Rate-induced Tipping in Applied Dynamical Systems: Multi-dimensional Flows and Maps

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  • Kiers, Claire Elizabeth
    • Affiliation: College of Arts and Sciences, Department of Mathematics
Abstract
  • We derive conditions under which we expect to see or not to see rate-induced tipping (R-tipping) in dynamical systems, both continuous-time (flows) and discrete-time (maps). This has already been studied in 1-dimensional flows, and we extend the results from this setting to flows and maps of arbitrary dimension. In particular, we show that when a path is not forward basin stable in a certain way, we can expect R-tipping in flows of any dimension. However, forward basin stability does not prevent R-tipping in flows of dimension greater than 1. Instead, we introduce a new condition, forward inflowing stability, which does prevent R-tipping in flows of any dimension. Although forward inflowing stability is a natural condition, it is difficult to verify in many concrete examples, so we focus on a particular class of flows called monotone systems, for which forward inflowing stability is implied by an easily verifiable condition. Predicting R-tipping in such a system is similar to predicting R-tipping in 1-dimensional flows. We then investigate two earth-climate models: the layer greenhouse model and the FAST tropical cyclone model, which are both monotone systems. For each, we give extensive explanation of its phase space and autonomous dynamics and then give some results about when to expect R-tipping. Finally, we define R-tipping in maps. We show that forward basin stability does not prevent R-tipping in maps of any dimension but that forward inflowing stability does. Also, if a path is not forward basin stable in a certain way, we can expect R-tipping, much like in flows. We also highlight some of the differences between R-tipping in flows and maps. Specifically, we show that what happens to a pullback attractor as the parameter rate of change approaches infinity is different in flows and maps. Also, we demonstrate that there can be different tipping behavior even when a map is obtained from a flow by evaluating at discrete time intervals. We finish by demonstrating an instance of R-tipping in the Ikeda map.
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Advisor
  • Jones, Christopher K.R.T.
  • Christianson, Hans
  • Gemmer, John
  • Newhall, Katie
  • Silber, Mary
Degree
  • Doctor of Philosophy
Degree granting institution
  • University of North Carolina at Chapel Hill Graduate School
Graduation year
  • 2020
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