Rate-induced tipping: thresholds, edge states and connecting orbits

Abstract

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 R-tipping 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 R-tipping in terms of computationally verifiable (heteroclinic) connections to regular R-tipping edge states in the autonomous compactified system.

Rate-induced tipping to metastable Zombie fires

“…Owing to the explicit time dependence of the external input, the ensuing dynamical system is nonautonomous. This means that analysis of tipping points requires, in general, techniques beyond classical autonomous stability theory [6][7][8]. Nonetheless, it is useful to consider the corresponding autonomous frozen system with fixed-in-time inputs.…”

“…Another, arguably more interesting example is when the external input changes faster than some critical rate, the nonautonomous system deviates too far from the changing base state, crosses some threshold or quasi-threshold [11] and tips to an alternative state. Such tipping is caused solely by the rate of change of the external input and we say the system undergoes rate-induced tipping, or in short Rtipping [6,8]. Crucially, unlike B-tipping, R-tipping can occur to an alternative transient state that lasts for a finite time, 1 after which the system returns to the base state [12][13][14][15].…”

“…Crucially, unlike B-tipping, R-tipping can occur to an alternative transient state that lasts for a finite time, 1 after which the system returns to the base state [12][13][14][15]. Such R-tipping is referred to as 'reversible' in [8]. Systems that exhibit reversible R-tipping are also known as excitable systems [16].…”

“…In the second, mathematical analysis part of the paper in § §4-7, we identify non-obvious dynamical mechanisms that are responsible for the R-tipping instability to the hot metastable state. The main obstacles to analysis of this instability are twofold: the conceptual soil-carbon model is a nonautonomous dynamical system, so it does not have any equilibria or compact invariant sets, and the R-tipping is a quantitative change ( [8], Sec.3.3) due to crossing an elusive quasi-threshold in the phase space [13,36]. To overcome these obstacles, we combine three different strategies.…”

“…These two strategies alone work for R-tipping due to crossing regular thresholds that are anchored at infinity by unstable compact invariant sets called regular R-tipping edge states ([8], Sec.4). However, quasi-thresholds do not contain such edge states ( [8], Sec.8). Therefore, thirdly, we exploit large timescale separation in the model and apply concepts from geometric singular perturbation theory (GSPT).…”