4,985 research outputs found
Wave packet autocorrelation functions for quantum hard-disk and hard-sphere billiards in the high-energy, diffraction regime
We consider the time evolution of a wave packet representing a quantum
particle moving in a geometrically open billiard that consists of a number of
fixed hard-disk or hard-sphere scatterers. Using the technique of multiple
collision expansions we provide a first-principle analytical calculation of the
time-dependent autocorrelation function for the wave packet in the high-energy
diffraction regime, in which the particle's de Broglie wave length, while being
small compared to the size of the scatterers, is large enough to prevent the
formation of geometric shadow over distances of the order of the particle's
free flight path. The hard-disk or hard-sphere scattering system must be
sufficiently dilute in order for this high-energy diffraction regime to be
achievable. Apart from the overall exponential decay, the autocorrelation
function exhibits a generally complicated sequence of relatively strong peaks
corresponding to partial revivals of the wave packet. Both the exponential
decay (or escape) rate and the revival peak structure are predominantly
determined by the underlying classical dynamics. A relation between the escape
rate, and the Lyapunov exponents and Kolmogorov-Sinai entropy of the
counterpart classical system, previously known for hard-disk billiards, is
strengthened by generalization to three spatial dimensions. The results of the
quantum mechanical calculation of the time-dependent autocorrelation function
agree with predictions of the semiclassical periodic orbit theory.Comment: 24 pages, 13 figure
Delayed Self-Synchronization in Homoclinic Chaos
The chaotic spike train of a homoclinic dynamical system is self-synchronized
by re-inserting a small fraction of the delayed output. Due to the sensitive
nature of the homoclinic chaos to external perturbations, stabilization of very
long periodic orbits is possible. On these orbits, the dynamics appears chaotic
over a finite time, but then it repeats with a recurrence time that is slightly
longer than the delay time. The effect, called delayed self-synchronization
(DSS), displays analogies with neurodynamic events which occur in the build-up
of long term memories.Comment: Submitted to Phys. Rev. Lett., 13 pages, 7 figure
Chaos in the segments from Korean traditional singing and western singing
We investigate the time series of the segments from a Korean traditional song
``Gwansanyungma'' and a western song ``La Mamma Morta'' using chaotic analysis
techniques.
It is found that the phase portrait in the reconstructed state space of the
time series of the segment from the Korean traditional song has a more complex
structure in comparison with the segment from the western songs. The segment
from the Korean traditional song has the correlation dimension 4.4 and two
positive Lyapunov exponents which show that the dynamic related to the Korean
traditional song is a high dimensional hyperchaotic process. On the other hand,
the segment from the western song with only one positive Lyapunov exponent and
the correlation dimension 2.5 exhibits low dimensional chaotic behavior.Comment: 23 pages including 10 eps figures, latex, to appear in J. Acoust.
Soc. A
Time lagged ordinal partition networks for capturing dynamics of continuous dynamical systems
We investigate a generalised version of the recently proposed ordinal
partition time series to network transformation algorithm. Firstly we introduce
a fixed time lag for the elements of each partition that is selected using
techniques from traditional time delay embedding. The resulting partitions
define regions in the embedding phase space that are mapped to nodes in the
network space. Edges are allocated between nodes based on temporal succession
thus creating a Markov chain representation of the time series. We then apply
this new transformation algorithm to time series generated by the R\"ossler
system and find that periodic dynamics translate to ring structures whereas
chaotic time series translate to band or tube-like structures -- thereby
indicating that our algorithm generates networks whose structure is sensitive
to system dynamics. Furthermore we demonstrate that simple network measures
including the mean out degree and variance of out degrees can track changes in
the dynamical behaviour in a manner comparable to the largest Lyapunov
exponent. We also apply the same analysis to experimental time series generated
by a diode resonator circuit and show that the network size, mean shortest path
length and network diameter are highly sensitive to the interior crisis
captured in this particular data set
Turbulence-induced melting of a nonequilibrium vortex crystal in a forced thin fluid film
To develop an understanding of recent experiments on the turbulence-induced
melting of a periodic array of vortices in a thin fluid film, we perform a
direct numerical simulation of the two-dimensional Navier-Stokes equations
forced such that, at low Reynolds numbers, the steady state of the film is a
square lattice of vortices. We find that, as we increase the Reynolds number,
this lattice undergoes a series of nonequilibrium phase transitions, first to a
crystal with a different reciprocal lattice and then to a sequence of crystals
that oscillate in time. Initially the temporal oscillations are periodic; this
periodic behaviour becomes more and more complicated, with increasing Reynolds
number, until the film enters a spatially disordered nonequilibrium statistical
steady that is turbulent. We study this sequence of transitions by using
fluid-dynamics measures, such as the Okubo-Weiss parameter that distinguishes
between vortical and extensional regions in the flow, ideas from nonlinear
dynamics, e.g., \Poincare maps, and theoretical methods that have been
developed to study the melting of an equilibrium crystal or the freezing of a
liquid and which lead to a natural set of order parameters for the crystalline
phases and spatial autocorrelation functions that characterise short- and
long-range order in the turbulent and crystalline phases, respectively.Comment: 31 pages, 56 figures, movie files not include
Two-Particle Circular Billiards Versus Randomly Perturbed One-Particle Circular Billiards
We study a two-particle circular billiard containing two finite-size circular
particles that collide elastically with the billiard boundary and with each
other. Such a two-particle circular billiard provides a clean example of an
"intermittent" system. This billiard system behaves chaotically, but the time
scale on which chaos manifests can become arbitrarily long as the sizes of the
confined particles become smaller. The finite-time dynamics of this system
depends on the relative frequencies of (chaotic) particle-particle collisions
versus (integrable) particle-boundary collisions, and investigating these
dynamics is computationally intensive because of the long time scales involved.
To help improve understanding of such two-particle dynamics, we compare the
results of diagnostics used to measure chaotic dynamics for a two-particle
circular billiard with those computed for two types of one-particle circular
billiards in which a confined particle undergoes random perturbations.
Importantly, such one-particle approximations are much less computationally
demanding than the original two-particle system, and we expect them to yield
reasonable estimates of the extent of chaotic behavior in the two-particle
system when the sizes of confined particles are small. Our computations of
recurrence-rate coefficients, finite-time Lyapunov exponents, and
autocorrelation coefficients support this hypothesis and suggest that studying
randomly perturbed one-particle billiards has the potential to yield insights
into the aggregate properties of two-particle billiards, which are difficult to
investigate directly without enormous computation times (especially when the
sizes of the confined particles are small).Comment: 9 pages, 7 figures (some with multiple parts); published in Chao
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