Transport in Transitory Dynamical Systems

We introduce the concept of a "transitory" dynamical system---one whose time-dependence is confined to a compact interval---and show how to quantify transport between two-dimensional Lagrangian coherent structures for the Hamiltonian case. This requires knowing only the "action" of relevant heteroclinic orbits at the intersection of invariant manifolds of "forward" and "backward" hyperbolic orbits. These manifolds can be easily computed by leveraging the autonomous nature of the vector fields on either side of the time-dependent transition. As illustrative examples we consider a two-dimensional fluid flow in a rotating double-gyre configuration and a simple one-and-a-half degree of freedom model of a resonant particle accelerator. We compare our results to those obtained using finite-time Lyapunov exponents and to adiabatic theory, discussing the benefits and limitations of each method.Comment: Updated and corrected version. LaTeX, 29 pages, 21 figure

Transport in Transitory, Three-Dimensional, Liouville Flows

We derive an action-flux formula to compute the volumes of lobes quantifying transport between past- and future-invariant Lagrangian coherent structures of n-dimensional, transitory, globally Liouville flows. A transitory system is one that is nonautonomous only on a compact time interval. This method requires relatively little Lagrangian information about the codimension-one surfaces bounding the lobes, relying only on the generalized actions of loops on the lobe boundaries. These are easily computed since the vector fields are autonomous before and after the time-dependent transition. Two examples in three-dimensions are studied: a transitory ABC flow and a model of a microdroplet moving through a microfluidic channel mixer. In both cases the action-flux computations of transport are compared to those obtained using Monte Carlo methods.Comment: 30 pages, 16 figures, 1 table, submitted to SIAM J. Appl. Dyn. Sy

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 non-dispersive over finite periods of time, despite the chaotic nature of the system. We develop a novel probabilistic methodology based upon transfer operators that automatically detects 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

Computational Method for Phase Space Transport with Applications to Lobe Dynamics and Rate of Escape

Lobe dynamics and escape from a potential well are general frameworks introduced to study phase space transport in chaotic dynamical systems. While the former approach studies how regions of phase space are transported by reducing the flow to a two-dimensional map, the latter approach studies the phase space structures that lead to critical events by crossing periodic orbit around saddles. Both of these frameworks require computation with curves represented by millions of points-computing intersection points between these curves and area bounded by the segments of these curves-for quantifying the transport and escape rate. We present a theory for computing these intersection points and the area bounded between the segments of these curves based on a classification of the intersection points using equivalence class. We also present an alternate theory for curves with nontransverse intersections and a method to increase the density of points on the curves for locating the intersection points accurately.The numerical implementation of the theory presented herein is available as an open source software called Lober. We used this package to demonstrate the application of the theory to lobe dynamics that arises in fluid mechanics, and rate of escape from a potential well that arises in ship dynamics.Comment: 33 pages, 17 figure

Local stable and unstable manifolds and their control in nonautonomous finite-time flows

It is well-known that stable and unstable manifolds strongly influence fluid motion in unsteady flows. These emanate from hyperbolic trajectories, with the structures moving nonautonomously in time. The local directions of emanation at each instance in time is the focus of this article. Within a nearly autonomous setting, it is shown that these time-varying directions can be characterised through the accumulated effect of velocity shear. Connections to Oseledets spaces and projection operators in exponential dichotomies are established. Availability of data for both infinite and finite time-intervals is considered. With microfluidic flow control in mind, a methodology for manipulating these directions in any prescribed time-varying fashion by applying a local velocity shear is developed. The results are verified for both smoothly and discontinuously time-varying directions using finite-time Lyapunov exponent fields, and excellent agreement is obtained.Comment: Under consideration for publication in the Journal of Nonlinear Science

The Lagrangian description of aperiodic flows: a case study of the Kuroshio Current

This article reviews several recently developed Lagrangian tools and shows how their combined use succeeds in obtaining a detailed description of purely advective transport events in general aperiodic flows. In particular, because of the climate impact of ocean transport processes, we illustrate a 2D application on altimeter data sets over the area of the Kuroshio Current, although the proposed techniques are general and applicable to arbitrary time dependent aperiodic flows. The first challenge for describing transport in aperiodical time dependent flows is obtaining a representation of the phase portrait where the most relevant dynamical features may be identified. This representation is accomplished by using global Lagrangian descriptors that when applied for instance to the altimeter data sets retrieve over the ocean surface a phase portrait where the geometry of interconnected dynamical systems is visible. The phase portrait picture is essential because it evinces which transport routes are acting on the whole flow. Once these routes are roughly recognised it is possible to complete a detailed description by the direct computation of the finite time stable and unstable manifolds of special hyperbolic trajectories that act as organising centres of the flow.Comment: 40 pages, 24 figure

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

From large deviations to semidistances of transport and mixing: coherence analysis for finite Lagrangian data

One way to analyze complicated non-autonomous flows is through trying to understand their transport behavior. In a quantitative, set-oriented approach to transport and mixing, finite time coherent sets play an important role. These are time-parametrized families of sets with unlikely transport to and from their surroundings under small or vanishing random perturbations of the dynamics. Here we propose, as a measure of transport and mixing for purely advective (i.e., deterministic) flows, (semi)distances that arise under vanishing perturbations in the sense of large deviations. Analogously, for given finite Lagrangian trajectory data we derive a discrete-time and space semidistance that comes from the "best" approximation of the randomly perturbed process conditioned on this limited information of the deterministic flow. It can be computed as shortest path in a graph with time-dependent weights. Furthermore, we argue that coherent sets are regions of maximal farness in terms of transport and mixing, hence they occur as extremal regions on a spanning structure of the state space under this semidistance---in fact, under any distance measure arising from the physical notion of transport. Based on this notion we develop a tool to analyze the state space (or the finite trajectory data at hand) and identify coherent regions. We validate our approach on idealized prototypical examples and well-studied standard cases.Comment: J Nonlinear Sci, 201