Detecting transient rate-tipping using Steklov averages and Lyapunov vectors

• Main Authors: Hoyer-Leitzel, Alanna, Nadeau, Alice, Roberts, Andrew, Steyer, Andrew
• Format: Report
• Language: unknown
• Published: arXiv 2017
• Subjects: Dynamical Systems math.DS; FOS Mathematics; 37C60 Primary, 37E99, 37M25 Secondary; Arctic; Sea ice
• Online Access: https://dx.doi.org/10.48550/arxiv.1702.02955; https://arxiv.org/abs/1702.02955

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. : 20 pages, 7 figures, 2 tables