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Gary Froyland

Publications and source records attributed to Gary Froyland.

At least 55 records · Page 3Linked to original sources
arXiv2018

Optimal mixing enhancement

We introduce a general-purpose method for optimising the mixing rate of advective fluid flows. An existing velocity field is perturbed in a $C^1$ neighborhood to maximize the mixing rate for flows generated by velocity fields in this neighborhood. Our numerical approach is based on the infinitesimal generator of the flow and is solved by standard linear programming methods. The perturbed flow may be easily constrained to preserve the same steady state distribution as the original flow, and various natural geometric constraints can also be simply applied. The same technique can also be used to optimize the mixing rate of advection-diffusion flow models by manipulating the drift term in a small neighborhood.

physics.flu-dyn↗
arXiv2018

Optimal linear responses for Markov chains and stochastically perturbed dynamical systems

The linear response of a dynamical system refers to changes to properties of the system when small external perturbations are applied. We consider the little-studied question of selecting an optimal perturbation so as to (i) maximise the linear response of the equilibrium distribution of the system, (ii) maximise the linear response of the expectation of a specified observable, and (iii) maximise the linear response of the rate of convergence of the system to the equilibrium distribution. We also consider the inhomogeneous or time-dependent situation where the governing dynamics is not stationary and one wishes to select a sequence of small perturbations so as to maximise the overall linear response at some terminal time. We develop the theory for finite-state Markov chains, provide explicit solutions for some illustrative examples, and numerically apply our theory to stochastically perturbed dynamical systems, where the Markov chain is replaced by a matrix representation of an approximate annealed transfer operator for the random dynamical system.

math.DS↗
arXiv2017

A spectral approach for quenched limit theorems for random expanding dynamical systems

We prove quenched versions of (i) a large deviations principle (LDP), (ii) a central limit theorem (CLT), and (iii) a local central limit theorem (LCLT) for non-autonomous dynamical systems. A key advance is the extension of the spectral method, commonly used in limit laws for deterministic maps, to the general random setting. We achieve this via multiplicative ergodic theory and the development of a general framework to control the regularity of Lyapunov exponents of \emph{twisted transfer operator cocycles} with respect to a twist parameter. While some versions of the LDP and CLT have previously been proved with other techniques, the local central limit theorem is, to our knowledge, a completely new result, and one that demonstrates the strength of our method. Applications include non-autonomous (piecewise) expanding maps, defined by random compositions of the form $T_{σ^{n-1}ω}\circ\cdots\circ T_{σω}\circ T_ω$. An important aspect of our results is that we only assume ergodicity and invertibility of the random driving $σ:Ω\toΩ$; in particular no expansivity or mixing properties are

math.DS↗
arXiv2017

A dynamic Laplacian for identifying Lagrangian coherent structures on weighted Riemannian manifolds

Transport and mixing in dynamical systems are important properties for many physical, chemical, biological, and engineering processes. The detection of transport barriers for dynamics with general time dependence is a difficult, but important problem, because such barriers control how rapidly different parts of phase space (which might correspond to different chemical or biological agents) interact. The key factor is the growth of interfaces that partition phase space into separate regions. The paper [Froyland, Nonlinearity 2015] introduced the notion of \textit{dynamic isoperimetry}: the study of sets with persistently small boundary size (the interface) relative to enclosed volume, when evolved by the dynamics. Sets with this minimal boundary size to volume ratio were identified as level sets of dominant eigenfunctions of a \textit{dynamic Laplace operator}. In this present work we extend the results of [Froyland, Nonlinearity 2015] to the situation where the dynamics (i) is not necessarily volume-preserving, (ii) acts on initial agent concentrations different from uniform concentrations, and (iii) occurs on a possibly curved phase space. Our main results include generalised versions of the dynamic isoperimetric problem, the dynamic Laplacian, Cheeger's inequality, and the Federer-Fleming theorem. We illustrate the computational approach with some simple numerical examples.

math.DS↗
arXiv2017

A Critical Comparison of Lagrangian Methods for Coherent Structure Detection

We review and test twelve different approaches to the detection of finite-time coherent material structures in two-dimensional, temporally aperiodic flows. We consider both mathematical methods and diagnostic scalar fields, comparing their performance on three benchmark examples: the quasiperiodically forced Bickley jet, a two-dimensional turbulence simulation, and an observational wind velocity field from Jupiter's atmosphere. A close inspection of the results reveals that the various methods often produce very different predictions for coherent structures, once they are evaluated beyond heuristic visual assessment. As we find by passive advection of the coherent set candidates, false positives and negatives can be produced even by some of the mathematically justified methods due to the ineffectiveness of their underlying coherence principles in certain flow configurations. We summarize the inferred strengths and weaknesses of each method, and make general recommendations for minimal self-consistency requirements that any Lagrangian coherence detection technique should satisfy.

physics.flu-dyn↗
arXiv2017

Hilbert Space Lyapunov Exponent stability

We study cocycles of compact operators acting on a separable Hilbert space, and investigate the stability of the Lyapunov exponents and Oseledets spaces when the operators are subjected to additive Gaussian noise. We show that as the noise is shrunk to 0, the Lyapunov exponents of the perturbed cocycle converge to those of the unperturbed cocycle; and the Oseledets spaces converge in probability to those of the unperturbed cocycle. This is, to our knowledge, the first result of this type with cocycles taking values in operators on infinite-dimensional spaces. The infinite dimensionality gives rise to a number of substantial difficulties that are not present in the finite-dimensional case.

math.DS↗
arXiv2017

Estimating long-term behavior of periodically driven flows without trajectory integration

Periodically driven flows are fundamental models of chaotic behavior and the study of their transport properties is an active area of research. A well-known analytic construction is the augmentation of phase space with an additional time dimension; in this augmented space, the flow becomes autonomous or time-independent. We prove several results concerning the connections between the original time-periodic representation and the time-extended representation, focusing on transport properties. In the deterministic setting, these include single-period outflows and time-asymptotic escape rates from time-parameterized families of sets. We also consider stochastic differential equations with time-periodic advection term. In this stochastic setting one has a time-periodic generator (the differential operator given by the right-hand-side of the corresponding time-periodic Fokker-Planck equation). We define in a natural way an autonomous generator corresponding to the flow on time-extended phase space. We prove relationships between these two generator representations and use these to quantify decay rates of observables and to determine time-periodic families of sets with slow escape rate. Finally, we use the generator on the time-extended phase space to create efficient numerical schemes to implement the various theoretical constructions. These ideas build on the work of Froyland et al (SINUM, 2013), and no expensive time integration is required. We introduce an efficient new hybrid approach, which treats the space and time dimensions separately.

math.DS↗
arXiv2016

Hölder continuity of Oseledets splittings for semi-invertible operator cocycles

For Hölder continuous cocycles over an invertible, Lipschitz base, we establish the Hölder continuity of Oseledets subspaces on compact sets of arbitrarily large measure. This extends a result of Araújo, Bufetov, and Filip by considering possibly noninvertible cocycles, which in addition may take values in the space of compact operators on a Hilbert space. As a by-product of our work, we also show that a noninvertible cocycle with nonvanishing Lyapunov exponents exhibits nonuniformly hyperbolic behaviour (in the sense of Pesin) on a set of full measure.

math.DS↗
arXiv2015

On fast computation of finite-time coherent sets using radial basis functions

Finite-time coherent sets inhibit mixing over finite times. The most expensive part of the transfer operator approach to detecting coherent sets is the construction of the operator itself. We present a numerical method based on radial basis function collocation and apply it to a recent transfer operator construction that has been designed specifically for purely advective dynamics. The construction is based on a "dynamic" Laplacian operator and minimises the boundary size of the coherent sets relative to their volume. The main advantage of our new approach is a substantial reduction in the number of Lagrangian trajectories that need to be computed, leading to large speedups in the transfer operator analysis when this computation is costly.

math.NA↗
arXiv2015

Dynamic isoperimetry and the geometry of Lagrangian coherent structures

The study of transport and mixing processes in dynamical systems is particularly important for the analysis of mathematical models of physical systems. We propose a novel, direct geometric method to identify subsets of phase space that remain strongly coherent over a finite time duration. This new method is based on a dynamic extension of classical (static) isoperimetric problems; the latter are concerned with identifying submanifolds with the smallest boundary size relative to their volume. The present work introduces \emph{dynamic} isoperimetric problems; the study of sets with small boundary size relative to volume \emph{as they are evolved by a general dynamical system}. We formulate and prove dynamic versions of the fundamental (static) isoperimetric (in)equalities; a dynamic Federer-Fleming theorem and a dynamic Cheeger inequality. We introduce a new dynamic Laplacian operator and describe a computational method to identify coherent sets based on eigenfunctions of the dynamic Laplacian. Our results include formal mathematical statements concerning geometric properties of finite-time coherent sets, whose boundaries can be regarded as Lagrangian coherent structures. The computational advantages of our new approach are a well-separated spectrum for the dynamic Laplacian, and flexibility in appropriate numerical approximation methods. Finally, we demonstrate that the dynamic Laplacian operator can be realised as a zero-diffusion limit of a newly advanced probabilistic transfer operator method (Froyland, 2013) for finding coherent sets, which is based on small diffusion. Thus, the present approach sits naturally alongside the probabilistic approach (Froyland, 2013), and adds a formal geometric interpretation.

math.DS↗
arXiv2015

A rough-and-ready cluster-based approach for extracting finite-time coherent sets from sparse and incomplete trajectory data

We present a numerical method to identify regions of phase space that are approximately retained in a mobile compact neighbourhood over a finite time duration. Our approach is based on spatio-temporal clustering of trajectory data. The main advantages of the approach are the ability to produce useful results (i) when there are relatively few trajectories and (ii) when there are gaps in observation of the trajectories as can occur with real data. The method is easy to implement, works in any dimension, and is fast to run.

math.DS↗
arXiv2014

A trajectory-free framework for analysing multiscale systems

We develop algorithms built around properties of the transfer operator and Koopman operator which 1) test for possible multiscale dynamics in a given dynamical system, 2) estimate the magnitude of the time-scale separation, and finally 3) distill the reduced slow dynamics on a suitably designed subspace. By avoiding trajectory integration, the developed techniques are highly computationally efficient. We corroborate our findings with numerical simulations of a test problem.

math.DS↗
arXiv2014

A computational method to extract macroscopic variables and their dynamics in multiscale systems

This paper introduces coordinate-independent methods for analysing multiscale dynamical systems using numerical techniques based on the transfer operator and its adjoint. In particular, we present a method for testing whether an arbitrary dynamical system exhibits multiscale behaviour and for estimating the time-scale separation. For systems with such behaviour, we establish techniques for analysing the fast dynamics in isolation, extracting slow variables for the system, and accurately simulating these slow variables at a large time step. We illustrate our method with numerical examples and show how the reduced slow dynamics faithfully represents statistical features of the full dynamics which are not coordinate dependent.

math.DS↗
arXiv2013

Ulam's method for Lasota-Yorke maps with holes

Ulam's method is a rigorous numerical scheme for approximating invariant densities of dynamical systems. The phase space is partitioned into connected sets and an inter-set transition matrix is computed from the dynamics; an approximate invariant density is read off as the leading left eigenvector of this matrix. When a hole in phase space is introduced, one instead searches for \emph{conditional} invariant densities and their associated escape rates. For Lasota-Yorke maps with holes we prove that a simple adaptation of the standard Ulam scheme provides convergent sequences of escape rates (from the leading eigenvalue), conditional invariant densities (from the corresponding left eigenvector), and quasi-conformal measures (from the corresponding right eigenvector). We also immediately obtain a convergent sequence for the invariant measure supported on the survivor set. Our approach allows us to consider relatively large holes. We illustrate the approach with several families of examples, including a class of Lorenz maps.

math.DS↗
arXiv2012

Stability and approximation of random invariant densities for Lasota-Yorke map cocycles

We establish stability of random absolutely continuous invariant measures (acims) for cocycles of random Lasota-Yorke maps under a variety of perturbations. Our family of random maps need not be close to a fixed map; thus, our results can handle very general driving mechanisms. We consider (i) perturbations via convolutions, (ii) perturbations arising from finite-rank transfer operator approximation schemes and (iii) static perturbations, perturbing to a nearby cocycle of Lasota-Yorke maps. The former two results provide a rigorous framework for the numerical approximation of random acims using a Fourier-based approach and Ulam's method, respectively; we also demonstrate the efficacy of these schemes.

math.DS↗
arXiv2012

An analytic framework for identifying finite-time coherent sets in time-dependent dynamical systems

The study of transport and mixing processes in dynamical systems is particularly important for the analysis of mathematical models of physical systems. Barriers to transport, which mitigate mixing, are currently the subject of intense study. In the autonomous setting, the use of transfer operators (Perron-Frobenius operators) to identify invariant and almost-invariant sets has been particularly successful. In the nonautonomous (time-dependent) setting, \emph{coherent sets}, a time-parameterised family of minimally dispersive sets, are a natural extension of almost-invariant sets. The present work introduces a new analytic transfer operator construction that enables the calculation of \emph{finite-time} coherent sets (sets are that minimally dispersive over a finite time interval). This new construction also elucidates the role of diffusion in the calculation and we show how properties such as the spectral gap and the regularity of singular vectors scale with noise amplitude. The construction can also be applied to general Markov processes on continuous state space.

math.DS↗
arXiv2012

Metastability, Lyapunov exponents, escape rates, and topological entropy in random dynamical systems

We explore the concept of metastability in random dynamical systems, focussing on connections between random Perron-Frobenius operator cocycles and escape rates of random maps, and on topological entropy of random shifts of finite type. The Lyapunov spectrum of the random Perron-Frobenius cocycle and the random adjacency matrix cocycle is used to decompose the random system into two disjoint random systems with rigorous upper and lower bounds on (i) the escape rate in the setting of random maps, and (ii) topological entropy in the setting of random shifts of finite type, respectively.

math.DS↗