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A Brief Introduction to Rough PathsOnline: attend

Rough Paths are a suitable generalization of smooth paths in settings where classical (Riemann-Stieltjes, Young, ...) integration breaks down, most notably for integral formulations of (stochastic/controlled) differential equations. We'll give a short account of the key objects, spaces, properties involved, as well as some subtleties of the theory. The focus of the talk is going to be on the motivation via the "Rough Path principle" and rough integration as well as on "the big picture".

A path towards classifying rate-induced tipping as a nonautonomous bifurcationMI 03.06.011 (Boltzmannstr. 3, 85748 Garching)

Critical transitions are sudden changes in the dynamics of complex systems, often with catastrophic consequences. There are several mechanisms leading to a critical transition and we focus on those caused by the rate of a time-dependent drift of parameters (which are usually fixed or at most varied “adiabatically”), so-called rate-induced tipping. Particularly, in the case of a rate-induced tipping, a system evolves in time into another with possibly same topological properties of stability. However, depending on the rate at which such “transition” takes place, a local attractor of the past system can fail to track the “corresponding” local attractor of the future system. This encompasses various real scenarios for example in ecology, climate, biology and quantum mechanics. I will review some of the results on rate-induced tipping obtained with several collaborators in the past three years. The aim is to show that while the autonomous bifurcation theory can not explain the occurrence of rate-induced tipping, the nonautonomous one is the most adequate framework to do so.

The results in this presentation stem from collaborations with Christian Kuehn, Technical University Munich, Carmen Nunez, University of Valladolid, Rafael Obaya, University of Valladolid and Martin Rasmussen, Imperial College London.

References [1] C. Kuehn, I.P. Longo: Estimating rate-induced tipping via asymptotic series and a Melnikov-like method. Submitted, (2020). [2] I.P. Longo, C. Núñez, R. Obaya, M. Rasmussen, Rate-induced tipping and saddlenode bifurcation for quadratic differential equations with nonautonomous asymptotic dynamics, SIAM J. Appl. Dyn. Sys., 20 (1) (2021), 500–540. [3] I.P. Longo, C. Núñez, R. Obaya, Critical transitions in piecewise uniformly continuous concave quadratic ordinary differential equations, Submitted, (2021).