**Project Heads**

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

**Project Members**

Luzie Helfmann (FU)

**Project Duration**

01.02.2019 – 31.01.2022

**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 for discrete processes fulfill a linear system of equations or in the continuous setting a boundary-value PDE. 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.

*Figure: 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 to time-dependent dynamics, that are either periodically-varying or generally time-dependent on a finite time interval [1] (the code is available on GitHub).

*Figure: 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.*

Next, we wanted to demonstrate the broad applicability of TPT for understanding transition dynamics. In [2], we applied Transition Path Theory to the stationary dynamics of marine debris particles in order to understand their paths from the coast to accumulation sites in the ocean. Below we show the found transition flow from the coast (black boxes) to the various accumulation patches (red boxes).

In [3], we are studying tipping cascades in high-dimensional agent-based models. For TPT to be feasible, we first had to find a reduced representation of the dynamics. By applying a nonlinear dimension reduction technique, Diffusion Maps, to a realization, we could significantly reduce the dimension and estimate a transition matrix on the reduced space. The agent-based model dynamics on modular interaction networks are very metastable with a large number of metastable regions. With TPT we could reveal the cascading tipping dynamics between the different metastable regions.

*Figure: Tipping Analysis of the oscillatory complex contagion model from [3]: (a) Forward committor. We decomposed the transition current into the current coming from cycle-free, productive paths (b) and the current from unproductive cycles (c).*

**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.”
*Chaos: An Interdisciplinary Journal of Nonlinear Science.*2021. - L. Helfmann, N. Djurdjevac Conrad, A. Djurdjevac, S. Winkelmann, Ch. Schütte. “From interacting agents to density-based modeling with stochastic PDEs.”
*Communications in Applied Mathematics and Computational Science.*2021 - L. Helfmann, J. Heitzig, P. Koltai, J. Kurths, Ch. Schütte. “Statistical Analysis of Tipping Pathways in Agent-Based Models.”
*The European Physical Journal Special Topics.*2021. - N. Wulkow, P. Koltai, and Ch. Schütte. “Memory-based reduced modelling and data-based estimation of opinion spreading.”
*Journal of Nonlinear Science*. 2021.

**Selected Pictures
**

Please insert any kind of pictures (photos, diagramms, simulations, graphics) related to the project in the above right field (Image with Text), by choosing the green plus image on top of the text editor. (You will be directed to the media library where you can add new files.)

(We need pictures for a lot of purposes in different contexts, like posters, scientific reports, flyers, website,…

Please upload pictures that might be just nice to look at, illustrate, explain or summarize your work.)

As Title in the above form please add a copyright.

And please give a short description of the picture and the context in the above textbox.

Don’t forget to press the “Save changes” button at the bottom of the box.

If you want to add more pictures, please use the “clone”-button at the right top of the above grey box.