629 research outputs found
Computing Lyapunov spectra with continuous Gram-Schmidt orthonormalization
We present a straightforward and reliable continuous method for computing the
full or a partial Lyapunov spectrum associated with a dynamical system
specified by a set of differential equations. We do this by introducing a
stability parameter beta>0 and augmenting the dynamical system with an
orthonormal k-dimensional frame and a Lyapunov vector such that the frame is
continuously Gram-Schmidt orthonormalized and at most linear growth of the
dynamical variables is involved. We prove that the method is strongly stable
when beta > -lambda_k where lambda_k is the k'th Lyapunov exponent in
descending order and we show through examples how the method is implemented. It
extends many previous results.Comment: 14 pages, 10 PS figures, ioplppt.sty, iopl12.sty, epsfig.sty 44 k
Non-integrability of the mixmaster universe
We comment on an analysis by Contopoulos et al. which demonstrates that the
governing six-dimensional Einstein equations for the mixmaster space-time
metric pass the ARS or reduced Painlev\'{e} test. We note that this is the case
irrespective of the value, , of the generating Hamiltonian which is a
constant of motion. For we find numerous closed orbits with two
unstable eigenvalues strongly indicating that there cannot exist two additional
first integrals apart from the Hamiltonian and thus that the system, at least
for this case, is very likely not integrable. In addition, we present numerical
evidence that the average Lyapunov exponent nevertheless vanishes. The model is
thus a very interesting example of a Hamiltonian dynamical system, which is
likely non-integrable yet passes the reduced Painlev\'{e} test.Comment: 11 pages LaTeX in J.Phys.A style (ioplppt.sty) + 6 PostScript figures
compressed and uuencoded with uufiles. Revised version to appear in J Phys.
The resonance spectrum of the cusp map in the space of analytic functions
We prove that the Frobenius--Perron operator of the cusp map
, (which is an approximation of the
Poincar\'e section of the Lorenz attractor) has no analytic eigenfunctions
corresponding to eigenvalues different from 0 and 1. We also prove that for any
the spectrum of in the Hardy space in the disk
\{z\in\C:|z-q|<1+q\} is the union of the segment and some finite or
countably infinite set of isolated eigenvalues of finite multiplicity.Comment: Submitted to JMP; The description of the spectrum in some Hardy
spaces is adde
Microscopic expressions for the thermodynamic temperature
We show that arbitrary phase space vector fields can be used to generate
phase functions whose ensemble averages give the thermodynamic temperature. We
describe conditions for the validity of these functions in periodic boundary
systems and the Molecular Dynamics (MD) ensemble, and test them with a
short-ranged potential MD simulation.Comment: 21 pages, 2 figures, Revtex. Submitted to Phys. Rev.
Microcanonical temperature for a classical field: application to Bose-Einstein condensation
We show that the projected Gross-Pitaevskii equation (PGPE) can be mapped
exactly onto Hamilton's equations of motion for classical position and momentum
variables. Making use of this mapping, we adapt techniques developed in
statistical mechanics to calculate the temperature and chemical potential of a
classical Bose field in the microcanonical ensemble. We apply the method to
simulations of the PGPE, which can be used to represent the highly occupied
modes of Bose condensed gases at finite temperature. The method is rigorous,
valid beyond the realms of perturbation theory, and agrees with an earlier
method of temperature measurement for the same system. Using this method we
show that the critical temperature for condensation in a homogeneous Bose gas
on a lattice with a UV cutoff increases with the interaction strength. We
discuss how to determine the temperature shift for the Bose gas in the
continuum limit using this type of calculation, and obtain a result in
agreement with more sophisticated Monte Carlo simulations. We also consider the
behaviour of the specific heat.Comment: v1: 9 pages, 5 figures, revtex 4. v2: additional text in response to
referee's comments, now 11 pages, to appear in Phys. Rev.
Efficient estimation of energy transfer efficiency in light-harvesting complexes
The fundamental physical mechanisms of energy transfer in photosynthetic
complexes is not yet fully understood. In particular, the degree of efficiency
or sensitivity of these systems for energy transfer is not known given their
non-perturbative and non-Markovian interactions with proteins backbone and
surrounding photonic and phononic environments. One major problem in studying
light-harvesting complexes has been the lack of an efficient method for
simulation of their dynamics in biological environments. To this end, here we
revisit the second-order time-convolution (TC2) master equation and examine its
reliability beyond extreme Markovian and perturbative limits. In particular, we
present a derivation of TC2 without making the usual weak system-bath coupling
assumption. Using this equation, we explore the long time behaviour of exciton
dynamics of Fenna-Matthews-Olson (FMO) protein complex. Moreover, we introduce
a constructive error analysis to estimate the accuracy of TC2 equation in
calculating energy transfer efficiency, exhibiting reliable performance for
environments with weak and intermediate memory and strength. Furthermore, we
numerically show that energy transfer efficiency is optimal and robust for the
FMO protein complex of green sulphur bacteria with respect to variations in
reorganization energy and bath correlation time-scales.Comment: 16 pages, 9 figures, modified version, updated appendices and
reference lis
Continued-fraction expansion of eigenvalues of generalized evolution operators in terms of periodic orbits
A new expansion scheme to evaluate the eigenvalues of the generalized
evolution operator (Frobenius-Perron operator) relevant to the
fluctuation spectrum and poles of the order- power spectrum is proposed. The
``partition function'' is computed in terms of unstable periodic orbits and
then used in a finite pole approximation of the continued fraction expansion
for the evolution operator. A solvable example is presented and the approximate
and exact results are compared; good agreement is found.Comment: CYCLER Paper 93mar00
Hopf's last hope: spatiotemporal chaos in terms of unstable recurrent patterns
Spatiotemporally chaotic dynamics of a Kuramoto-Sivashinsky system is
described by means of an infinite hierarchy of its unstable spatiotemporally
periodic solutions. An intrinsic parametrization of the corresponding invariant
set serves as accurate guide to the high-dimensional dynamics, and the periodic
orbit theory yields several global averages characterizing the chaotic
dynamics.Comment: Latex, ioplppt.sty and iopl10.sty, 18 pages, 11 PS-figures,
compressed and encoded with uufiles, 170 k
Resonances of the cusp family
We study a family of chaotic maps with limit cases the tent map and the cusp
map (the cusp family). We discuss the spectral properties of the corresponding
Frobenius--Perron operator in different function spaces including spaces of
analytic functions. A numerical study of the eigenvalues and eigenfunctions is
performed.Comment: 14 pages, 3 figures. Submitted to J.Phys.
- …