135 research outputs found
Instability and Chaos in Non-Linear Wave Interaction: a simple model
We analyze stability of a system which contains an harmonic oscillator
non-linearly coupled to its second harmonic, in the presence of a driving
force. It is found that there always exists a critical amplitude of the driving
force above which a loss of stability appears. The dependence of the critical
input power on the physical parameters is analyzed. For a driving force with
higher amplitude chaotic behavior is observed. Generalization to interactions
which include higher modes is discussed.
Keywords: Non-Linear Waves, Stability, Chaos.Comment: 16 pages, 4 figure
Winding number instability in the phase-turbulence regime of the Complex Ginzburg-Landau Equation
We give a statistical characterization of states with nonzero winding number
in the Phase Turbulence (PT) regime of the one-dimensional Complex
Ginzburg-Landau equation. We find that states with winding number larger than a
critical one are unstable, in the sense that they decay to states with smaller
winding number. The transition from Phase to Defect Turbulence is interpreted
as an ergodicity breaking transition which occurs when the range of stable
winding numbers vanishes. Asymptotically stable states which are not
spatio-temporally chaotic are described within the PT regime of nonzero winding
number.Comment: 4 pages,REVTeX, including 4 Figures. Latex (or postscript) version
with figures available at http://formentor.uib.es/~montagne/textos/nupt
Lyapunov spectral analysis of a nonequilibrium Ising-like transition
By simulating a nonequilibrium coupled map lattice that undergoes an
Ising-like phase transition, we show that the Lyapunov spectrum and related
dynamical quantities such as the dimension correlation length~ are
insensitive to the onset of long-range ferromagnetic order. As a function of
lattice coupling constant~ and for certain lattice maps, the Lyapunov
dimension density and other dynamical order parameters go through a minimum.
The occurrence of this minimum as a function of~ depends on the number of
nearest neighbors of a lattice point but not on the lattice symmetry, on the
lattice dimensionality or on the position of the Ising-like transition. In
one-space dimension, the spatial correlation length associated with magnitude
fluctuations and the length~ are approximately equal, with both
varying linearly with the radius of the lattice coupling.Comment: 29 pages of text plus 15 figures, uses REVTeX macros. Submitted to
Phys. Rev. E
A Non-Equilibrium Defect-Unbinding Transition: Defect Trajectories and Loop Statistics
In a Ginzburg-Landau model for parametrically driven waves a transition
between a state of ordered and one of disordered spatio-temporal defect chaos
is found. To characterize the two different chaotic states and to get insight
into the break-down of the order, the trajectories of the defects are tracked
in detail. Since the defects are always created and annihilated in pairs the
trajectories form loops in space time. The probability distribution functions
for the size of the loops and the number of defects involved in them undergo a
transition from exponential decay in the ordered regime to a power-law decay in
the disordered regime. These power laws are also found in a simple lattice
model of randomly created defect pairs that diffuse and annihilate upon
collision.Comment: 4 pages 5 figure
The dynamics of thin vibrated granular layers
We describe a series of experiments and computer simulations on vibrated
granular media in a geometry chosen to eliminate gravitationally induced
settling. The system consists of a collection of identical spherical particles
on a horizontal plate vibrating vertically, with or without a confining lid.
Previously reported results are reviewed, including the observation of
homogeneous, disordered liquid-like states, an instability to a `collapse' of
motionless spheres on a perfect hexagonal lattice, and a fluctuating,
hexagonally ordered state. In the presence of a confining lid we see a variety
of solid phases at high densities and relatively high vibration amplitudes,
several of which are reported for the first time in this article. The phase
behavior of the system is closely related to that observed in confined
hard-sphere colloidal suspensions in equilibrium, but with modifications due to
the effects of the forcing and dissipation. We also review measurements of
velocity distributions, which range from Maxwellian to strongly non-Maxwellian
depending on the experimental parameter values. We describe measurements of
spatial velocity correlations that show a clear dependence on the mechanism of
energy injection. We also report new measurements of the velocity
autocorrelation function in the granular layer and show that increased
inelasticity leads to enhanced particle self-diffusion.Comment: 11 pages, 7 figure
Phase Diffusion in Localized Spatio-Temporal Amplitude Chaos
We present numerical simulations of coupled Ginzburg-Landau equations
describing parametrically excited waves which reveal persistent dynamics due to
the occurrence of phase slips in sequential pairs, with the second phase slip
quickly following and negating the first. Of particular interest are solutions
where these double phase slips occur irregularly in space and time within a
spatially localized region. An effective phase diffusion equation utilizing the
long term phase conservation of the solution explains the localization of this
new form of amplitude chaos.Comment: 4 pages incl. 5 figures uucompresse
Phase chaos in the anisotropic complex Ginzburg-Landau Equation
Of the various interesting solutions found in the two-dimensional complex
Ginzburg-Landau equation for anisotropic systems, the phase-chaotic states show
particularly novel features. They exist in a broader parameter range than in
the isotropic case, and often even broader than in one dimension. They
typically represent the global attractor of the system. There exist two
variants of phase chaos: a quasi-one dimensional and a two-dimensional
solution. The transition to defect chaos is of intermittent type.Comment: 4 pages RevTeX, 5 figures, little changes in figures and references,
typos removed, accepted as Rapid Commun. in Phys. Rev.
Extensive Chaos in the Nikolaevskii Model
We carry out a systematic study of a novel type of chaos at onset ("soft-mode
turbulence") based on numerical integration of the simplest one dimensional
model. The chaos is characterized by a smooth interplay of different spatial
scales, with defect generation being unimportant. The Lyapunov exponents are
calculated for several system sizes for fixed values of the control parameter
. The Lyapunov dimension and the Kolmogorov-Sinai entropy are
calculated and both shown to exhibit extensive and microextensive scaling. The
distribution functional is shown to satisfy Gaussian statistics at small
wavenumbers and small frequency.Comment: 4 pages (including 5 figures) LaTeX file. Submitted to Phys. Rev.
Let
Wound-up phase turbulence in the Complex Ginzburg-Landau equation
We consider phase turbulent regimes with nonzero winding number in the
one-dimensional Complex Ginzburg-Landau equation. We find that phase turbulent
states with winding number larger than a critical one are only transients and
decay to states within a range of allowed winding numbers. The analogy with the
Eckhaus instability for non-turbulent waves is stressed. The transition from
phase to defect turbulence is interpreted as an ergodicity breaking transition
which occurs when the range of allowed winding numbers vanishes. We explain the
states reached at long times in terms of three basic states, namely
quasiperiodic states, frozen turbulence states, and riding turbulence states.
Justification and some insight into them is obtained from an analysis of a
phase equation for nonzero winding number: rigidly moving solutions of this
equation, which correspond to quasiperiodic and frozen turbulence states, are
understood in terms of periodic and chaotic solutions of an associated system
of ordinary differential equations. A short report of some of our results has
been published in [Montagne et al., Phys. Rev. Lett. 77, 267 (1996)].Comment: 22 pages, 15 figures included. Uses subfigure.sty (included) and
epsf.tex (not included). Related research in
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