67 research outputs found
Transition between Two Oscillation Modes
A model for the symmetric coupling of two self-oscillators is presented. The
nonlinearities cause the system to vibrate in two modes of different
symmetries. The transition between these two regimes of oscillation can occur
by two different scenarios. This might model the release of vortices behind
circular cylinders with a possible transition from a symmetric to an
antisymmetric Benard-von Karman vortex street.Comment: 12 pages, 0 figure
Weakly nonlinear theory of grain boundary motion in patterns with crystalline symmetry
We study the motion of a grain boundary separating two otherwise stationary
domains of hexagonal symmetry. Starting from an order parameter equation
appropriate for hexagonal patterns, a multiple scale analysis leads to an
analytical equation of motion for the boundary that shares many properties with
that of a crystalline solid. We find that defect motion is generically opposed
by a pinning force that arises from non-adiabatic corrections to the standard
amplitude equation. The magnitude of this force depends sharply on the
mis-orientation angle between adjacent domains so that the most easily pinned
grain boundaries are those with an angle between four and eight degrees.
Although pinning effects may be small, they do not vanish asymptotically near
the onset of this subcritical bifurcation, and can be orders of magnitude
larger than those present in smectic phases that bifurcate supercritically
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
Dynamical Properties of Multi-Armed Global Spirals in Rayleigh-Benard Convection
Explicit formulas for the rotation frequency and the long-wavenumber
diffusion coefficients of global spirals with arms in Rayleigh-Benard
convection are obtained. Global spirals and parallel rolls share exactly the
same Eckhaus, zigzag and skewed-varicose instability boundaries. Global spirals
seem not to have a characteristic frequency or a typical size ,
but their product is a constant under given experimental
conditions. The ratio of the radii of any two dislocations (,
) inside a multi-armed spiral is also predicted to be constant. Some of
these results have been tested by our numerical work.Comment: To appear in Phys. Rev. E as Rapid Communication
Logarithmic periodicities in the bifurcations of type-I intermittent chaos
The critical relations for statistical properties on saddle-node bifurcations
are shown to display undulating fine structure, in addition to their known
smooth dependence on the control parameter. A piecewise linear map with the
type-I intermittency is studied and a log-periodic dependence is numerically
obtained for the average time between laminar events, the Lyapunov exponent and
attractor moments. The origin of the oscillations is built in the natural
probabilistic measure of the map and can be traced back to the existence of
logarithmically distributed discrete values of the control parameter giving
Markov partition. Reinjection and noise effect dependences are discussed and
indications are given on how the oscillations are potentially applicable to
complement predictions made with the usual critical exponents, taken from data
in critical phenomena.Comment: 4 pages, 6 figures, accepted for publication in PRL (2004
Invariant manifolds and equilibrium states for non-uniformly hyperbolic horseshoes
In this paper we consider horseshoes containing an orbit of homoclinic
tangency accumulated by periodic points. We prove a version of the Invariant
Manifolds Theorem, construct finite Markov partitions and use them to prove the
existence and uniqueness of equilibrium states associated to H\"older
continuous potentials.Comment: 33 pages, 6 figure
Generalized Arcsine Law and Stable Law in an Infinite Measure Dynamical System
Limit theorems for the time average of some observation functions in an
infinite measure dynamical system are studied. It is known that intermittent
phenomena, such as the Rayleigh-Benard convection and Belousov-Zhabotinsky
reaction, are described by infinite measure dynamical systems.We show that the
time average of the observation function which is not the function,
whose average with respect to the invariant measure is finite, converges to
the generalized arcsine distribution. This result leads to the novel view that
the correlation function is intrinsically random and does not decay. Moreover,
it is also numerically shown that the time average of the observation function
converges to the stable distribution when the observation function has the
infinite mean.Comment: 8 pages, 8 figure
Type-III intermittency in a four-level coherently pumped laser
We study a homogeneously broadened four-level model for a coherently pumped laser with pump and laser fields having crossed linear polarizations. For a parameter range of the type explored in the experiments by Tang et al. [Phys. Rev. A 44, R35 (1991)] the system exhibits a family of type-III-intermittency transitions to chaos in which the onset of intermittency is preceded by period-2, period-3, or period-4 states. We find similarities but also differences between the results of our theory and their experimental results
Grain boundary pinning and glassy dynamics in stripe phases
We study numerically and analytically the coarsening of stripe phases in two
spatial dimensions, and show that transient configurations do not achieve long
ranged orientational order but rather evolve into glassy configurations with
very slow dynamics. In the absence of thermal fluctuations, defects such as
grain boundaries become pinned in an effective periodic potential that is
induced by the underlying periodicity of the stripe pattern itself. Pinning
arises without quenched disorder from the non-adiabatic coupling between the
slowly varying envelope of the order parameter around a defect, and its fast
variation over the stripe wavelength. The characteristic size of ordered
domains asymptotes to a finite value $R_g \sim \lambda_0\
\epsilon^{-1/2}\exp(|a|/\sqrt{\epsilon})\epsilon\ll 1\lambda_0a$ a constant of order unity. Random fluctuations allow defect motion to
resume until a new characteristic scale is reached, function of the intensity
of the fluctuations. We finally discuss the relationship between defect pinning
and the coarsening laws obtained in the intermediate time regime.Comment: 17 pages, 8 figures. Corrected version with one new figur
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