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#### Institute

We consider complex dynamical systems showing metastable behavior but no local
separation of fast and slow time scales. The article raises the question of whether
such systems exhibit a low-dimensional manifold supporting its effective dynamics.
For answering this question, we aim at finding nonlinear coordinates, called reaction
coordinates, such that the projection of the dynamics onto these coordinates preserves
the dominant time scales of the dynamics. We show that, based on a specific
reducibility property, the existence of good low-dimensional reaction coordinates
preserving the dominant time scales is guaranteed. Based on this theoretical framework,
we develop and test a novel numerical approach for computing good reaction
coordinates. The proposed algorithmic approach is fully local and thus not prone to
the curse of dimension with respect to the state space of the dynamics. Hence, it is
a promising method for data-based model reduction of complex dynamical systems
such as molecular dynamics.

We investigate opinion dynamics based on an agent-based model, and are
interested in predicting the evolution of the percentages of the entire
agent population that share an opinion. Since these opinion percentages
can be seen as an aggregated observation of the full system state, the
individual opinions of each agent, we view this in the framework of the
Mori-Zwanzig projection formalism. More specifically, we show how to
estimate a nonlinear autoregressive model (NAR) with memory from data
given by a time series of opinion percentages, and discuss its prediction
capacities for various specific topologies of the agent interaction
network. We demonstrate that the inclusion of memory terms significantly
improves the prediction quality on examples with different network
topologies.

We investigate metastable dynamical systems subject to non-stationary forcing as they appear in molecular dynamics for systems driven by external fields. We show, that if the strength of the forcing is inversely proportional to the length of the slow metastable time scales of the unforced system, then the effective behavior of the forced system on slow time scales can be described by a low-dimensional reduced master equation. Our construction is explicit and uses the multiscale perturbation expansion method called two-timing, or method of multiple scales. The reduced master equation—a Markov state model—can be assembled by constructing two equilibrium Markov state models; one for the unforced system, and one for a slightly perturbed one.

Given a time-dependent stochastic process with trajectories x(t) in a space $\Omega$, there may be sets such that the corresponding trajectories only very rarely cross the boundaries of these sets. We can analyze such a process in terms of metastability or coherence. Metastable sets M are defined in space $M\subset\Omega$, coherent sets $M(t)\subset\Omega$ are defined in space and time. Hence, if we extend the space by the time-variable t, coherent sets are metastable sets in $\Omega\times[0,\infty]$. This relation can be exploited, because there already exist spectral algorithms for the identification of metastable sets. In this article we show that these well-established spectral algorithms (like PCCA+) also identify coherent sets of non-autonomous dynamical systems. For the identification of coherent sets, one has to compute a discretization (a matrix T) of the transfer operator of the process using a space-timediscretization scheme. The article gives an overview about different time-discretization schemes and shows their applicability in two different fields of application.

We prove the Fréchet differentiability with respect to the drift of Perron–Frobenius and Koopman operators associated to time-inhomogeneous ordinary stochastic differential equations. This result relies on a similar differentiability result for pathwise expectations of path functionals of the solution of the stochastic differential equation, which we establish using Girsanov's formula. We demonstrate the significance of our result in the context of dynamical systems and operator theory, by proving continuously differentiable drift dependence of the simple eigen- and singular values and the corresponding eigen- and singular functions of the stochastic Perron–Frobenius and Koopman operators.

We used transition path theory (TPT) to infer "reactive" pathways of floating marine debris trajectories. The TPT analysis was applied on a pollution-aware time-homogeneous Markov chain model constructed from trajectories produced by satellite-tracked undrogued buoys from the NOAA Global Drifter Program. The latter involved coping with the openness of the system in physical space, which further required an adaptation of the standard TPT setting. Directly connecting pollution sources along coastlines with garbage patches of varied strengths, the unveiled reactive pollution routes represent alternative targets for ocean cleanup efforts. Among our specific findings we highlight: constraining a highly probable pollution source for the Great Pacific Garbage Patch; characterizing the weakness of the Indian Ocean gyre as a trap for plastic waste; and unveiling a tendency of the subtropical gyres to export garbage toward the coastlines rather than to other gyres in the event of anomalously intense winds.