5,160 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
Geometrical properties of local dynamics in Hamiltonian systems: the Generalized Alignment Index (GALI) method
We investigate the detailed dynamics of multidimensional Hamiltonian systems
by studying the evolution of volume elements formed by unit deviation vectors
about their orbits. The behavior of these volumes is strongly influenced by the
regular or chaotic nature of the motion, the number of deviation vectors, their
linear (in)dependence and the spectrum of Lyapunov exponents. The different
time evolution of these volumes can be used to identify rapidly and efficiently
the nature of the dynamics, leading to the introduction of quantities that
clearly distinguish between chaotic behavior and quasiperiodic motion on
-dimensional tori. More specifically we introduce the Generalized Alignment
Index of order (GALI) as the volume of a generalized parallelepiped,
whose edges are initially linearly independent unit deviation vectors from
the studied orbit whose magnitude is normalized to unity at every time step.
The GALI is a generalization of the Smaller Alignment Index (SALI)
(GALI SALI). However, GALI provides significantly more
detailed information on the local dynamics, allows for a faster and clearer
distinction between order and chaos than SALI and works even in cases where the
SALI method is inconclusive.Comment: 45 pages, 10 figures, accepted for publication in Physica
Secular dynamics of a planar model of the Sun-Jupiter-Saturn-Uranus system; effective stability into the light of Kolmogorov and Nekhoroshev theories
We investigate the long-time stability of the Sun-Jupiter-Saturn-Uranus
system by considering a planar secular model, that can be regarded as a major
refinement of the approach first introduced by Lagrange. Indeed, concerning the
planetary orbital revolutions, we improve the classical circular approximation
by replacing it with a solution that is invariant up to order two in the
masses; therefore, we investigate the stability of the secular system for
rather small values of the eccentricities. First, we explicitly construct a
Kolmogorov normal form, so as to find an invariant KAM torus which approximates
very well the secular orbits. Finally, we adapt the approach that is at basis
of the analytic part of the Nekhoroshev's theorem, so as to show that there is
a neighborhood of that torus for which the estimated stability time is larger
than the lifetime of the Solar System. The size of such a neighborhood,
compared with the uncertainties of the astronomical observations, is about ten
times smaller.Comment: 31 pages, 2 figures. arXiv admin note: text overlap with
arXiv:1010.260
Thermalization, Error-Correction, and Memory Lifetime for Ising Anyon Systems
We consider two-dimensional lattice models that support Ising anyonic
excitations and are coupled to a thermal bath. We propose a phenomenological
model for the resulting short-time dynamics that includes pair-creation,
hopping, braiding, and fusion of anyons. By explicitly constructing topological
quantum error-correcting codes for this class of system, we use our
thermalization model to estimate the lifetime of the quantum information stored
in the encoded spaces. To decode and correct errors in these codes, we adapt
several existing topological decoders to the non-Abelian setting. We perform
large-scale numerical simulations of these two-dimensional Ising anyon systems
and find that the thresholds of these models range between 13% to 25%. To our
knowledge, these are the first numerical threshold estimates for quantum codes
without explicit additive structure.Comment: 34 pages, 9 figures; v2 matches the journal version and corrects a
misstatement about the detailed balance condition of our Metropolis
simulations. All conclusions from v1 are unaffected by this correctio
Control of stochasticity in magnetic field lines
We present a method of control which is able to create barriers to magnetic
field line diffusion by a small modification of the magnetic perturbation. This
method of control is based on a localized control of chaos in Hamiltonian
systems. The aim is to modify the perturbation locally by a small control term
which creates invariant tori acting as barriers to diffusion for Hamiltonian
systems with two degrees of freedom. The location of the invariant torus is
enforced in the vicinity of the chosen target. Given the importance of
confinement in magnetic fusion devices, the method is applied to two examples
with a loss of magnetic confinement. In the case of locked tearing modes, an
invariant torus can be restored that aims at showing the current quench and
therefore the generation of runaway electrons. In the second case, the method
is applied to the control of stochastic boundaries allowing one to define a
transport barrier within the stochastic boundary and therefore to monitor the
volume of closed field lines
Interpolating vector fields for near identity maps and averaging
For a smooth near identity map, we introduce the notion of an interpolating
vector field written in terms of iterates of the map. Our construction is based
on Lagrangian interpolation and provides an explicit expressions for autonomous
vector fields which approximately interpolate the map. We study properties of
the interpolating vector fields and explore their applications to the study of
dynamics. In particular, we construct adiabatic invariants for symplectic near
identity maps. We also introduce the notion of a Poincar\'e section for a near
identity map and use it to visualise dynamics of four dimensional maps. We
illustrate our theory with several examples, including the Chirikov standard
map and a symplectic map in dimension four, an example motivated by the theory
of Arnold diffusion.Comment: 28 pages, 9 Figure
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