12,530 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
On Codazzi tensors on a hyperbolic surface and flat Lorentzian geometry
Using global considerations, Mess proved that the moduli space of globally
hyperbolic flat Lorentzian structures on is the tangent
bundle of the Teichm\"uller space of , if is a closed surface. One of
the goals of this paper is to deepen this surprising occurrence and to make
explicit the relation between the Mess parameters and the embedding data of any
Cauchy surface. This relation is pointed out by using some specific properties
of Codazzi tensors on hyperbolic surfaces. As a by-product we get a new
Lorentzian proof of Goldman's celebrated result about the coincidence of the
Weil-Petersson symplectic form and the Goldman pairing.
In the second part of the paper we use this machinery to get a classification
of globally hyperbolic flat space-times with particles of angles in
containing a uniformly convex Cauchy surface. The analogue of Mess' result is
achieved showing that the corresponding moduli space is the tangent bundle of
the Teichm\"uller space of a punctured surface. To generalize the theory in the
case of particles, we deepen the study of Codazzi tensors on hyperbolic
surfaces with cone singularities, proving that the well-known decomposition of
a Codazzi tensor in a harmonic part and a trivial part can be generalized in
the context of hyperbolic metrics with cone singularities.Comment: 49 pages, 4 figure
Characterizing the Delaunay decompositions of compact hyperbolic surfaces
Given a Delaunay decomposition of a compact hyperbolic surface, one may
record the topological data of the decomposition, together with the
intersection angles between the `empty disks' circumscribing the regions of the
decomposition. The main result of this paper is a characterization of when a
given topological decomposition and angle assignment can be realized as the
data of an actual Delaunay decomposition of a hyperbolic surface.Comment: Published by Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol6/paper12.abs.htm
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