137 research outputs found
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
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
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
Coherent sets for nonautonomous dynamical systems
We describe a mathematical formalism and numerical algorithms for identifying
and tracking slowly mixing objects in nonautonomous dynamical systems. In the
autonomous setting, such objects are variously known as almost-invariant sets,
metastable sets, persistent patterns, or strange eigenmodes, and have proved to
be important in a variety of applications. In this current work, we explain how
to extend existing autonomous approaches to the nonautonomous setting. We call
the new time-dependent slowly mixing objects coherent sets as they represent
regions of phase space that disperse very slowly and remain coherent. The new
methods are illustrated via detailed examples in both discrete and continuous
time
A semi-invertible Oseledets Theorem with applications to transfer operator cocycles
Oseledets' celebrated Multiplicative Ergodic Theorem (MET) is concerned with
the exponential growth rates of vectors under the action of a linear cocycle on
R^d. When the linear actions are invertible, the MET guarantees an
almost-everywhere pointwise splitting of R^d into subspaces of distinct
exponential growth rates (called Lyapunov exponents). When the linear actions
are non-invertible, Oseledets' MET only yields the existence of a filtration of
subspaces, the elements of which contain all vectors that grow no faster than
exponential rates given by the Lyapunov exponents. The authors recently
demonstrated that a splitting over R^d is guaranteed even without the
invertibility assumption on the linear actions. Motivated by applications of
the MET to cocycles of (non-invertible) transfer operators arising from random
dynamical systems, we demonstrate the existence of an Oseledets splitting for
cocycles of quasi-compact non-invertible linear operators on Banach spaces.Comment: 26 page
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