Detecting rate-dependent tipping with numerical stability spectra

Alice Nadeau (Mathematics, Cornell University)

Rate-dependent tipping is heuristically characterized by a change in the qualitative behavior of solutions of a nonautonomous dynamical system due to the rate at which parameters are changing in time. Importantly, the change in solutions can be caused without passing through a traditional bifurcation value of the parameters. In this talk, I’ll give a brief introduction to rate-dependent tipping and the dynamical systems techniques currently used to study the phenomenon. Unfortunately, these current methods assume that the parameter changes are asymptotically constant and do not necessarily apply when those assumptions are relaxed. Instead, we’ll introduce nonautonomous stability spectra in the form of Steklov averages as a more general way to study rate-dependent tipping. In one dimension, we can prove that Steklov averages are indicative of rate-dependent tipping in a specific family of equations. Adapting these techniques for more generic nonautonomous equations or systems in two dimensions and higher is more subtle and we’ll end with a discussion of the challenges that arise when we attempt to do so.