**Project Heads**

*Jobst Heitzig, Péter Koltai, Jürgen Kurths, Christof Schütte*

**Project Members**

Luzie Helfmann (FU)

**Project Duration**

01.01.2019 – 31.12.2021

**Located at**

FU Berlin

Many complex systems exhibit tipping behaviour, i.e., drastic changes of a system from one rather stable regime to another. Understanding these transitions is often of interest, e.g., in social systems (the appearance of trends, hypes, opinion shifts in social populations) or in the climate system (where sub-components of the earth can tip to undesired states).

Motivated by the desire to better analyse and understand tipping processes and critical transitions in real-world systems, this project aims at developing tools for studying transition pathways in stochastic, possibly high-dimensional and time-dependent systems.

Our approach builds on Transition Path Theory (TPT), an existing theory for quantitatively studying the noise-induced transition behaviour between fixed sets A and B in a stationary Markov process. The main object of TPT are the forward committor function, telling us the probability of next hitting set B and not A given the system is in some state, and the backward committor function, that gives the probability that the process last came from A and not B. These functions naturally subdivide the state space into regions of similar behaviour with respect to transitioning between the sets A and B and fulfill a linear system of equations (for discrete processes) or boundary-value PDE (in the continuous setting). Moreover, from the committor probabilities one can derive meaningful A → B transition quantities, e.g., the most important transition pathways, the rate of transitions, the mean transition length.

*Transitions in the stationary, overdamped Langevin process. (a) Energy landscape V defining two strongly (wells A and B) and one weakly (top well) attracting region with two exemplary transitions from A to B. (b) Forward committor. (c) Transition current from A to B. (d) Backward committor.*

We extended the classical Transition Path Theory for time-dependent systems, that are either periodically-varying or generally time-dependent on a finite time interval [1] (the code is available on GitHub).

*Adding a periodically-varying force (anti-clockwise rotation during the first half of the period followed by a clockwise rotation) to the dynamics in the triple well landscape, results in a time-dependent transition flow from A to B that sometimes prefers the lower channel and at other times the upper channel through the top well.*

The next steps are tackling high-dimensional state spaces, where numerically solving the committor equations poses a challenge, and studying the committors and transition channels under parameter changes. We will apply the developed theory to a range of models of social and climate dynamics that exhibit tipping.

**Project Webpages**

**Selected Publications
**

- L. Helfmann, E. Ribera Borrell, Ch. Schütte, and P. Koltai. “Extending Transition Path Theory: Periodically Driven and Finite-Time Dynamics.”
*Journal of Nonlinear Science.*2020. - P. Miron, F. Beron-Vera, L. Helfmann, and P. Koltai. “Transition paths of marine debris and the stability of the garbage patches.”
*Preprint on arXiv: 2009.11234.*2020

**Selected Pictures
**

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