We generalize the concept of basin of attraction of a stable state in order to facilitate the analysis of dynamical systems with noise and to assess stability properties of metastable states and long transients. To this end we examine the notions of mean sojourn times and absorption probabilities for Markov chains and study their relation to the basins of attraction. Our approach is directly applicable to all systems that can be approximated as Markov chains, including stochastic and deterministic differential equations. We also provide a sampling based generalization of basin stability that works without resorting to the Markov approximation by sampling trajectories directly.We discuss two far reaching generalizations of the basin of attraction of an attractor. These apply to general sets in the phase space of nondeterministic systems. The first is based on absorption probabilities, the second on expected mean sojourn times with respect to a finite time horizon. By casting the problem in the transfer operator language, we are able to give a simple formula for the first generalization along the lines of committor functions.We show that the two notions of generalized basin coincide in the limit of a vanishing absorption probability and an infinite time horizon respectively. Importantly, for well-behaved deterministic systems this limit recovers the usual notion of basin of attraction. Finally we point out that derived quantities like the volume of the generalized basin are accessible through sampling trajectories at the same computational cost as evaluating basin stability for deterministic systems.

We study the Lagrangian dynamics of passive tracers in a simple model of a driven two-dimensional vortex resembling real-world geophysical flow patterns. Using a discrete approximation of the system's transfer operator, we construct a directed network that characterizes the exchange of mass between distinct regions of the flow domain. By studying different measures characterizing flow network connectivity at different time-scales, we are able to identify the location of dynamically invariant structures and regions of maximum dispersion. Specifically, our approach allows to delimit co-existing flow regimes with different dynamics. To validate our findings, we compare several network characteristics to the well-established finite-time Lyapunov exponents and apply a receiver operating characteristic (ROC) analysis to identify network measures that are particularly useful for unveiling the skeleton of Lagrangian chaos.

The prediction of dynamical stability of power grids becomes more important and challenging with increasing shares of renewable energy sources due to their decentralized structure, reduced inertia and volatility. We investigate the feasibility of applying graph neural networks (GNN) to predict dynamic stability of synchronisation in complex power grids using the single-node basin stability (SNBS) as a measure. To do so, we generate two synthetic datasets for grids with 20 and 100 nodes respectively and estimate SNBS using Monte-Carlo sampling. Those datasets are used to train and evaluate the performance of eight different GNN-models. All models use the full graph without simplifications as input and predict SNBS in a nodal-regression-setup. We show that SNBS can be predicted in general and the performance significantly changes using different GNN-models. Furthermore, we observe interesting transfer capabilities of our approach: GNN-models trained on smaller grids can directly be applied on larger grids without the need of retraining.

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