1,051 research outputs found
Fractal asymptotics
Recent advances in the periodic orbit theory of stochastically perturbed
systems have permitted a calculation of the escape rate of a noisy chaotic map
to order 64 in the noise strength. Comparison with the usual asymptotic
expansions obtained from integrals and with a previous calculation of the
electrostatic potential of exactly selfsimilar fractal charge distributions,
suggests a remarkably accurate form for the late terms in the expansion, with
parameters determined independently from the fractal repeller and the critical
point of the map. Two methods give a precise meaning to the asymptotic
expansion, Borel summation and Shafer approximants. These can then be compared
with the escape rate as computed by alternative methods.Comment: 15 pages, 5 postscript figures incorporated into the text; v2:
Quadratic Pade (Shafer) method added, also a few reference
Entire Fredholm determinants for Evaluation of Semi-classical and Thermodynamical Spectra
Proofs that Fredholm determinants of transfer operators for hyperbolic flows
are entire can be extended to a large new class of multiplicative evolution
operators. We construct such operators both for the Gutzwiller semi-classical
quantum mechanics and for classical thermodynamic formalism, and introduce a
new functional determinant which is expected to be entire for Axiom A flows,
and whose zeros coincide with the zeros of the Gutzwiler-Voros zeta function.Comment: 4 pages, Revtex + one PS figure attached to the end of the text cut
before you run revtex
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
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
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
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
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
The role of singularities in chaotic spectroscopy
We review the status of the semiclassical trace formula with emphasis on the
particular types of singularities that occur in the Gutzwiller-Voros zeta
function for bound chaotic systems. To understand the problem better we extend
the discussion to include various classical zeta functions and we contrast
properties of axiom-A scattering systems with those of typical bound systems.
Singularities in classical zeta functions contain topological and dynamical
information, concerning e.g. anomalous diffusion, phase transitions among
generalized Lyapunov exponents, power law decay of correlations. Singularities
in semiclassical zeta functions are artifacts and enters because one neglects
some quantum effects when deriving them, typically by making saddle point
approximation when the saddle points are not enough separated. The discussion
is exemplified by the Sinai billiard where intermittent orbits associated with
neutral orbits induce a branch point in the zeta functions. This singularity is
responsible for a diverging diffusion constant in Lorentz gases with unbounded
horizon. In the semiclassical case there is interference between neutral orbits
and intermittent orbits. The Gutzwiller-Voros zeta function exhibit a branch
point because it does not take this effect into account. Another consequence is
that individual states, high up in the spectrum, cannot be resolved by
Berry-Keating technique.Comment: 22 pages LaTeX, figures available from autho
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