5,536 research outputs found
Variational method for locating invariant tori
We formulate a variational fictitious-time flow which drives an initial guess
torus to a torus invariant under given dynamics. The method is general and
applies in principle to continuous time flows and discrete time maps in
arbitrary dimension, and to both Hamiltonian and dissipative systems.Comment: 10 page
The invariant tori of knot type and the interlinked invariant tori in the Nos\'e-Hoover system
We revisit the famous Nos\'e-Hoover system in this paper and show the
existence of some averagely conservative regions which are filled with an
infinite sequence of nested tori. Depending on initial conditions, some
invariant tori are of trefoil knot type, while the others are of trivial knot
type. Moreover, we present a variety of interlinked invariant tori whose
initial conditions are chosen from different averagely conservative regions and
give all the interlinking numbers of those interlinked tori, showing that this
quadratic system possesses so rich dynamic properties.Comment: 8 pages,8 figure
Uncontrolled spacecraft formations on two-dimensional invariant tori
Within the class of natural motions near libration point regions quasi-periodic trajectories evolving on invariant tori are studied. Those orbits prove beneficial for relative spacecraft configurations with large distances among satellites. In this study properties of invariant tori are outlined, and non-resonant and resonant tori around the Sun/Earth libration point L1 are computed. A numerical approach to obtain the frequency base and to parametrize a torus in angular phase space is introduced. Initial states for spacecraft formations on the torusâ surface are defined. The formation naturally evolve along its surface such that the relative positions within a formation are unaltered and the relative distances and the orientation are closely bounded. An in-plane coordinate frame together with a modified torus motion is introduced and the inner and outer behaviour of the formationâs geometry is investigated
Fast iteration of cocyles over rotations and Computation of hyperbolic bundles
In this paper, we develop numerical algorithms that use small requirements of
storage and operations for the computation of hyperbolic cocycles over a
rotation. We present fast algorithms for the iteration of the quasi-periodic
cocycles and the computation of the invariant bundles, which is a preliminary
step for the computation of invariant whiskered tori
Hydrogen atom in crossed electric and magnetic fields: Phase space topology and torus quantization via periodic orbits
A hierarchical ordering is demonstrated for the periodic orbits in a strongly
coupled multidimensional Hamiltonian system, namely the hydrogen atom in
crossed electric and magnetic fields. It mirrors the hierarchy of broken
resonant tori and thereby allows one to characterize the periodic orbits by a
set of winding numbers. With this knowledge, we construct the action variables
as functions of the frequency ratios and carry out a semiclassical torus
quantization. The semiclassical energy levels thus obtained agree well with
exact quantum calculations
Are ghost surfaces quadratic-flux-minimizing?
Two candidates for "almost-invariant" toroidal surfaces passing through
magnetic islands, namely quadratic-flux-minimizing (QFMin) surfaces and ghost
surfaces, use families of periodic pseudo-orbits (i.e. paths for which the
action is not exactly extremal). QFMin pseudo-orbits, which are
coordinate-dependent, are field lines obtained from a modified magnetic field,
and ghost-surface pseudo-orbits are obtained by displacing closed field lines
in the direction of steepest descent of magnetic action, . A generalized Hamiltonian definition of ghost
surfaces is given and specialized to the usual Lagrangian definition. A
modified Hamilton's Principle is introduced that allows the use of Lagrangian
integration for calculation of the QFMin pseudo-orbits. Numerical calculations
show QFMin and Lagrangian ghost surfaces give very similar results for a
chaotic magnetic field perturbed from an integrable case, and this is explained
using a perturbative construction of an auxiliary poloidal angle for which
QFMin and Lagrangian ghost surfaces are the same up to second order. While
presented in the context of 3-dimensional magnetic field line systems, the
concepts are applicable to defining almost-invariant tori in other
degree-of-freedom nonintegrable Lagrangian/Hamiltonian systems.Comment: 8 pages, 3 figures. Revised version includes post-publication
corrections in text, as described in Appendix C Erratu
Construction of invariant whiskered tori by a parameterization method. Part I: Maps and flows in finite dimensions
We present theorems which provide the existence of invariant whiskered tori
in finite-dimensional exact symplectic maps and flows. The method is based on
the study of a functional equation expressing that there is an invariant torus.
We show that, given an approximate solution of the invariance equation which
satisfies some non-degeneracy conditions, there is a true solution nearby. We
call this an {\sl a posteriori} approach.
The proof of the main theorems is based on an iterative method to solve the
functional equation.
The theorems do not assume that the system is close to integrable nor that it
is written in action-angle variables (hence we can deal in a unified way with
primary and secondary tori). It also does not assume that the hyperbolic
bundles are trivial and much less that the hyperbolic motion can be reduced to
constant.
The a posteriori formulation allows us to justify approximate solutions
produced by many non-rigorous methods (e.g. formal series expansions, numerical
methods). The iterative method is not based on transformation theory, but
rather on succesive corrections. This makes it possible to adapt the method
almost verbatim to several infinite-dimensional situations, which we will
discuss in a forthcoming paper. We also note that the method leads to fast and
efficient algorithms. We plan to develop these improvements in forthcoming
papers.Comment: To appear in JD
Chaos in the Gauge/Gravity Correspondence
We study the motion of a string in the background of the Schwarzschild black
hole in AdS_5 by applying the standard arsenal of dynamical systems. Our
description of the phase space includes: the power spectrum, the largest
Lyapunov exponent, Poincare sections and basins of attractions. We find
convincing evidence that the motion is chaotic. We discuss the implications of
some of the quantities associated with chaotic systems for aspects of the
gauge/gravity correspondence. In particular, we suggest some potential
relevance for the information loss paradox.Comment: 29 pages, 11 figure
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