222 research outputs found
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
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
Reducing or enhancing chaos using periodic orbits
A method to reduce or enhance chaos in Hamiltonian flows with two degrees of
freedom is discussed. This method is based on finding a suitable perturbation
of the system such that the stability of a set of periodic orbits changes
(local bifurcations). Depending on the values of the residues, reflecting their
linear stability properties, a set of invariant tori is destroyed or created in
the neighborhood of the chosen periodic orbits. An application on a
paradigmatic system, a forced pendulum, illustrates the method
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
Wave Chaos in Elastodynamic Cavity Scattering
The exact elastodynamic scattering theory is constructed to describe the
spectral properties of two- and more-cylindrical cavity systems, and compared
to an elastodynamic generalization of the semi-classical Gutzwiller unstable
periodic orbits formulas. In contrast to quantum mechanics, complex periodic
orbits associated with the surface Rayleigh waves dominate the low-frequency
spectrum, and already the two-cavity system displays chaotic features.Comment: 7 pages, 5 eps figures, latex (with epl.cls
Lyapunov exponents and anomalous diffusion of a Lorentz gas with infinite horizon using approximate zeta functions
We compute the Lyapunov exponent, generalized Lyapunov exponents and the
diffusion constant for a Lorentz gas on a square lattice, thus having infinite
horizon. Approximate zeta functions, written in terms of probabilities rather
than periodic orbits, a re used in order to avoid the convergence problems of
cycle expansions. The emphasis is on the relation between the analytic
structure of the zeta function, where a branch cut plays an important role, and
the asymptotic dynamics of the system. We find a diverging diffusion constant
and a phase transition for the generalized Lyapunov
exponents.Comment: 14 pages LaTeX, figs 2-3 on .uu file, fig 1 available from autho
On the state space geometry of the Kuramoto-Sivashinsky flow in a periodic domain
The continuous and discrete symmetries of the Kuramoto-Sivashinsky system
restricted to a spatially periodic domain play a prominent role in shaping the
invariant sets of its chaotic dynamics. The continuous spatial translation
symmetry leads to relative equilibrium (traveling wave) and relative periodic
orbit (modulated traveling wave) solutions. The discrete symmetries lead to
existence of equilibrium and periodic orbit solutions, induce decomposition of
state space into invariant subspaces, and enforce certain structurally stable
heteroclinic connections between equilibria. We show, on the example of a
particular small-cell Kuramoto-Sivashinsky system, how the geometry of its
dynamical state space is organized by a rigid `cage' built by heteroclinic
connections between equilibria, and demonstrate the preponderance of unstable
relative periodic orbits and their likely role as the skeleton underpinning
spatiotemporal turbulence in systems with continuous symmetries. We also offer
novel visualizations of the high-dimensional Kuramoto-Sivashinsky state space
flow through projections onto low-dimensional, PDE representation independent,
dynamically invariant intrinsic coordinate frames, as well as in terms of the
physical, symmetry invariant energy transfer rates.Comment: 31 pages, 17 figures; added references, corrected typos. Due to file
size restrictions some figures in this preprint are of low quality. A high
quality copy may be obtained from
http://www.cns.gatech.edu/~predrag/papers/preprints.html#rp
An exactly solvable self-convolutive recurrence
We consider a self-convolutive recurrence whose solution is the sequence of
coefficients in the asymptotic expansion of the logarithmic derivative of the
confluent hypergeometic function . By application of the Hilbert
transform we convert this expression into an explicit, non-recursive solution
in which the th coefficient is expressed as the th moment of a
measure, and also as the trace of the th iterate of a linear operator.
Applications of these sequences, and hence of the explicit solution provided,
are found in quantum field theory as the number of Feynman diagrams of a
certain type and order, in Brownian motion theory, and in combinatorics
Stability ordering of cycle expansions
We propose that cycle expansions be ordered with respect to stability rather
than orbit length for many chaotic systems, particularly those exhibiting
crises. This is illustrated with the strong field Lorentz gas, where we obtain
significant improvements over traditional approaches.Comment: Revtex, 5 incorporated figures, total size 200
Group theory factors for Feynman diagrams
We present algorithms for the group independent reduction of group theory
factors of Feynman diagrams. We also give formulas and values for a large
number of group invariants in which the group theory factors are expressed.
This includes formulas for various contractions of symmetric invariant tensors,
formulas and algorithms for the computation of characters and generalized
Dynkin indices and trace identities. Tables of all Dynkin indices for all
exceptional algebras are presented, as well as all trace identities to order
equal to the dual Coxeter number. Further results are available through
efficient computer algorithms (see http://norma.nikhef.nl/~t58/ and
http://norma.nikhef.nl/~t68/ ).Comment: Latex (using axodraw.sty), 47 page
- âŠ