58 research outputs found
Lagrangian Descriptors: A Method for Revealing Phase Space Structures of General Time Dependent Dynamical Systems
In this paper we develop new techniques for revealing geometrical structures
in phase space that are valid for aperiodically time dependent dynamical
systems, which we refer to as Lagrangian descriptors. These quantities are
based on the integration, for a finite time, along trajectories of an intrinsic
bounded, positive geometrical and/or physical property of the trajectory
itself. We discuss a general methodology for constructing Lagrangian
descriptors, and we discuss a "heuristic argument" that explains why this
method is successful for revealing geometrical structures in the phase space of
a dynamical system. We support this argument by explicit calculations on a
benchmark problem having a hyperbolic fixed point with stable and unstable
manifolds that are known analytically. Several other benchmark examples are
considered that allow us the assess the performance of Lagrangian descriptors
in revealing invariant tori and regions of shear. Throughout the paper
"side-by-side" comparisons of the performance of Lagrangian descriptors with
both finite time Lyapunov exponents (FTLEs) and finite time averages of certain
components of the vector field ("time averages") are carried out and discussed.
In all cases Lagrangian descriptors are shown to be both more accurate and
computationally efficient than these methods. We also perform computations for
an explicitly three dimensional, aperiodically time-dependent vector field and
an aperiodically time dependent vector field defined as a data set. Comparisons
with FTLEs and time averages for these examples are also carried out, with
similar conclusions as for the benchmark examples.Comment: 52 pages, 25 figure
Finite-time Lagrangian transport analysis: Stable and unstable manifolds of hyperbolic trajectories and finite-time Lyapunov exponents
We consider issues associated with the Lagrangian characterisation of flow
structures arising in aperiodically time-dependent vector fields that are only
known on a finite time interval. A major motivation for the consideration of
this problem arises from the desire to study transport and mixing problems in
geophysical flows where the flow is obtained from a numerical solution, on a
finite space-time grid, of an appropriate partial differential equation model
for the velocity field. Of particular interest is the characterisation,
location, and evolution of "transport barriers" in the flow, i.e. material
curves and surfaces. We argue that a general theory of Lagrangian transport has
to account for the effects of transient flow phenomena which are not captured
by the infinite-time notions of hyperbolicity even for flows defined for all
time. Notions of finite-time hyperbolic trajectories, their finite time stable
and unstable manifolds, as well as finite-time Lyapunov exponent (FTLE) fields
and associated Lagrangian coherent structures have been the main tools for
characterizing transport barriers in the time-aperiodic situation. In this
paper we consider a variety of examples, some with explicit solutions, that
illustrate, in a concrete manner, the issues and phenomena that arise in the
setting of finite-time dynamical systems. Of particular significance for
geophysical applications is the notion of "flow transition" which occurs when
finite-time hyperbolicity is lost, or gained. The phenomena discovered and
analysed in our examples point the way to a variety of directions for rigorous
mathematical research in this rapidly developing, and important, new area of
dynamical systems theory
The turnstile mechanism across the Kuroshio current: analysis of dynamics in altimeter velocity fields
In this article we explore the ability of dynamical systems tools to describe
transport in oceanic flows characterized by data sets measured from satellite.
In particular we have studied the geometrical skeleton describing transport in
the Kuroshio region. For this purpose we have computed special hyperbolic
trajectories, recognized as distinguished hyperbolic trajectories, that act as
organizing centres of the flow. We have computed their stable and unstable
manifolds, and they reveal that the turnstile mechanism is at work during
several spring months in the year 2003 across the Kuroshio current. We have
found that near the hyperbolic trajectories takes place a filamentous transport
front-cross the current that mixes waters at both sides.Comment: Nonlinear Processes in Geophysics 17, 2010 (in press
Lagrangian transport through an ocean front in the North-Western Mediterranean Sea
We analyze with the tools of lobe dynamics the velocity field from a
numerical simulation of the surface circulation in the Northwestern
Mediterranean Sea. We identify relevant hyperbolic trajectories and their
manifolds, and show that the transport mechanism known as the `turnstile',
previously identified in abstract dynamical systems and simplified model flows,
is also at work in this complex and rather realistic ocean flow. In addition
nonlinear dynamics techniques are shown to be powerful enough to identify the
key geometric structures in this part of the Mediterranean. In particular the
North Balearic Front, the westernmost part of the transition zone between
saltier and fresher waters in the Western Mediterranean is interpreted in terms
of the presence of a semipermanent ``Lagrangian barrier'' across which little
transport occurs. Our construction also reveals the routes along which this
transport happens. Topological changes in that picture, associated with the
crossing by eddies and that may be interpreted as the breakdown of the front,
are also observed during the simulation.Comment: 34 pages, 6 (multiple) figures. Version with higher quality figures
available from
http://www.imedea.uib.es/physdept/publications/showpaper_en.php?indice=1764 .
Problems with paper size fixe
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
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
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
Nonautonomous dynamical systems: from theory to applications
Tesis doctoral inédita leÃda en la Universidad Autónoma de Madrid, Facultad de Ciencias, Departamento de Matemáticas. Fecha de lectura: 27-04-201
Distinguished trajectories in time dependent vector fields
We introduce a new definition of distinguished trajectory that generalises
the concepts of fixed point and periodic orbit to aperiodic dynamical systems.
This new definition is valid for identifying distinguished trajectories with
hyperbolic and non-hyperbolic types of stability. The definition is implemented
numerically and the procedure consist in determining a path of limit
coordinates. It has been successfully applied to known examples of
distinguished trajectories. In the context of highly aperiodic realistic flows
our definition characterises distinguished trajectories in finite time
intervals, and states that outside these intervals trajectories are no longer
distinguished.Comment: Chaos 19 (2009), 013111-1-013111-1
- …