392 research outputs found
Scarred Patterns in Surface Waves
Surface wave patterns are investigated experimentally in a system geometry
that has become a paradigm of quantum chaos: the stadium billiard. Linear waves
in bounded geometries for which classical ray trajectories are chaotic are
known to give rise to scarred patterns. Here, we utilize parametrically forced
surface waves (Faraday waves), which become progressively nonlinear beyond the
wave instability threshold, to investigate the subtle interplay between
boundaries and nonlinearity. Only a subset (three main types) of the computed
linear modes of the stadium are observed in a systematic scan. These correspond
to modes in which the wave amplitudes are strongly enhanced along paths
corresponding to certain periodic ray orbits. Many other modes are found to be
suppressed, in general agreement with a prediction by Agam and Altshuler based
on boundary dissipation and the Lyapunov exponent of the associated orbit.
Spatially asymmetric or disordered (but time-independent) patterns are also
found even near onset. As the driving acceleration is increased, the
time-independent scarred patterns persist, but in some cases transitions
between modes are noted. The onset of spatiotemporal chaos at higher forcing
amplitude often involves a nonperiodic oscillation between spatially ordered
and disordered states. We characterize this phenomenon using the concept of
pattern entropy. The rate of change of the patterns is found to be reduced as
the state passes temporarily near the ordered configurations of lower entropy.
We also report complex but highly symmetric (time-independent) patterns far
above onset in the regime that is normally chaotic.Comment: 9 pages, 10 figures (low resolution gif files). Updated and added
references and text. For high resolution images:
http://physics.clarku.edu/~akudrolli/stadium.htm
The boundary of hyperbolicity for Henon-like families
We consider C^{2} Henon-like families of diffeomorphisms of R^{2} and study
the boundary of the region of parameter values for which the nonwandering set
is uniformly hyperbolic. Assuming sufficient dissipativity, we show that the
loss of hyperbolicity is caused by a first homoclinic or heteroclinic tangency
and that uniform hyperbolicity estimates hold uniformly in the parameter up to
this bifurcation parameter and even, to some extent, at the bifurcation
parameter.Comment: 32 pages, 11 figures. Several minor revisions, additional figures,
clarifications of some argument
Nonchaotic Stagnant Motion in a Marginal Quasiperiodic Gradient System
A one-dimensional dynamical system with a marginal quasiperiodic gradient is
presented as a mathematical extension of a nonuniform oscillator. The system
exhibits a nonchaotic stagnant motion, which is reminiscent of intermittent
chaos. In fact, the density function of residence times near stagnation points
obeys an inverse-square law, due to a mechanism similar to type-I
intermittency. However, unlike intermittent chaos, in which the alternation
between long stagnant phases and rapid moving phases occurs in a random manner,
here the alternation occurs in a quasiperiodic manner. In particular, in case
of a gradient with the golden ratio, the renewal of the largest residence time
occurs at positions corresponding to the Fibonacci sequence. Finally, the
asymptotic long-time behavior, in the form of a nested logarithm, is
theoretically derived. Compared with the Pomeau-Manneville intermittency, a
significant difference in the relaxation property of the long-time average of
the dynamical variable is found.Comment: 11pages, 5figure
Predictability: a way to characterize Complexity
Different aspects of the predictability problem in dynamical systems are
reviewed. The deep relation among Lyapunov exponents, Kolmogorov-Sinai entropy,
Shannon entropy and algorithmic complexity is discussed. In particular, we
emphasize how a characterization of the unpredictability of a system gives a
measure of its complexity. Adopting this point of view, we review some
developments in the characterization of the predictability of systems showing
different kind of complexity: from low-dimensional systems to high-dimensional
ones with spatio-temporal chaos and to fully developed turbulence. A special
attention is devoted to finite-time and finite-resolution effects on
predictability, which can be accounted with suitable generalization of the
standard indicators. The problems involved in systems with intrinsic randomness
is discussed, with emphasis on the important problems of distinguishing chaos
from noise and of modeling the system. The characterization of irregular
behavior in systems with discrete phase space is also considered.Comment: 142 Latex pgs. 41 included eps figures, submitted to Physics Reports.
Related information at this http://axtnt2.phys.uniroma1.i
Entropy production and Lyapunov instability at the onset of turbulent convection
Computer simulations of a compressible fluid, convecting heat in two
dimensions, suggest that, within a range of Rayleigh numbers, two distinctly
different, but stable, time-dependent flow morphologies are possible. The
simpler of the flows has two characteristic frequencies: the rotation frequency
of the convecting rolls, and the vertical oscillation frequency of the rolls.
Observables, such as the heat flux, have a simple-periodic (harmonic) time
dependence. The more complex flow has at least one additional characteristic
frequency -- the horizontal frequency of the cold, downward- and the warm,
upward-flowing plumes. Observables of this latter flow have a broadband
frequency distribution. The two flow morphologies, at the same Rayleigh number,
have different rates of entropy production and different Lyapunov exponents.
The simpler "harmonic" flow transports more heat (produces entropy at a greater
rate), whereas the more complex "chaotic" flow has a larger maximum Lyapunov
exponent (corresponding to a larger rate of phase-space information loss). A
linear combination of these two rates is invariant for the two flow
morphologies over the entire range of Rayleigh numbers for which the flows
coexist, suggesting a relation between the two rates near the onset of
convective turbulence.Comment: 5 pages, 4 figure
Avalanches in the Weakly Driven Frenkel-Kontorova Model
A damped chain of particles with harmonic nearest-neighbor interactions in a
spatially periodic, piecewise harmonic potential (Frenkel-Kontorova model) is
studied numerically. One end of the chain is pulled slowly which acts as a weak
driving mechanism. The numerical study was performed in the limit of infinitely
weak driving. The model exhibits avalanches starting at the pulled end of the
chain. The dynamics of the avalanches and their size and strength distributions
are studied in detail. The behavior depends on the value of the damping
constant. For moderate values a erratic sequence of avalanches of all sizes
occurs. The avalanche distributions are power-laws which is a key feature of
self-organized criticality (SOC). It will be shown that the system selects a
state where perturbations are just able to propagate through the whole system.
For strong damping a regular behavior occurs where a sequence of states
reappears periodically but shifted by an integer multiple of the period of the
external potential. There is a broad transition regime between regular and
irregular behavior, which is characterized by multistability between regular
and irregular behavior. The avalanches are build up by sound waves and shock
waves. Shock waves can turn their direction of propagation, or they can split
into two pulses propagating in opposite directions leading to transient
spatio-temporal chaos. PACS numbers: 05.70.Ln,05.50.+q,46.10.+zComment: 33 pages (RevTex), 15 Figures (available on request), appears in
Phys. Rev.
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