arXiv subjects

Gary Froyland

Publications and source records attributed to Gary Froyland.

At least 37 records · Page 2Linked to original sources

arXiv2023

Revealing trends and persistent cycles of non-autonomous systems with operator-theoretic techniques: Applications to past and present climate dynamics

An important problem in modern applied science is characterizing the behavior of systems with complex internal dynamics subjected to external forcings from their environment. While a great variety of techniques has been developed to analyze such non-autonomous systems, many approaches rely on the availability of ensembles of experiments or simulations in order to generate sufficient information to encapsulate the external forcings. This makes them unsuitable to study important classes of natural systems such as climate dynamics where only a single realization is observed. Here, we show that operator-theoretic techniques previously developed to identify slowly decaying observables of autonomous dynamical systems provide a powerful means for identifying trends and persistent cycles of non-autonomous systems using data from a \emph{single} trajectory of the system. Using systematic mathematical analysis and prototype examples, we demonstrate that eigenfunctions of Koopman and transfer operators provide coordinates that simultaneously capture nonlinear trends and coherent modes of internal variability. In addition, we apply our framework to two real-world examples from present and past climate dynamics: Variability of sea surface temperature (SST) over the industrial era and the mid-Pleistocene transition (MPT) of Quaternary glaciation cycles. Our results provide a nonparametric representation of SST and surface air temperature (SAT) trends over the industrial era, while also capturing the response of the seasonal precipitation cycle to these trends. In addition, our paleo-climate analysis reveals the dominant glaciation cycles over the past 3 million years and the MPT with a high level of granularity.

math.DS↗

arXiv2023

Bathymetry imposes a global pattern of cross-front transport in the Southern Ocean

The Southern Ocean plays an integral role in the global climate system, exchanging heat, salt, and carbon throughout the major ocean basins via the deep, fast-flowing Antarctic Circumpolar Current. The Antarctic Circumpolar Current is bounded by spatially and temporally varying fronts that partition distinct water masses. Locating and quantifying cross-front transport is crucial for understanding global patterns of inter-basin exchange; however, this is challenging because fronts are typically defined by hydrographic properties, such as temperature or salinity, which are subject to external sources and sinks, rather than by properties of the flow itself. Here we show that, when characterized by material contours that minimize deformation and cross-contour mixing, Southern Ocean fronts exhibit a global pattern of alternating poleward and equatorward transport caused by frontal meandering, which, in turn is influenced by prominent sea-floor obstacles. These results highlight the importance of bathymetric features in controlling Southern Ocean dynamics and inter-basin exchange.

physics.ao-ph↗

arXiv2022

Angular values of nonautonomous and random linear dynamical systems: Part I -- Fundamentals

We introduce the notion of angular values for deterministic linear difference equations and random linear cocycles. We measure the principal angles between subspaces of fixed dimension as they evolve under nonautonomous or random linear dynamics. The focus is on long-term averages of these principal angles, which we call angular values: we demonstrate relationships between different types of angular values and prove their existence for random dynamical systems. For one-dimensional subspaces in two-dimensional systems our angular values agree with the classical theory of rotation numbers for orientation-preserving circle homeomorphisms if the matrix has positive determinant and does not rotate vectors by more than $\fracπ{2}$. Because our notion of angular values ignores orientation by looking at subspaces rather than vectors, our results apply to dynamical systems of any dimension and to subspaces of arbitrary dimension. The second part of the paper delves deeper into the theory of the autonomous case. We explore the relation to (generalized) eigenspaces, provide some explicit formulas for angular values, and set up a general numerical algorithm for computing angular values via Schur decompositions.

math.DS↗

arXiv2022

Perturbation formulae for quenched random dynamics with applications to open systems and extreme value theory

We consider quasi-compact linear operator cocycles $\mathcal{L}^{n}_ω:=\mathcal{L}_{σ^{n-1}ω}\circ\cdots\circ\mathcal{L}_{σω}\circ \mathcal{L}_ω$ driven by an invertible ergodic process $σ:Ω\toΩ$, and their small perturbations $\mathcal{L}_{ω,ε}^{n}$. We prove an abstract $ω$-wise first-order formula for the leading Lyapunov multipliers. We then consider the situation where $\mathcal{L}_ω^{n}$ is a transfer operator cocycle for a random map cocycle $T_ω^{n}:=T_{σ^{n-1}ω}\circ\cdots\circ T_{σω}\circ T_ω$ and the perturbed transfer operators $\mathcal{L}_{ω,ε}$ are defined by the introduction of small random holes $H_{ω,ε}$ in $[0,1]$, creating a random open dynamical system. We obtain a first-order perturbation formula in this setting, which reads $λ_{ω,ε}=λ_ω-θ_ωμ_ω(H_{ω,ε})+o(μ_ω(H_{ω,ε})),$ where $μ_ω$ is the unique equivariant random measure (and equilibrium state) for the original closed random dynamics. Our new machinery is then deployed to create a spectral approach for a quenched extreme value theory that considers random dynamics with general ergodic invertible driving, and random observations. An extreme value law is derived using the first-order terms $θ_ω$. Further, in the setting of random piecewise expanding interval maps, we establish the existence of random equilibrium states and conditionally invariant measures for random open systems via a random perturbative approach. Finally we prove quenched statistical limit theorems for random equilibrium states arising from contracting potentials. We illustrate the theory with a variety of explicit examples.

math.DS↗

arXiv2022

Equilibrium states for non-transitive random open and closed dynamical systems

We prove a random Ruelle--Perron--Frobenius theorem and the existence of relative equilibrium states for a class of random open and closed interval maps, without imposing transitivity requirements, such as mixing and covering conditions, which are prevalent in the literature. This theorem provides existence and uniqueness of random conformal and invariant measures with exponential decay of correlations, and allows us to expand the class of examples of (random) dynamical systems amenable to multiplicative ergodic theory and the thermodynamic formalism. Applications include open and closed non-transitive random maps, and a connection between Lyapunov exponents and escape rates through random holes. We are also able to treat random intermittent maps with geometric potentials.

math.DS↗

arXiv2021

Persistence and material coherence of a mesoscale ocean eddy

Ocean eddies play an important role in the transport and mixing processes of the ocean due to their ability to transport material, heat, salt, and other tracers across large distances. They exhibit at least two timescales; an Eulerian lifetime associated with persistent identifiable signatures in gridded fields such as vorticity or sea-surface height, and multiple Lagrangian or material coherence timescales that are typically much shorter. We propose a method to study the multi-timescale material transport, leakage, and entrainment by eddies with their surroundings by constructing sequences of finite-time coherent sets, computed as superlevel sets of dominant eigenfunctions of dynamic Laplace operators. The dominant eigenvalues of dynamic Laplace operators defined on time intervals of varying length allows us to identify a maximal coherence timescale that minimizes the rate of mass loss over a domain, per unit flow time. We apply the method to examine the persistence and material coherence of an Agulhas ring, an ocean eddy in the South Atlantic ocean, using particle trajectories derived from a $0.1^\circ$ global numerical ocean simulation. Using a sequence of sliding windows, the method is able to identify and track a persistent eddy feature for a time much longer than the maximal coherence timescale, and with considerably larger material transport than the corresponding eddy feature identified from purely Eulerian information. Furthermore, the median residence times of fluid in the identified feature far exceed the timescale over which fully material motion is guaranteed. Through residence time calculations, we find that this particular eddy does not exhibit a long-lived coherent inner core and that the bulk of material transport is performed by the quasi-coherent outer ring of the eddy.

physics.flu-dyn↗

arXiv2021

Deep Lagrangian connectivity in the global ocean inferred from Argo floats

We describe the application of a new technique from nonlinear dynamical systems to infer the Lagrangian connectivity of the deep global ocean. We approximate the dynamic Laplacian using Argo trajectories from January 2011 to January 2017 and extract the eight dominant coherent (or dynamically self-connected) regions at 1500m depth. Our approach overcomes issues such as sparsity of observed data, and floats continually leaving and entering the dataset; only 10\% of floats record for the full six years. The identified coherent regions maximally trap water within them over the six-year time frame, providing a distinct analysis of the deep global ocean, and relevant information for planning future float deployment. While our study is concerned with ocean circulation at a multi-year, global scale, the dynamic Laplacian approach may be applied at any temporal or spatial scale to identify coherent structures in ocean flow from positional time series information arising from observations or models.

math.DS↗

arXiv2021

Spectral analysis of climate dynamics with operator-theoretic approaches

The Earth's climate system is a classical example of a multiscale, multiphysics dynamical system with an extremely large number of active degrees of freedom, exhibiting variability on scales ranging from micrometers and seconds in cloud microphysics, to thousands of kilometers and centuries in ocean dynamics. Yet, despite this dynamical complexity, climate dynamics is known to exhibit coherent modes of variability. A primary example is the El Niño Southern Oscillation (ENSO), the dominant mode of interannual (3-5 yr) variability in the climate system. The objective and robust characterization of this and other important phenomena presents a long-standing challenge in Earth system science, the resolution of which would lead to improved scientific understanding and prediction of climate dynamics, as well as assessment of their impacts on human and natural systems. Here, we show that the spectral theory of dynamical systems, combined with techniques from data science, provides an effective means for extracting coherent modes of climate variability from high-dimensional model and observational data, requiring no frequency prefiltering, but recovering multiple timescales and their interactions. Lifecycle composites of ENSO are shown to improve upon results from conventional indices in terms of dynamical consistency and physical interpretability. In addition, the role of combination modes between ENSO and the annual cycle in ENSO diversity is elucidated.

physics.ao-ph↗

arXiv2021

Thermodynamic Formalism for Random Weighted Covering Systems

We develop a quenched thermodynamic formalism for random dynamical systems generated by countably branched, piecewise-monotone mappings of the interval that satisfy a random covering condition. Given a random contracting potential $φ$ (in the sense of Liverani-Saussol-Vaienti), we prove there exists a unique random conformal measure $ν_φ$ and unique random equilibrium state $μ_φ$. Further, we prove quasi-compactness of the associated transfer operator cocycle and exponential decay of correlations for $μ_φ$. Our random driving is generated by an invertible, ergodic, measure-preserving transformation $σ$ on a probability space $(Ω,\mathscr{F},m)$; for each $ω\inΩ$ we associate a piecewise-monotone, surjective map $T_ω:I\to I$. We consider general potentials $φ_ω:I\to\mathbb R\cup\{-\infty\}$ such that the weight function $g_ω=e^{φ_ω}$ is of bounded variation. We provide several examples of our general theory. In particular, our results apply to linear and non-linear systems including random $β$-transformations, randomly translated random $β$-transformations, random Gauss-Renyi maps, random non-uniformly expanding maps such as intermittent maps and maps with contracting branches, and a large class of random Lasota-Yorke maps.

math.DS↗

arXiv2021

Linear response for the dynamic Laplacian and finite-time coherent sets

Finite-time coherent sets represent minimally mixing objects in general nonlinear dynamics, and are spatially mobile features that are the most predictable in the medium term. When the dynamical system is subjected to small parameter change, one can ask about the rate of change of (i) the location and shape of the coherent sets, and (ii) the mixing properties (how much more or less mixing), with respect to the parameter. We answer these questions by developing linear response theory for the eigenfunctions of the dynamic Laplace operator, from which one readily obtains the linear response of the corresponding coherent sets. We construct efficient numerical methods based on a recent finite-element approach and provide numerical examples.

math.DS↗

arXiv2021

Optimal linear response for Markov Hilbert-Schmidt integral operators and stochastic dynamical systems

We consider optimal control problems for discrete-time random dynamical systems, finding unique perturbations that provoke maximal responses of statistical properties of the system. We treat systems whose transfer operator has an $L^2$ kernel, and we consider the problems of finding (i) the infinitesimal perturbation maximising the expectation of a given observable and (ii) the infinitesimal perturbation maximising the spectral gap, and hence the exponential mixing rate of the system. Our perturbations are either (a) perturbations of the kernel or (b) perturbations of a deterministic map subjected to additive noise. We develop a general setting in which these optimisation problems have a unique solution and construct explicit formulae for the unique optimal perturbations. We apply our results to a Pomeau-Manneville map and an interval exchange map, both subjected to additive noise, to explicitly compute the perturbations provoking maximal responses.

math.DS↗

arXiv2021

Thermodynamic Formalism for Random Interval Maps with Holes

We develop a quenched thermodynamic formalism for open random dynamical systems generated by finitely branched, piecewise-monotone mappings of the interval. The openness refers to the presence of holes in the interval, which terminate trajectories once they enter; the holes may also be random. Our random driving is generated by an invertible, ergodic, measure-preserving transformation $σ$ on a probability space $(Ω,\mathscr{F},m)$. For each $ω\inΩ$ we associate a piecewise-monotone, surjective map $T_ω:I\to I$, and a hole $H_ω\subset I$; the map $T_ω$, the random potential $φ_ω$, and the hole $H_ω$ generate the corresponding open transfer operator $\mathcal{L}_ω$. For a contracting potential, under a condition on the open random dynamics in the spirit of Liverani--Maume-Deschamps, we prove there exists a unique random probability measure $ν_ω$ supported on the survivor set ${X}_{ω,\infty}$ satisfying $ν_{σ(ω)}(\mathcal{L}_ωf)=λ_ων_ω(f)$. We also prove the existence of a unique random family of functions $q_ω$ that satisfy $\mathcal{L}_ωq_ω=λ_ωq_{σ(ω)}$. These yield an ergodic random invariant measure $μ=νq$ supported on the global survivor set, while $q$ combined with the random closed conformal measure yields a unique random absolutely continuous conditional invariant measure (RACCIM) $η$ supported on $I$. We prove quasi-compactness of the transfer operator cocycle and exponential decay of correlations for $μ$. Finally, the escape rates of the random closed conformal measure and the RACCIM $η$ coincide, and are given in terms of the expected pressure, as is the Hausdorff dimension of the surviving set $X_{ω,\infty}$. We provide examples of our general theory, including random $β$-transformations and random Lasota-Yorke maps.

math.DS↗

arXiv2020

Computation and optimal perturbation of finite-time coherent sets for aperiodic flows without trajectory integration

Understanding the macroscopic behavior of dynamical systems is an important tool to unravel transport mechanisms in complex flows. A decomposition of the state space into coherent sets is a popular way to reveal this essential macroscopic evolution. To compute coherent sets from an aperiodic time-dependent dynamical system we consider the relevant transfer operators and their infinitesimal generators on an augmented space-time manifold. This space-time generator approach avoids trajectory integration, and creates a convenient linearization of the aperiodic evolution. This linearization can be further exploited to create a simple and effective spectral optimization methodology for diminishing or enhancing coherence. We obtain explicit solutions for these optimization problems using Lagrange multipliers and illustrate this technique by increasing and decreasing mixing of spatial regions through small velocity field perturbations.

math.DS↗

arXiv2019

Higher-Order Finite Element Approximation of the Dynamic Laplacian

The dynamic Laplace operator arises from extending problems of isoperimetry from fixed manifolds to manifolds evolved by general nonlinear dynamics. Eigenfunctions of this operator are used to identify and track finite-time coherent sets, which physically manifest in fluid flows as jets, vortices, and more complicated structures. Two robust and efficient finite-element discretisation schemes for numerically computing the dynamic Laplacian were proposed in Froyland & Junge (2018). In this work we consider higher-order versions of these two numerical schemes and analyse them experimentally. We also prove the numerically computed eigenvalues and eigenvectors converge to the true objects for both schemes under certain assumptions. We provide an efficient implementation of the higher-order element schemes in an accompanying Julia package.

math.DS↗

arXiv2019

Stability and approximation of statistical limit laws for multidimensional piecewise expanding maps

The unpredictability of chaotic nonlinear dynamics leads naturally to statistical descriptions, including probabilistic limit laws such as the central limit theorem and large deviation principle. A key tool in the Nagaev-Guivarc'h spectral method for establishing statistical limit theorems is a "twisted" transfer operator. In the abstract setting of Keller-Liverani we prove that derivatives of all orders of the leading eigenvalues and eigenprojections of the twisted transfer operators with respect to the twist parameter are stable when subjected to a broad class of perturbations. As a result, we demonstrate stability of the variance in the central limit theorem and the rate function from a large deviation principle with respect to deterministic and stochastic perturbations of the dynamics and perturbations induced by numerical schemes. We apply these results to piecewise expanding maps in one and multiple dimensions, including new convergence results for Ulam projections on quasi-Hölder spaces.

math.DS↗

arXiv2019

Fourier approximation of the statistical properties of Anosov maps on tori

We study the stability of statistical properties of Anosov maps on tori by examining the stability of the spectrum of an analytically twisted Perron-Frobenius operator on the anisotropic Banach spaces of Gouëzel and Liverani. By extending our previous work in [8], we obtain the stability of various statistical properties (the variance of a CLT and the rate function of an LDP) of Anosov maps to general perturbations, including new classes of numerical approximations. In particular, we obtain new results on the stability of the rate function under deterministic perturbations. As a key application, we focus on perturbations arising from numerical schemes and develop two new Fourier-analytic method for efficiently computing approximations of the aforementioned statistical properties. This includes the first example of a rigorous scheme for approximating the peripheral spectral data of the Perron-Frobenius operator of an Anosov map without mollification. Using the two schemes we obtain the first rigorous estimates of the variance and rate function for Anosov maps.

math.DS↗

arXiv2018

Sparse eigenbasis approximation: multiple feature extraction across spatiotemporal scales with application to coherent set identification

The output of spectral clustering is a collection of eigenvalues and eigenvectors that encode important connectivity information about a graph or a manifold. This connectivity information is often not cleanly represented in the eigenvectors and must be disentangled by some secondary procedure. We propose the use of an approximate sparse basis for the space spanned by the leading eigenvectors as a natural, robust, and efficient means of performing this separation. The use of sparsity yields a natural cutoff in this disentanglement procedure and is particularly useful in practical situations when there is no clear eigengap. In order to select a suitable collection of vectors we develop a new Weyl-inspired eigengap heuristic and heuristics based on the sparse basis vectors. We develop an automated eigenvector separation procedure and illustrate its efficacy on examples from time-dependent dynamics on manifolds. In this context, transfer operator approaches are extensively used to find dynamically disconnected regions of phase space, known as almost-invariant sets or coherent sets. The dominant eigenvectors of transfer operators or related operators, such as the dynamic Laplacian, encode dynamic connectivity information. Our sparse eigenbasis approximation (SEBA) methodology streamlines the final stage of transfer operator methods, namely the extraction of almost-invariant or coherent sets from the eigenvectors. It is particularly useful when used on domains with large numbers of coherent sets, and when the coherent sets do not exhaust the phase space, such as in large geophysical datasets.

math.NA↗

arXiv2018

Robust FEM-based extraction of finite-time coherent sets using scattered, sparse, and incomplete trajectories

Transport and mixing properties of aperiodic flows are crucial to a dynamical analysis of the flow, and often have to be carried out with limited information. Finite-time coherent sets are regions of the flow that minimally mix with the remainder of the flow domain over the finite period of time considered. In the purely advective setting this is equivalent to identifying sets whose boundary interfaces remain small throughout their finite-time evolution. Finite-time coherent sets thus provide a skeleton of distinct regions around which more turbulent flow occurs. They manifest in geophysical systems in the forms of e.g.\ ocean eddies, ocean gyres, and atmospheric vortices. In real-world settings, often observational data is scattered and sparse, which makes the difficult problem of coherent set identification and tracking even more challenging. We develop three FEM-based numerical methods to efficiently approximate the dynamic Laplace operator, and introduce a new dynamic isoperimetric problem using Dirichlet boundary conditions. Using these FEM-based methods we rapidly and reliably extract finite-time coherent sets from models or scattered, possibly sparse, and possibly incomplete observed data.

math.DS↗