arXiv subjects

Gary Froyland

Publications and source records attributed to Gary Froyland.

At least 19 recordsLinked to original sources

arXiv2026

Data-driven methods for computation of optimal linear response in high-dimensional dynamical systems

We develop a data-driven framework for estimating optimal linear response of nonlinear dynamical systems. Our approach is based on kernel-smoothed approximations of the transfer/Koopman operators of the system, built from possibly high-dimensional observations along trajectories. Combining these operator approximations with the theory developed in [Antown et al. (2018), J. Stat. Phys., 170(6), 1051-1087], we formulate a computationally tractable optimization problem for the optimal infinitesimal perturbation realising any desired manipulation of the spectrum. We also introduce a notion of optimal-response vector fields for visualising, and physically interpreting, the system's response to the optimal perturbation under arbitrary observations. Our focus is on finding perturbations that optimally increase the frequency or optimally suppress the decay of correlations of almost-cycles or almost-invariant sets associated with the eigenvalues of the kernel-smoothed transfer operator. We illustrate our approach with applications to low-dimensional periodic and chaotic systems, as well as a high-dimensional example involving the El Nino Southern Oscillation in a comprehensive Earth system model. In these examples our approach discovers nontrivial optimal perturbations of the system, which are post hoc natural and consistent with the desired dynamical objectives.

math.DS↗

arXiv2026

Fixed-point approximation for self-consistent transfer operators with Newton's method

Self consistent transfer operators arise naturally in the study of mean-field coupled dynamical systems and are closely related to kinetic PDEs such as the Vlasov equation. Despite substantial progress on existence and uniqueness of fixed points for self-consistent transfer operators, the development of fast, reliable, and provably accurate numerical methods remains largely unresolved. In this work, we construct a nonlinear Fourier-Fej\'er discretisation and establish convergence of the resulting finite-dimensional fixed point to that of the original self-consistent transfer operator. Further, using the nonlinear Fourier-Fej\'er discretisation, we prove exponential convergence of a sequential iteration scheme and develop a Newton framework with quadratic convergence. We present numerical examples demonstrating the efficiency and flexibility of the above methods.

math.DS↗

arXiv2025

Transient almost-invariant sets reveal convective heat transfer patterns in plane-layer Rayleigh-B\'enard convection

Horizontally extended plane-layer convection flows are characterized by characteristic patterns of turbulent heat transfer due to the convective fluid motion consisting of a nearly-regular ridge network where hot fluid rises and cold fluid sinks. Here, we analyse this transport behavior by the so-called inflated generator framework, which identifies quasi-stationary families of almost-invariant sets, derived from leading inflated generator eigenvectors. We demonstrate the effectiveness of this data-driven analysis framework in three-dimensional turbulent flow, by extracting transient characteristic heat transfer patterns as families of almost-invariant sets subject to a transient evolution, which contribute least to the convective heat transfer.

physics.flu-dyn↗

arXiv2025

On the cardinality of measures of maximal relative entropy for smooth skew products

Let $\Omega$ and $M$ be compact smooth manifolds and let $\Theta:\Omega\times M\to\Omega\times M$ be a $\mathcal C^{1+\alpha}$ skew-product diffeomorphism over a transitive Anosov base. We show that $\Theta$ has at most countably many ergodic hyperbolic measures of maximal relative entropy. When $\dim M=2$, if $\Theta$ has positive relative topological entropy, then $\Theta$ has at most countably many ergodic measures of maximal relative entropy.

math.DS↗

arXiv2025

On the structure of complex spectra and eigenfunctions of transfer and Koopman operators

Complex eigenspectra of transfer and Koopman operators describe rotational motion in dynamical systems. A particularly relevant situation in applications is when the rotation speed depends on the state-space position of the dynamics. We consider a canonical model of such dynamics in the presence of small noise, and provide precise characterisations of the eigenspectrum and eigenfunctions of the corresponding transfer operators. Further, we study the limiting behaviour of the eigenspectrum and eigenfunctions in the zero-noise limit, including their quadratic and linear response. Our results clarify the structure of transfer and Koopman operator eigenspectra, and provide new interpretations relevant to applications. Our theorems on support localisation of the eigenfunctions yield simple algorithms to detect the existence and state-space location of approximately cyclic motion with distinct periods. Our numerical results verify that information on the cycle periods and their locations determined by the operator eigendata is insensitive to noise level in the linear response regime. We believe that the dynamic mechanisms underlying the eigendata and their properties apply rather broadly and enhance our understanding of approximate cycle detection in dynamical systems with operator methods.

math.DS↗

arXiv2025

Learning dynamically inspired bases for Koopman and transfer operator approximation

Transfer and Koopman operator methods offer a framework for representing complex, nonlinear dynamical systems via linear transformations, enabling a deeper understanding of the underlying dynamics. The spectra of these operators provide important insights into system predictability and emergent behaviour, although efficiently estimating them from data can be challenging. We approach this issue through the lens of general operator and representational learning, in which we approximate these linear operators using efficient finite-dimensional representations. Specifically, we machine-learn orthonormal basis functions that are dynamically tailored to the system. This learned basis provides a particularly accurate approximation of the operator's action and enables efficient recovery of eigenfunctions and invariant measures. We illustrate our approach with examples that showcase the retrieval of spectral properties from the estimated operator, and emphasise the dynamically adaptive quality of the machine-learned basis.

math.DS↗

arXiv2025

Optimal linear response for Anosov diffeomorphisms

It is well known that an Anosov diffeomorphism $T$ enjoys linear response of its SRB measure with respect to infinitesimal perturbations $\dot{T}$. For a fixed observation function $c$, we develop a theory to optimise the response of the SRB-expectation of $c$. Our approach is based on the response of the transfer operator on the anisotropic Banach spaces of Gou\"ezel--Liverani. We prove that the optimising perturbation $\dot{T}$ is unique for non-degenerate response functions and provide explicit expressions for the Fourier coefficients of $\dot{T}$. We develop an efficient Fourier-based numerical scheme to approximate the optimal vector field $\dot{T}$, along with a proof of convergence. The utility of our approach is illustrated in two numerical examples, by localising SRB measures with small, optimally selected, perturbations.

math.DS↗

arXiv2024

Multi-set spectral clustering of time-evolving networks using the supra-Laplacian

Complex time-varying networks are prominent models for a wide variety of spatiotemporal phenomena. The functioning of networks depends crucially on their connectivity, yet reliable techniques for learning communities in time-evolving networks remain elusive. We adapt successful spectral techniques from continuous-time dynamics on manifolds to the graph setting to fill this gap. We consider the supra-Laplacian for graphs and develop a spectral theory to underpin the corresponding algorithmic realisations. We develop spectral clustering approaches for both multiplex and non-multiplex networks, based on the eigenvectors of the supra-Laplacian and specialised Sparse EigenBasis Approximation (SEBA) post-processing of these eigenvectors. We demonstrate that our approach can outperform the Leiden algorithm applied both in spacetime and layer-by-layer, and we analyse voting data from the US senate (where senators come and go as congresses evolve) to quantify increasing polarisation in time.

cs.SI↗

arXiv2024

A tutorial on the dynamic Laplacian

Spectral techniques are popular and robust approaches to data analysis. A prominent example is the use of eigenvectors of a Laplacian, constructed from data affinities, to identify natural data groupings or clusters, or to produce a simplified representation of data lying on a manifold. This tutorial concerns the dynamic Laplacian, which is a natural generalisation of the Laplacian to handle data that has a time component and lies on a time-evolving manifold. In this dynamic setting, clusters correspond to long-lived ``coherent'' collections. We begin with a gentle recap of spectral geometry before describing the dynamic generalisations. We also discuss computational methods and the automatic separation of many distinct features through the SEBA algorithm. The purpose of this tutorial is to bring together many results from the dynamic Laplacian literature into a single short document, written in an accessible style.

math.DS↗

arXiv2024

Identifying the onset and decay of quasi-stationary families of almost-invariant sets with an application to atmospheric blocking events

Macroscopic features of dynamical systems such as almost-invariant sets and coherent sets provide crucial high-level information on how the dynamics organises phase space. We introduce a method to identify time-parameterised families of almost-invariant sets in time-dependent dynamical systems, as well as the families' emergence and disappearance. In contrast to coherent sets, which may freely move about in phase space over time, our technique focuses on families of metastable sets that are quasi-stationary in space. Our straightforward approach extends successful transfer operator methods for almost-invariant sets to time-dependent dynamics and utilises the Ulam scheme for the generator of the transfer operator on a time-expanded domain. The new methodology is illustrated with an idealised fluid flow and with atmospheric velocity data. We identify atmospheric blocking events in the 2003 European heatwave and compare our technique to existing geophysical methods of blocking diagnosis.

math.DS↗

arXiv2024

An inflated dynamic Laplacian to track the emergence and disappearance of semi-material coherent sets

Lagrangian methods continue to stand at the forefront of the analysis of time-dependent dynamical systems. Most Lagrangian methods have criteria that must be fulfilled by trajectories as they are followed throughout a given finite flow duration. This key strength of Lagrangian methods can also be a limitation in more complex evolving environments. It places a high importance on selecting a time window that produces useful results, and these results may vary significantly with changes in the flow duration. We show how to overcome this drawback in the tracking of coherent flow features. Finite-time coherent sets (FTCS) are material objects that strongly resist mixing in complicated nonlinear flows. Like other materially coherent objects, by definition they must retain their coherence properties throughout the specified flow duration. Recent work [Froyland and Koltai, CPAM, 2023] introduced the notion of semi-material FTCS, whereby a balance is struck between the material nature and the coherence properties of FTCS. This balance provides the flexibility for FTCS to come and go, merge and separate, or undergo other changes as the governing unsteady flow experiences dramatic shifts. The purpose of this work is to illustrate the utility of the inflated dynamic Laplacian introduced in [Froyland and Koltai, CPAM, 2023] in a range of dynamical systems that are challenging to analyse by standard Lagrangian means, and to provide an efficient meshfree numerical approach for the discretisation of the inflated dynamic Laplacian.

physics.flu-dyn↗

arXiv2024

Compound Poisson statistics for dynamical systems via spectral perturbation

We consider random transformations $T_ω^n:=T_{σ^{n-1}ω}\circ\cdots\circ T_{σω}\circ T_ω,$ where each map $T_ω$ acts on a complete metrizable space $M$. The randomness comes from an invertible ergodic driving map $σ:Ω\toΩ$ acting on a probability space $(Ω,\mathcal{F},m).$ For a family of random target sets $H_{ω, n}\subset M$ that shrink as $n\to\infty$, we consider quenched compound Poisson statistics of returns of random orbits to these random targets. We develop a spectral approach to such statistics: associated with the random map cocycle is a transfer operator cocycle $\mathcal{L}^{n}_{ω,0}:=\mathcal{L}_{σ^{n-1}ω,0}\circ\cdots\circ\mathcal{L}_{σω,0}\circ\mathcal{L}_{ω,0}$, where $\mathcal{L}_{ω,0}$ is the transfer operator for the map $T_ω$. We construct a perturbed cocycle with generator $\mathcal{L}_{ω,n,s}(\cdot):=\mathcal{L}_{ω,0}(\cdot e^{is\mathbb{1}_{H_{ω,n}}})$ and an associated random variable $S_{ω,n,k}(x):=\sum_{j=0}^{k-1}\mathbb{1}_{H_{σ^jω,n}}(T_ω^jx)$, which counts the number of visits to random targets in an orbit of length $k$. Under suitable assumptions, we show that in the $n\to\infty$ limit, the random variables $S_{ω,n,n}$ converge in distribution to a compound Poisson distributed random variable. We provide several explicit examples for piecewise monotone interval maps in both the deterministic and random settings.

math.DS↗

arXiv2023

Universal Gap Growth for Lyapunov Exponents of Perturbed Matrix Products

We study the quantitative simplicity of the Lyapunov spectrum of $d$-dimensional bounded matrix cocycles subjected to additive random perturbations. In dimensions 2 and 3, we establish explicit lower bounds on the gaps between consecutive Lyapunov exponents of the perturbed cocycle, depending only on the scale of the perturbation. In arbitrary dimensions, we show existence of a universal lower bound on these gaps. A novelty of this work is that the bounds provided are uniform over all choices of the original sequence of matrices. Furthermore, we make no stationarity assumptions on this sequence. Hence, our results apply to random and sequential dynamical systems alike.

math.DS↗

arXiv2023

Optimal linear response for expanding circle maps

We consider the problem of optimal linear response for deterministic expanding maps of the circle. To each infinitesimal perturbation $\dot{T}$ of a circle map $T$ we consider (i) the response of the expectation of an observation function and (ii) the response of isolated spectral points of the transfer operator of $T$. In each case, under mild conditions on the set of feasible perturbations $\dot{T}$ we show there is a unique optimal feasible infinitesimal perturbation $\dot{T}_{\rm optimal}$, maximising the increase of the expectation of the given observation function or maximising the increase of the spectral gap of the transfer operator associated to the system. We derive expressions for the unique maximiser $\dot{T}_{\rm optimal}$ in terms of its Fourier coefficients. We also devise a Fourier-based computational scheme and apply it to illustrate our theory.

math.DS↗

arXiv2023

Detecting the birth and death of finite-time coherent sets

Finite-time coherent sets (FTCSs) are distinguished regions of phase space that resist mixing with the surrounding space for some finite period of time; physical manifestations include eddies and vortices in the ocean and atmosphere, respectively. The boundaries of finite-time coherent sets are examples of Lagrangian coherent structures (LCSs). The selection of the time duration over which FTCS and LCS computations are made in practice is crucial to their success. If this time is longer than the lifetime of coherence of individual objects then existing methods will fail to detect the shorter-lived coherence. It is of clear practical interest to determine the full lifetime of coherent objects, but in complicated practical situations, for example a field of ocean eddies with varying lifetimes, this is impossible with existing approaches. Moreover, determining the timing of emergence and destruction of coherent sets is of significant scientific interest. In this work we introduce new constructions to address these issues. The key components are an inflated dynamic Laplace operator and the concept of semi-material FTCSs. We make strong mathematical connections between the inflated dynamic Laplacian and the standard dynamic Laplacian [Froyland 2015], showing that the latter arises as a limit of the former. The spectrum and eigenfunctions of the inflated dynamic Laplacian directly provide information on the number, lifetimes, and evolution of coherent~sets.

math.DS↗

arXiv2023

A dynamic $p$-Laplacian

We generalise the dynamic Laplacian introduced in (Froyland, 2015) to a dynamic $p$-Laplacian, in analogy to the generalisation of the standard $2$-Laplacian to the standard $p$-Laplacian for $p>1$. Spectral properties of the dynamic Laplacian are connected to the geometric problem of finding "coherent" sets with persistently small boundaries under dynamical evolution, and we show that the dynamic $p$-Laplacian shares similar geometric connections. In particular, we prove that the first eigenvalue of the dynamic $p$-Laplacian with Dirichlet boundary conditions exists and converges to a dynamic version of the Cheeger constant introduced in (Froyland, 2015) as $p\rightarrow 1$. We develop a numerical scheme to estimate the leading eigenfunctions of the (nonlinear) dynamic $p$-Laplacian, and through a series of examples we investigate the behaviour of the level sets of these eigenfunctions. These level sets define the boundaries of sets in the domain of the dynamics that remain coherent under the dynamical evolution.

math.DS↗

arXiv2023

Higher Cheeger ratios of features in Laplace-Beltrami eigenfunctions

This paper investigates links between the eigenvalues and eigenfunctions of the Laplace-Beltrami operator, and the higher Cheeger constants of smooth Riemannian manifolds, possibly weighted and/or with boundary. The higher Cheeger constants give a loose description of the major geometric features of a manifold. We give a constructive upper bound on the higher Cheeger constants, in terms of the eigenvalue of any eigenfunction with the corresponding number of nodal domains. Specifically, we show that for each such eigenfunction, a positive-measure collection of its superlevel sets have their Cheeger ratios bounded above in terms of the corresponding eigenvalue. Some manifolds have their major features entwined across several eigenfunctions, and no single eigenfunction contains all the major features. In this case, there may exist carefully chosen linear combinations of the eigenfunctions, each with large values on a single feature, and small values elsewhere. We can then apply a soft-thresholding operator to these linear combinations to obtain new functions, each supported on a single feature. We show that the Cheeger ratios of the level sets of these functions also give an upper bound on the Laplace-Beltrami eigenvalues. We extend these level set results to nonautonomous dynamical systems, and show that the dynamic Laplacian eigenfunctions reveal sets with small dynamic Cheeger ratios.

math.DG↗

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↗