Detecting transient rate-tipping using Steklov averages and Lyapunov vectors

TL;DR

This paper introduces methods using Steklov averages and Lyapunov vectors to detect transient rate-tipping in dynamical systems, addressing limitations of traditional long-term analysis techniques.

Contribution

It proposes novel criteria based on Steklov averages and Lyapunov vectors for identifying rate-tipping in nonautonomous dynamical systems.

Findings

Steklov averages correlate with rate-tipping in systems with unique QSEs.

Comparison of Lyapunov vectors can serve as a detection criterion.

Methods are applicable to systems with parameters changing at a constant rate.

Abstract

A wide variety of physical systems ranging from the firing of neurons to eutrophication of lakes to the presence of Arctic summer sea ice exhibit a phenomenon known as tipping. In mathematical models, tipping can be caused by bifurcations, noise, and the rate at which parameters are changing in time [2]. Because traditional methods in dynamical systems are usually concerned with the long-term behavior of the system, these methods are not always able to detect the transient dynamics characteristic of rate-tipping. In this paper, we consider one- and two-dimensional dynamical systems with nonautonomous parameters that exhibit rate-tipping, as defined as not tracking the evolution of stable equilibria (QSEs) in the corresponding autonomous systems. We find that nonautonomous stability spectra in the form of Steklov averages and their derivatives appear to be correlated with transient rate-tipping in systems with unique QSEs or with parameters that change at a constant rate. Furthermore, for systems in two dimensions and higher, comparison of the angle between leading Lyapunov vectors of different trajectories admits a possible criterion for detecting rate-tipping. Our heuristic results add to the body of work dedicated to studying and understanding the phenomenon of rate-tipping.