291 research outputs found
Unstable recurrent patterns in Kuramoto-Sivashinsky dynamics
We undertake a systematic exploration of recurrent patterns in a
1-dimensional Kuramoto-Sivashinsky system. For a small, but already rather
turbulent system, the long-time dynamics takes place on a low-dimensional
invariant manifold. A set of equilibria offers a coarse geometrical partition
of this manifold. A variational method enables us to determine numerically a
large number of unstable spatiotemporally periodic solutions. The attracting
set appears surprisingly thin - its backbone are several Smale horseshoe
repellers, well approximated by intrinsic local 1-dimensional return maps, each
with an approximate symbolic dynamics. The dynamics appears decomposable into
chaotic dynamics within such local repellers, interspersed by rapid jumps
between them.Comment: 11 pages, 11 figure
Revealing the state space of turbulent pipe flow by symmetry reduction
Symmetry reduction by the method of slices is applied to pipe flow in order
to quotient the stream-wise translation and azimuthal rotation symmetries of
turbulent flow states. Within the symmetry-reduced state space, all travelling
wave solutions reduce to equilibria, and all relative periodic orbits reduce to
periodic orbits. Projections of these solutions and their unstable manifolds
from their -dimensional symmetry-reduced state space onto suitably
chosen 2- or 3-dimensional subspaces reveal their interrelations and the role
they play in organising turbulence in wall-bounded shear flows. Visualisations
of the flow within the slice and its linearisation at equilibria enable us to
trace out the unstable manifolds, determine close recurrences, identify
connections between different travelling wave solutions, and find, for the
first time for pipe flows, relative periodic orbits that are embedded within
the chaotic attractor, which capture turbulent dynamics at transitional
Reynolds numbers.Comment: 24 pages, 12 figure
Transport Properties of the Lorentz Gas in Terms of Periodic Orbits
We establish a formula relating global diffusion in a space periodic
dynamical system to cycles in the elementary cell which tiles the space under
translations.Comment: 8 pages, Postscript, A
Visualizing the geometry of state space in plane Couette flow
Motivated by recent experimental and numerical studies of coherent structures
in wall-bounded shear flows, we initiate a systematic exploration of the
hierarchy of unstable invariant solutions of the Navier-Stokes equations. We
construct a dynamical, 10^5-dimensional state-space representation of plane
Couette flow at Re = 400 in a small, periodic cell and offer a new method of
visualizing invariant manifolds embedded in such high dimensions. We compute a
new equilibrium solution of plane Couette flow and the leading eigenvalues and
eigenfunctions of known equilibria at this Reynolds number and cell size. What
emerges from global continuations of their unstable manifolds is a surprisingly
elegant dynamical-systems visualization of moderate-Reynolds turbulence. The
invariant manifolds tessellate the region of state space explored by
transiently turbulent dynamics with a rigid web of continuous and discrete
symmetry-induced heteroclinic connections.Comment: 32 pages, 13 figures submitted to Journal of Fluid Mechanic
Accelerating cycle expansions by dynamical conjugacy
Periodic orbit theory provides two important functions---the dynamical zeta
function and the spectral determinant for the calculation of dynamical averages
in a nonlinear system. Their cycle expansions converge rapidly when the system
is uniformly hyperbolic but greatly slowed down in the presence of
non-hyperbolicity. We find that the slow convergence can be associated with
singularities in the natural measure. A properly designed coordinate
transformation may remove these singularities and results in a dynamically
conjugate system where fast convergence is restored. The technique is
successfully demonstrated on several examples of one-dimensional maps and some
remaining challenges are discussed
Trace formula for noise corrections to trace formulas
We consider an evolution operator for a discrete Langevin equation with a
strongly hyperbolic classical dynamics and Gaussian noise. Using an integral
representation of the evolution operator we investigate the high order
corrections to the trace of arbitary power of the operator.
The asymptotic behaviour is found to be controlled by sub-dominant saddle
points previously neglected in the perturbative expansion. We show that a trace
formula can be derived to describe the high order noise corrections.Comment: 4 pages, 2 figure
A New Method for Computing Topological Pressure
The topological pressure introduced by Ruelle and similar quantities describe
dynamical multifractal properties of dynamical systems. These are important
characteristics of mesoscopic systems in the classical regime. Original
definition of these quantities are based on the symbolic description of the
dynamics. It is hard or impossible to find symbolic description and generating
partition to a general dynamical system, therefore these quantities are often
not accessible for further studies. Here we present a new method by which the
symbolic description can be omitted. We apply the method for a mixing and an
intermittent system.Comment: 8 pages LaTeX with revtex.sty, the 4 postscript figures are included
using psfig.tex to appear in PR
Complexity and non-separability of classical Liouvillian dynamics
We propose a simple complexity indicator of classical Liouvillian dynamics,
namely the separability entropy, which determines the logarithm of an effective
number of terms in a Schmidt decomposition of phase space density with respect
to an arbitrary fixed product basis. We show that linear growth of separability
entropy provides stricter criterion of complexity than Kolmogorov-Sinai
entropy, namely it requires that dynamics is exponentially unstable, non-linear
and non-markovian.Comment: Revised version, 5 pages (RevTeX), with 6 pdf-figure
Recycling Parrondo games
We consider a deterministic realization of Parrondo games and use periodic
orbit theory to analyze their asymptotic behavior.Comment: 12 pages, 9 figure
Alternative method to find orbits in chaotic systems
We present here a new method which applies well ordered symbolic dynamics to
find unstable periodic and non-periodic orbits in a chaotic system. The method
is simple and efficient and has been successfully applied to a number of
different systems such as the H\'enon map, disk billiards, stadium billiard,
wedge billiard, diamagnetic Kepler problem, colinear Helium atom and systems
with attracting potentials. The method seems to be better than earlier applied
methods.Comment: 5 pages, uuencoded compressed tar PostScript fil
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