Information about the behavior of dynamical systems can often be obtained by analyzing eigenvalues and corresponding eigenfunctions linear operators associated with a system. Examples such are Perron-Frobenius Koopman operator. In this paper, we will review different methods that have been developed over last decades to compute finite-dimensional approximations these infinite-dimensional - e.g. Ulam's method Extended Dynamic Mode Decomposition (EDMD) highlight similarities differences...

Markov state models (MSMs) and master equation are popular approaches to approximate molecular kinetics, equilibria, metastable states, reaction coordinates in terms of a space discretization usually obtained by clustering. Recently, powerful generalization MSMs has been introduced, the variational approach conformation dynamics/molecular kinetics (VAC) its special case time-lagged independent component analysis (TICA), which allow us slow collective variables linear combinations smooth...

The long-term distributions of trajectories a flow are described by invariant densities, i.e., fixed points an associated transfer operator. In addition, global slowly mixing structures, such as almost-invariant sets, which partition phase space into regions that almost dynamically disconnected, can also be identified certain eigenfunctions this Indeed, these structures often hard to obtain brute-force trajectory-based analyses. wide variety applications, operators have proven very efficient...

Dynamical systems often exhibit the emergence of long-lived coherent sets, which are regions in state space that keep their geometric integrity to a high extent and thus play an important role transport. In this article, we provide method for extracting sets from possibly sparse Lagrangian trajectory data. Our can be seen as extension diffusion maps space, it allows us construct "dynamical coordinates," reveal intrinsic low-dimensional organization data with respect The only priori knowledge...

We consider complex dynamical systems showing metastable behavior but no local separation of fast and slow time scales. The article raises the question whether such exhibit a low-dimensional manifold supporting its effective dynamics. For answering this question, we aim at finding nonlinear coordinates, called reaction that projection dynamics onto these coordinates preserves dominant scales show that, based on specific reducibility property, existence good preserving is guaranteed. Based...

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 the National Oceanic and Atmospheric Administration's Global Drifter Program. latter involved coping with openness system in physical space, which further required an adaptation standard setting. Directly connecting pollution...

There are multiple ways in which a stochastic system can be out of statistical equilibrium. It might subject to time-varying forcing; or transient phase on its way towards equilibrium; it even equilibrium without us noticing it, due insufficient observations; and failing admit an distribution at all. We review some the approaches that model effective behavior non-equilibrium dynamical systems, show both cases considered under unified framework optimal low-rank approximation so-called...

Abstract Finite‐time coherent sets (FTCSs) are distinguished regions of phase space that resist mixing with the surrounding for some finite period time; physical manifestations include eddies and vortices in ocean atmosphere, respectively. The boundaries FTCSs examples Lagrangian structures (LCSs). selection time duration over which FTCS LCS computations made practice is crucial to their success. If this longer than lifetime coherence individual objects then existing methods will fail detect...

Abstract. Atmospheric blocking exerts a major influence on mid-latitude atmospheric circulation and is known to be associated with extreme weather events. Previous work has highlighted the importance of origin air parcels that define region, especially respect non-adiabatic processes such as latent heating. So far, an objective method clustering individual Lagrangian trajectories passing through into larger and, more importantly, spatially coherent streams not been established. This focus...

The global macroscopic behavior of a dynamical system is encoded in the eigenfunctions associated Frobenius–Perron operator. For systems with low dimensional long term dynamics, efficient techniques exist for numerical approximation most important eigenfunctions; cf. [M. Dellnitz and O. Junge, SIAM J. Numer. Anal., 36 (1999), pp. 491–515]. They are based on projection operator onto space piecewise constant functions supported neighborhood attractor—Ulam's method. In this paper we develop...

Metastable behavior in dynamical systems may be a significant challenge for simulation-based analysis. In recent years, transfer operator--based approaches to problems exhibiting metastability have matured. order make these computationally feasible larger systems, various reduction techniques been proposed: For example, Schütte introduced spatial operator which acts on densities configuration space, while Weber proposed avoid trajectory simulation (like Froyland, Junge, and Koltai) by...

Unlike for systems in equilibrium, a straightforward definition of metastable set the non-stationary, non-equilibrium case may only be given case-by-case-and therefore it is not directly useful any more, particular cases where slowest relaxation time scales are comparable to at which external field driving system varies. We generalize concept metastability by relying on theory coherent sets. A pair sets and B called with respect interval [t1, t2] if (a) most trajectories starting t1 end up...

.Reaction coordinates (RCs) are indicators of hidden, low-dimensional mechanisms that govern the long-term behavior high-dimensional stochastic processes. We present a novel and general variational characterization optimal RCs provide conditions for their existence. Optimal minimizers certain loss function, reduced models based on them guarantee good approximation statistical properties original process. show slow-fast systems, metastable other systems with known RCs, theory reproduces...

Abstract We present a novel kernel-based machine learning algorithm for identifying the low-dimensional geometry of effective dynamics high-dimensional multiscale stochastic systems. Recently, authors developed mathematical framework computation optimal reaction coordinates such systems that is based on parameterization transition manifold in certain function space. In this article, we enhance approach by embedding and reproducing kernel Hilbert space, exploiting favorable properties...

Periodically driven flows are fundamental models of chaotic behavior and the study their transport properties is an active area research. A well-known analytic construction augmentation phase space with additional time dimension; in this augmented space, flow becomes autonomous or time-independent. We prove several results concerning connections between original time-periodic representation time-extended representation, focusing on properties. In deterministic setting, these include...

Markov-chain models are constructed for the probabilistic description of drift marine debris from Malaysian Airlines flight MH370. En route Kuala Lumpur to Beijing, MH370 mysteriously disappeared in southeastern Indian Ocean on 8 March 2014, somewhere along arc 7th ping ring around Inmarsat-3F1 satellite position when airplane lost contact. The obtained by discretizing motion undrogued satellite-tracked surface drifting buoys global historical data bank. A spectral analysis, Bayesian...

Given a time-dependent stochastic process with trajectories x(t) in space Ω, there may be sets such that the corresponding only very rarely cross boundaries of these sets. We can analyze terms metastability or coherence. Metastable setsM are defined M⊂Ω, and coherent setsM(t)⊂Ω time. Hence, if we extend Ω by time-variable t, metastable Ω×[0,∞) an appropriate space-time process. This relation exploited, because already exist spectral algorithms for identification In this article, show...

Understanding the macroscopic behavior of dynamical systems is an important tool to unravel transport mechanisms in complex flows. A decomposition state space into coherent sets a popular way reveal this essential evolution. To compute from aperiodic time-dependent system we consider relevant transfer operators and their infinitesimal generators on augmented space-time manifold. This generator approach avoids trajectory integration creates convenient linearization can be further exploited...