51 research outputs found
Anisotropies in magnetic field evolution and local Lyapunov exponents
The natural occurrence of small scale structures and the extreme anisotropy in the evolution of a magnetic field embedded in a conducting flow is interpreted in terms of the properties of the local Lyapunov exponents along the various local characteristic (un)stable directions for the Lagrangian flow trajectories. The local Lyapunov exponents and the characteristic directions are functions of Lagrangian coordinates and time, which are completely determined once the flow field is specified. The characteristic directions that are associated with the spatial anisotropy of the problem, are prescribed in both Lagrangian and Eulerian frames. Coordinate transformation techniques are employed to relate the spatial distributions of the magnetic field, the induced current density, and the Lorentz force, which are usually followed in Eulerian frame, to those of the local Lyapunov exponents, which are naturally defined in Lagrangian coordinates
Lyapunov Mode Dynamics in Hard-Disk Systems
The tangent dynamics of the Lyapunov modes and their dynamics as generated
numerically - {\it the numerical dynamics} - is considered. We present a new
phenomenological description of the numerical dynamical structure that
accurately reproduces the experimental data for the quasi-one-dimensional
hard-disk system, and shows that the Lyapunov mode numerical dynamics is linear
and separate from the rest of the tangent space. Moreover, we propose a new,
detailed structure for the Lyapunov mode tangent dynamics, which implies that
the Lyapunov modes have well-defined (in)stability in either direction of time.
We test this tangent dynamics and its derivative properties numerically with
partial success. The phenomenological description involves a time-modal linear
combination of all other Lyapunov modes on the same polarization branch and our
proposed Lyapunov mode tangent dynamics is based upon the form of the tangent
dynamics for the zero modes
Hill's Equation with Random Forcing Parameters: Determination of Growth Rates through Random Matrices
This paper derives expressions for the growth rates for the random 2 x 2
matrices that result from solutions to the random Hill's equation. The
parameters that appear in Hill's equation include the forcing strength and
oscillation frequency. The development of the solutions to this periodic
differential equation can be described by a discrete map, where the matrix
elements are given by the principal solutions for each cycle. Variations in the
forcing strength and oscillation frequency lead to matrix elements that vary
from cycle to cycle. This paper presents an analysis of the growth rates
including cases where all of the cycles are highly unstable, where some cycles
are near the stability border, and where the map would be stable in the absence
of fluctuations. For all of these regimes, we provide expressions for the
growth rates of the matrices that describe the solutions.Comment: 22 pages, 3 figure
Statistics of finite-time Lyapunov exponents in the Ulam map
The statistical properties of finite-time Lyapunov exponents at the Ulam
point of the logistic map are investigated. The exact analytical expression for
the autocorrelation function of one-step Lyapunov exponents is obtained,
allowing the calculation of the variance of exponents computed over time
intervals of length . The variance anomalously decays as . The
probability density of finite-time exponents noticeably deviates from the
Gaussian shape, decaying with exponential tails and presenting spikes
that narrow and accumulate close to the mean value with increasing . The
asymptotic expression for this probability distribution function is derived. It
provides an adequate smooth approximation to describe numerical histograms
built for not too small , where the finiteness of bin size trimmes the sharp
peaks.Comment: 6 pages, 4 figures, to appear in Phys. Rev.
Lyapunov exponent in quantum mechanics. A phase-space approach
Using the symplectic tomography map, both for the probability distributions
in classical phase space and for the Wigner functions of its quantum
counterpart, we discuss a notion of Lyapunov exponent for quantum dynamics.
Because the marginal distributions, obtained by the tomography map, are always
well defined probabilities, the correspondence between classical and quantum
notions is very clear. Then we also obtain the corresponding expressions in
Hilbert space. Some examples are worked out. Classical and quantum exponents
are seen to coincide for local and non-local time-dependent quadratic
potentials. For non-quadratic potentials classical and quantum exponents are
different and some insight is obtained on the taming effect of quantum
mechanics on classical chaos. A detailed analysis is made for the standard map.
Providing an unambiguous extension of the notion of Lyapunov exponent to
quantum mechnics, the method that is developed is also computationally
efficient in obtaining analytical results for the Lyapunov exponent, both
classical and quantum.Comment: 30 pages Late
Lyapunov exponents for products of complex Gaussian random matrices
The exact value of the Lyapunov exponents for the random matrix product with each , where
is a fixed positive definite matrix and a complex Gaussian matrix with entries standard complex normals, are
calculated. Also obtained is an exact expression for the sum of the Lyapunov
exponents in both the complex and real cases, and the Lyapunov exponents for
diffusing complex matrices.Comment: 15 page
Scalar Decay in Chaotic Mixing
I review the local theory of mixing, which focuses on infinitesimal blobs of
scalar being advected and stretched by a random velocity field. An advantage of
this theory is that it provides elegant analytical results. A disadvantage is
that it is highly idealised. Nevertheless, it provides insight into the
mechanism of chaotic mixing and the effect of random fluctuations on the rate
of decay of the concentration field of a passive scalar.Comment: 35 pages, 15 figures. Springer-Verlag conference style svmult.cls
(included). Published in "Transport in Geophysical Flows: Ten Years After,"
Proceedings of the Grand Combin Summer School, 14-24 June 2004, Valle
d'Aosta, Italy. Fixed some typo
Theory and computation of covariant Lyapunov vectors
Lyapunov exponents are well-known characteristic numbers that describe growth
rates of perturbations applied to a trajectory of a dynamical system in
different state space directions. Covariant (or characteristic) Lyapunov
vectors indicate these directions. Though the concept of these vectors has been
known for a long time, they became practically computable only recently due to
algorithms suggested by Ginelli et al. [Phys. Rev. Lett. 99, 2007, 130601] and
by Wolfe and Samelson [Tellus 59A, 2007, 355]. In view of the great interest in
covariant Lyapunov vectors and their wide range of potential applications, in
this article we summarize the available information related to Lyapunov vectors
and provide a detailed explanation of both the theoretical basics and numerical
algorithms. We introduce the notion of adjoint covariant Lyapunov vectors. The
angles between these vectors and the original covariant vectors are
norm-independent and can be considered as characteristic numbers. Moreover, we
present and study in detail an improved approach for computing covariant
Lyapunov vectors. Also we describe, how one can test for hyperbolicity of
chaotic dynamics without explicitly computing covariant vectors.Comment: 21 pages, 5 figure
Plankton lattices and the role of chaos in plankton patchiness
Spatiotemporal and interspecies irregularities in planktonic populations have been widely observed. Much research into the drivers of such plankton patches has been initiated over the past few decades but only recently have the dynamics of the interacting patches themselves been considered. We take a coupled lattice approach to model continuous-in-time plankton patch dynamics, as opposed to the more common continuum type reaction-diffusion-advection model, because it potentially offers a broader scope of application and numerical study with relative ease. We show that nonsynchronous plankton patch dynamics (the discrete analog of spatiotemporal irregularity) arise quite naturally for patches whose underlying dynamics are chaotic. However, we also observe that for parameters in a neighborhood of the chaotic regime, smooth generalized synchronization of nonidentical patches is more readily supported which reduces the incidence of distinct patchiness. We demonstrate that simply associating the coupling strength with measurements of (effective) turbulent diffusivity results in a realistic critical length of the order of 100 km, above which one would expect to observe unsynchronized behavior. It is likely that this estimate of critical length may be reduced by a more exact interpretation of coupling in turbulent flows
Disorder-assisted error correction in Majorana chains
It was recently realized that quenched disorder may enhance the reliability
of topological qubits by reducing the mobility of anyons at zero temperature.
Here we compute storage times with and without disorder for quantum chains with
unpaired Majorana fermions - the simplest toy model of a quantum memory.
Disorder takes the form of a random site-dependent chemical potential. The
corresponding one-particle problem is a one-dimensional Anderson model with
disorder in the hopping amplitudes. We focus on the zero-temperature storage of
a qubit encoded in the ground state of the Majorana chain. Storage and
retrieval are modeled by a unitary evolution under the memory Hamiltonian with
an unknown weak perturbation followed by an error-correction step. Assuming
dynamical localization of the one-particle problem, we show that the storage
time grows exponentially with the system size. We give supporting evidence for
the required localization property by estimating Lyapunov exponents of the
one-particle eigenfunctions. We also simulate the storage process for chains
with a few hundred sites. Our numerical results indicate that in the absence of
disorder, the storage time grows only as a logarithm of the system size. We
provide numerical evidence for the beneficial effect of disorder on storage
times and show that suitably chosen pseudorandom potentials can outperform
random ones.Comment: 50 pages, 7 figure
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