2,849 research outputs found
A Theoretical Framework for Lagrangian Descriptors
This paper provides a theoretical background for Lagrangian Descriptors
(LDs). The goal of achieving rigourous proofs that justify the ability of LDs
to detect invariant manifolds is simplified by introducing an alternative
definition for LDs. The definition is stated for -dimensional systems with
general time dependence, however we rigorously prove that this method reveals
the stable and unstable manifolds of hyperbolic points in four particular 2D
cases: a hyperbolic saddle point for linear autonomous systems, a hyperbolic
saddle point for nonlinear autonomous systems, a hyperbolic saddle point for
linear nonautonomous systems and a hyperbolic saddle point for nonlinear
nonautonomous systems. We also discuss further rigorous results which show the
ability of LDs to highlight additional invariants sets, such as -tori. These
results are just a simple extension of the ergodic partition theory which we
illustrate by applying this methodology to well-known examples, such as the
planar field of the harmonic oscillator and the 3D ABC flow. Finally, we
provide a thorough discussion on the requirement of the objectivity
(frame-invariance) property for tools designed to reveal phase space structures
and their implications for Lagrangian descriptors
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
Lagrangian Descriptors for Stochastic Differential Equations: A Tool for Revealing the Phase Portrait of Stochastic Dynamical Systems
In this paper we introduce a new technique for depicting the phase portrait
of stochastic differential equations. Following previous work for deterministic
systems, we represent the phase space by means of a generalization of the
method of Lagrangian descriptors to stochastic differential equations.
Analogously to the deterministic differential equations setting, the Lagrangian
descriptors graphically provide the distinguished trajectories and hyperbolic
structures arising within the stochastic dynamics, such as random fixed points
and their stable and unstable manifolds. We analyze the sense in which
structures form barriers to transport in stochastic systems. We apply the
method to several benchmark examples where the deterministic phase space
structures are well-understood. In particular, we apply our method to the noisy
saddle, the stochastically forced Duffing equation, and the stochastic double
gyre model that is a benchmark for analyzing fluid transport
Graded quantization for multiple description coding of compressive measurements
Compressed sensing (CS) is an emerging paradigm for acquisition of compressed
representations of a sparse signal. Its low complexity is appealing for
resource-constrained scenarios like sensor networks. However, such scenarios
are often coupled with unreliable communication channels and providing robust
transmission of the acquired data to a receiver is an issue. Multiple
description coding (MDC) effectively combats channel losses for systems without
feedback, thus raising the interest in developing MDC methods explicitly
designed for the CS framework, and exploiting its properties. We propose a
method called Graded Quantization (CS-GQ) that leverages the democratic
property of compressive measurements to effectively implement MDC, and we
provide methods to optimize its performance. A novel decoding algorithm based
on the alternating directions method of multipliers is derived to reconstruct
signals from a limited number of received descriptions. Simulations are
performed to assess the performance of CS-GQ against other methods in presence
of packet losses. The proposed method is successful at providing robust coding
of CS measurements and outperforms other schemes for the considered test
metrics
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
Finding NHIM: Identifying High Dimensional Phase Space Structures in Reaction Dynamics using Lagrangian Descriptors
Phase space structures such as dividing surfaces, normally hyperbolic
invariant manifolds, their stable and unstable manifolds have been an integral
part of computing quantitative results such as transition fraction, stability
erosion in multi-stable mechanical systems, and reaction rates in chemical
reaction dynamics. Thus, methods that can reveal their geometry in high
dimensional phase space (4 or more dimensions) need to be benchmarked by
comparing with known results. In this study, we assess the capability of one
such method called Lagrangian descriptor for revealing the types of high
dimensional phase space structures associated with index-1 saddle in
Hamiltonian systems. The Lagrangian descriptor based approach is applied to two
and three degree-of-freedom quadratic Hamiltonian systems where the high
dimensional phase space structures are known, that is as closed-form analytical
expressions. This leads to a direct comparison of features in the Lagrangian
descriptor plots and the phase space structures' intersection with an
isoenergetic two-dimensional surface and hence provides a validation of the
approach.Comment: 39 pages, 7 figures, Submitted to Communications in Nonlinear Science
and Numerical Simulatio
An Efficient Dual Approach to Distance Metric Learning
Distance metric learning is of fundamental interest in machine learning
because the distance metric employed can significantly affect the performance
of many learning methods. Quadratic Mahalanobis metric learning is a popular
approach to the problem, but typically requires solving a semidefinite
programming (SDP) problem, which is computationally expensive. Standard
interior-point SDP solvers typically have a complexity of (with
the dimension of input data), and can thus only practically solve problems
exhibiting less than a few thousand variables. Since the number of variables is
, this implies a limit upon the size of problem that can
practically be solved of around a few hundred dimensions. The complexity of the
popular quadratic Mahalanobis metric learning approach thus limits the size of
problem to which metric learning can be applied. Here we propose a
significantly more efficient approach to the metric learning problem based on
the Lagrange dual formulation of the problem. The proposed formulation is much
simpler to implement, and therefore allows much larger Mahalanobis metric
learning problems to be solved. The time complexity of the proposed method is
, which is significantly lower than that of the SDP approach.
Experiments on a variety of datasets demonstrate that the proposed method
achieves an accuracy comparable to the state-of-the-art, but is applicable to
significantly larger problems. We also show that the proposed method can be
applied to solve more general Frobenius-norm regularized SDP problems
approximately
- …