49 research outputs found
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
Dominant transport pathways in an atmospheric blocking event
A Lagrangian flow network is constructed for the atmospheric blocking of
eastern Europe and western Russia in summer 2010. We compute the most probable
paths followed by fluid particles which reveal the {\it Omega}-block skeleton
of the event. A hierarchy of sets of highly probable paths is introduced to
describe transport pathways when the most probable path alone is not
representative enough. These sets of paths have the shape of narrow coherent
tubes flowing close to the most probable one. Thus, even when the most probable
path is not very significant in terms of its probability, it still identifies
the geometry of the transport pathways.Comment: Appendix added with path calculations for a simple kinematic model
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Hydrodynamic provinces and oceanic connectivity from a transport network help designing marine reserves
Oceanic dispersal and connectivity have been identified as crucial factors
for structuring marine populations and designing Marine Protected Areas (MPAs).
Focusing on larval dispersal by ocean currents, we propose an approach coupling
Lagrangian transport and new tools from Network Theory to characterize marine
connectivity in the Mediterranean basin. Larvae of different pelagic durations
and seasons are modeled as passive tracers advected in a simulated oceanic
surface flow from which a network of connected areas is constructed.
Hydrodynamical provinces extracted from this network are delimited by frontiers
which match multi-scale oceanographic features. By examining the repeated
occurrence of such boundaries, we identify the spatial scales and geographic
structures that would control larval dispersal across the entire seascape.
Based on these hydrodynamical units, we study novel connectivity metrics for
existing reserves. Our results are discussed in the context of ocean
biogeography and MPAs design, having ecological and managerial implications
Stochastic Stability Analysis of Discrete Time System Using Lyapunov Measure
In this paper, we study the stability problem of a stochastic, nonlinear,
discrete-time system. We introduce a linear transfer operator-based Lyapunov
measure as a new tool for stability verification of stochastic systems. Weaker
set-theoretic notion of almost everywhere stochastic stability is introduced
and verified, using Lyapunov measure-based stochastic stability theorems.
Furthermore, connection between Lyapunov functions, a popular tool for
stochastic stability verification, and Lyapunov measures is established. Using
the duality property between the linear transfer Perron-Frobenius and Koopman
operators, we show the Lyapunov measure and Lyapunov function used for the
verification of stochastic stability are dual to each other. Set-oriented
numerical methods are proposed for the finite dimensional approximation of the
Perron-Frobenius operator; hence, Lyapunov measure is proposed. Stability
results in finite dimensional approximation space are also presented. Finite
dimensional approximation is shown to introduce further weaker notion of
stability referred to as coarse stochastic stability. The results in this paper
extend our earlier work on the use of Lyapunov measures for almost everywhere
stability verification of deterministic dynamical systems ("Lyapunov Measure
for Almost Everywhere Stability", {\it IEEE Trans. on Automatic Control}, Vol.
53, No. 1, Feb. 2008).Comment: Proceedings of American Control Conference, Chicago IL, 201
Simultaneous Coherent Structure Coloring facilitates interpretable clustering of scientific data by amplifying dissimilarity
The clustering of data into physically meaningful subsets often requires
assumptions regarding the number, size, or shape of the subgroups. Here, we
present a new method, simultaneous coherent structure coloring (sCSC), which
accomplishes the task of unsupervised clustering without a priori guidance
regarding the underlying structure of the data. sCSC performs a sequence of
binary splittings on the dataset such that the most dissimilar data points are
required to be in separate clusters. To achieve this, we obtain a set of
orthogonal coordinates along which dissimilarity in the dataset is maximized
from a generalized eigenvalue problem based on the pairwise dissimilarity
between the data points to be clustered. This sequence of bifurcations produces
a binary tree representation of the system, from which the number of clusters
in the data and their interrelationships naturally emerge. To illustrate the
effectiveness of the method in the absence of a priori assumptions, we apply it
to three exemplary problems in fluid dynamics. Then, we illustrate its capacity
for interpretability using a high-dimensional protein folding simulation
dataset. While we restrict our examples to dynamical physical systems in this
work, we anticipate straightforward translation to other fields where existing
analysis tools require ad hoc assumptions on the data structure, lack the
interpretability of the present method, or in which the underlying processes
are less accessible, such as genomics and neuroscience