autonomous dynamics of the future limit system to analyse Rtipping in the nonautonomous system. We compactify the original nonautonomous system to
include the limiting autonomous dynamics. Considering regular R-tipping edge states that are
equilibria allows us to prove two results. First, we give sufficient conditions for the occurrence of
R-tipping in terms of easily testable properties of the frozen system and input variation. Second,
we give necessary and sufficient conditions for the occurrence of reversible and irreversible Rtipping in terms of computationally verifiable (heteroclinic) connections to regular R-tipping
edge states in the autonomous compactified system.
