Identifying almost invariant sets in stochastic dynamical systems

Stable invariant models via Koopman spectra

Weight-tied models have attracted attention in the modern development of neural networks. The deep equilibrium model (DEQ) represents infinitely deep neural networks with weight-tying, and recent studies have shown the potential of this type of approach. DEQs are needed to iteratively solve root-finding problems in training and are built on the assumption that the underlying dynamics determined by the models converge to a fixed point. In this paper, we present the stable invariant model (SIM), a new class of deep models that in principle approximates DEQs under stability and extends the dynamics to more general ones converging to an invariant set (not restricted in a fixed point). The key ingredient in deriving SIMs is a representation of the dynamics with the spectra of the Koopman and Perron-Frobenius operators. This perspective approximately reveals stable dynamics with DEQs and then derives two variants of SIMs. We also propose an implementation of SIMs that can be learned in the same way as feedforward models. We illustrate the empirical performance of SIMs with experiments and demonstrate that SIMs achieve comparative or superior performance against DEQs in several learning tasks.

Stochastic basins of attraction and generalized committor functions

We study two generalizations of the basin of attraction of a stable state, to the case of stochastic dynamics, arbitrary regions, and finite-time horizons. This is done by introducing generalized committor functions and studying soujourn times. We show that the volume of the generalized basin, the basin stability, can be efficiently estimated using Monte Carlo-like techniques, making this concept amenable to the study of high-dimension stochastic systems. Finally, we illustrate in a set of examples that stochastic basins efficiently capture the realm of attraction of metastable sets, which parts of phase space go into long transients in deterministic systems, that they allow us to deal with numerical noise, and can detect the collapse of metastability in high-dimensional systems. We discuss two far-reaching generalizations of the basin of attraction of an attractor. The basin of attraction of an attractor are those states that eventually will get to the attractor. In a generic stochastic system, all regions will be left again; no attraction is permanent. To obtain the equivalent of the basin of attraction of a region we need to generalize the notion to cover finite-time horizons and finite regions. We do so by considering soujourn times, the fraction of time that a trajectory spends in a set, and by generalizing committor functions which arise in the study of hitting probabilities. In a simplified setting we show that these two notions reduce to the normal notions of the basin of attraction in the appropriate limits. We also show that the volume of these stochastic basins can be efficiently estimated for high-dimensional systems at computational cost comparable to that for deterministic systems. To fully illustrate the properties captured by the stochastic basins, we show a set of examples ranging from simple conceptual models to high-dimensional inhomogeneous oscillator chains. These show that stochastic basins efficiently capture metastable attraction, the presence of long transients, that they allow us to deal with numerical and approximation noise, and can detect the collapse of metastability with increasing noise in high-dimensional systems.

Toward efficient navigation in uncertain gyre-like flows

We present the development and experimental validation of an autonomous surface/underwater vehicle control strategy that leverages the environmental dynamics and uncertainty to navigate in a stochastic fluidic environment. We assume that the workspace is composed of the union of a collection of disjoint regions, each bounded by Lagrangian coherent structures (LCSs). LCSs are dynamical features in the flow field that behave like invariant manifolds in general time-invariant dynamical systems and delineate the boundaries of attraction basins. We analyze a passive particle’s noise-induced transition between adjacent LCS-bounded regions and show how most probable escape trajectories with respect to the transition probability between adjacent LCS-bounded regions can be determined. Additionally, we show how the likelihood of transition can be controlled through minimal actuation. The result is an energy efficient navigation strategy that leverages the inherent dynamics of the surrounding flow field for mobile sensors operating in a noisy fluidic environment. We experimentally validate the proposed vehicle control strategy and analyze its theoretical properties. Our results show that the single vehicle control parameter exhibits a predictable exponential scaling with respect to the escape times and is effective even in situations where the structure of the flow is not fully known and control effort is costly.

Topological chaos, braiding and bifurcation of almost-cyclic sets

In certain two-dimensional time-dependent flows, the braiding of periodic orbits provides a way to analyze chaos in the system through application of the Thurston-Nielsen classification theorem (TNCT). We expand upon earlier work that introduced the application of the TNCT to braiding of almost-cyclic sets, which are individual components of almost-invariant sets [Stremler et al., "Topological chaos and periodic braiding of almost-cyclic sets," Phys. Rev. Lett. 106, 114101 (2011)]. In this context, almost-cyclic sets are periodic regions in the flow with high local residence time that act as stirrers or "ghost rods" around which the surrounding fluid appears to be stretched and folded. In the present work, we discuss the bifurcation of the almost-cyclic sets as a system parameter is varied, which results in a sequence of topologically distinct braids. We show that, for Stokes' flow in a lid-driven cavity, these various braids give good lower bounds on the topological entropy over the respective parameter regimes in which they exist. We make the case that a topological analysis based on spatiotemporal braiding of almost-cyclic sets can be used for analyzing chaos in fluid flows. Hence, we further develop a connection between set-oriented statistical methods and topological methods, which promises to be an important analysis tool in the study of complex systems.

Topological chaos and periodic braiding of almost-cyclic sets

In certain (2+1)-dimensional dynamical systems, the braiding of periodic orbits provides a framework for analyzing chaos in the system through application of the Thurston-Nielsen classification theorem. Periodic orbits generated by the dynamics can behave as physical obstructions that "stir" the surrounding domain and serve as the basis for this topological analysis. We provide evidence that, even in the absence of periodic orbits, almost-cyclic regions identified using a transfer operator approach can reveal an underlying structure that enables topological analysis of chaos in the domain.

Transport in time-dependent dynamical systems: finite-time coherent sets

We study the transport properties of nonautonomous chaotic dynamical systems over a finite-time duration. We are particularly interested in those regions that remain coherent and relatively nondispersive over finite periods of time, despite the chaotic nature of the system. We develop a novel probabilistic methodology based upon transfer operators that automatically detect maximally coherent sets. The approach is very simple to implement, requiring only singular vector computations of a matrix of transitions induced by the dynamics. We illustrate our new methodology on an idealized stratospheric flow and in two and three-dimensional analyses of European Centre for Medium Range Weather Forecasting (ECMWF) reanalysis data.

Escape from attracting sets in randomly perturbed systems

The dynamics of escape from an attractive state due to random perturbations is of central interest to many areas in science. Previous studies of escape in chaotic systems have rather focused on the case of unbounded noise, usually assumed to have Gaussian distribution. In this paper, we address the problem of escape induced by bounded noise. We show that the dynamics of escape from an attractor's basin is equivalent to that of a closed system with an appropriately chosen "hole." Using this equivalence, we show that there is a minimum noise amplitude above which escape takes place, and we derive analytical expressions for the scaling of the escape rate with noise amplitude near the escape transition. We verify our analytical predictions through numerical simulations of two well-known two-dimensional maps with noise.