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Reaction-diffusion
approach to Nano-localized structure in absorbed monoatomic
layers:
We study the robust dynamical behaviors of reaction diffusion
systems where the transport gives rise to non Fickian diffusion.
A prototype model describing the deposition of molecules in
a surface is used to show the generic appearance of Turing
structures which can coexist with homogeneous states giving
rise to localized structures through the pinning mechanism.
The characteristic lengths of these structures are in the
nanometer region in agreement with recent experimental observations
(Participants:
M. Trejo and E. Tirapegui).
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Quasi-reversible instabilities : Hamiltonian and
time reversible dynamical systems present two generic linear
instabilities for a given equilibrium: The stationary instability
or resonance at zero frequency and the 1:1 resonance or confusion
of frequencies.
We have studied dynamical systems, in which time reversal
symmetry is weakly broken in presence of a neutral mode through
which energy is injected in the system, that is, we have considered
systems in the neighborhood of those time reversible.
We have shown that the normal form of the stationary instability
when one has reflection symmetry is the Lorenz model and
the normal form of 1:1 resonance is the set of Maxwell-Bloch
equations, which describes the dynamics of two level atom
in an optical cavity. These two well know sets of equations
turns out to be then universal equations.
We have exhibited numerous examples of these situations. An
interested system is a simple mechanical pendulum oscillating
respect to a turning support submitted to a constant torque
(see Figure and Animation Lorenz Pendulum) which shows
Lorenz type chaotic behavior. We have called Lorenz pendulum
to this simple system.
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Near the critical point the persistence of a homoclinic
solution allow us to find an analytical prediction of
chaotic behavior, preliminary experimental results agree
with the theoretical prediction. We have also characterized
the generic quasi-reversible instabilities of closed
orbits or periodic solutions. We have shown that after
a period change of variables the asymptotic normal form
of doubling period is the Lorenz model. The quasi-reversible
2:1 resonance is simple example of this (Participants:
P. Coullet and E. Tirapegui).
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Localized
peak in bistable pattern forming systems:
We have developed an unifying description close to a spatial bifurcation
of localized states, appearing as large amplitude peaks nucleating
over a pattern of lower amplitude. Localized states are pinned over
a lattice spontaneously generated by the system itself. The smallest
localized stated we have termed Localized peak (cf. figure).
We show that the phenomenon is generic and requires only the coexistence
of two spatially periodic states. At the onset of the spatial bifurcation,
a forced amplitude equation is derived for the critical modes, which
accounts for the appearance of localized peaks (Participant:
U. Bortolozzo, C. Falcon, S.
Residori, and R. Rojas ).
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| Latent
Bifurcation of Mechanical System: There
are two fundamental codimension-one spectral instability for
the Hamiltonian and time reversible systems. The stationary
instability and 1:1 resonance. Spectral instability implies
linearly instability, but linearly instability does not implies
spectral instability. We have studied the dynamics and perturbations
of systems that are gyroscopically (spectrally) stable, yet
have a saddle point in their energy. We call such situations
Latent bifurcation since interesting physical perturbations
can cause movements of eigenvalues across the
imaginary axis.
This
bifurcation requires long time to manifest.
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Latent bifurcation
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A indication of this is the phenomenon denominated
dissipation induced instability, that is, when one
consider small dissipation effects the equilibrium becomes
spectrally unstable. The latent bifurcation is a consequence
of the fact a conservative quantitative becomes non definite
at equilibrium, which allows that the equilibrium perturbations
explore a larger region of phase space (see Fig. Latent
bifurcation). Physical systems that exhibit this bifurcation
are: Laser with slightly pumping (active medium), Baroclinic
instability, simple mechanic systems (Double spherical pendulum),
movement of planets in Celestial mechanics, intramolecular
dynamics, for mention a few (Participant: J.
E. Marsden).
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LCLV: Experimental setup, bistability and front
propagation.
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Bouncing localized structures in a Liquid-Crystal-Light-Valve
experiment
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First-order
Freédericksz transition in LCLV:
One of the most well studied phenomena in the physics of liquid
crystal is field induced distortion of a nematic liquid crystal,
called Freédericksz transition. Which is usually a second
order or supercritical transition. This transition can become
first order for a planar aligned nematic film in which a feedback
mechanism leads to a dependence of applied electric field on the
liquid crystal director. Experimentally, we have realized this
feedback by means of a liquid crystal light valve (see Fig.LCLV).
Starting from Frank free energy, that includes the effect of feedback
as well as the usual nonlinear elastic terms, we have deduced
an amplitude equation. Which shows that depending on the mutual
orientation of the light polarization and liquid crystal director
the transition can become of first order. Our theoretical description
is in a fair qualitative agreement with the experimental observations
(Participants: S.
Residori, C.S. Riera, A. Petrosyan).
Bouncing
localized structures:
The liquid crystal light valve with optical feedback exhibits
localized structures (cf. Fig.). Due to non variational nature
of this system, we observe experimentally permanet dynamics as
bouncing localized structures. Oscillations in the position of
the localized states are described by a consistent amplitude equation,
which we call the Lifshitz normal form equation, in analogy
with phase transitions. Localized structures are shown to arise
close to the Lifshtiz point, where non-variational terms drive
the dynamics into complex and oscillatory behaviors (Participants:
S.
Residori, A. Petrosyan).
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dynamics in liquid crystal:
We have experimentally observed pattern instabilities of an
Ising wall in a nematic or cholesteric liquid crystal. In the
framework of nonlinear elastic theory of liquid crystal, we
have deduced an amplitude equation, relevant close to the Freedericksz
transition. In the case of zigzag instability (see Fig. Zigzag),
this model is characterized by a conservative and variational
order parameter whose gradient satisfies a Cahn-Hilliard equation.
The dynamical behaviors is described by coarsening dynamic of
bubbles. Three
opposite facets form a
bubble (zig-zag-zig). For a gas of diluted bubbles (cf. Fig.
Bubbles interaction), we have found an ordinary differential
equations describing their interaction, which permit us to describe
the ulterior dynamic of the system in a very good qualitative
agreement with the experiments. We have also investigated
the influence of slightly broken symmetries, the lack of translation
invariance or reflection symmetry along the wall can induce
new interfacial patterns which have been both experimentally
and theoretically pointed out (Participants: C. Chevallard,
P. Coullet, and J. M. Gilli; A.
Argentina, C. Calisto, R. Rojas, and E. Tirapegui). |
Zig-Zag
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Bubles interaction
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Chaotic Alternation |
Chaotic
Alternation of Waves in Ring Lasers: Periodic and chaotic
alternation of right and left traveling waves close to threshold
appear to be the more robust dynamical behavior in a Ring
Laser (see Fig. Chaotic Alternation). In the framework of semiclassical
description of the laser, we have deduced a new set of
amplitude equations characterized by two parameters valid close
to the laser instability. This model allows us to study in great
detail the mechanism of the transition from traveling waves
to alternating waves and the nature of the chaotic behaviors.
Particularity, experiments
in c-class laser have a great
qualitative agreement our theoretical description. Stable standing
waves are predicted in a narrow parameters region close to laser
instability. This kind of waves can play a fundamental role
in the design of micro-gyroscope (Participant: P. Coullet) |
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Phase transition in granular media: The
theory of non-ideal gases at thermodynamic equilibrium,
for instance the van derWaals gasmodel, has played a central
role in our understanding of coexisting phases, as well
as the transitions between them. In contrast, the theory
fails with granular matter because collisions between the
grains dissipate energy, and their macroscopic size renders
thermal fluctuations negligible. When a mass of grains is
subjected to mechanical vibration, it can make a transition
to a fluid state. In this state, granular matter exhibits
patterns and instabilities that resemble those of molecular
fluids. Here, we report a granular solid–liquid phase
transition in a vibrating granular monolayer. Unexpectedly,
the transition is mediated by waves and is triggered by
a negative compressibility, as for van der Waals phase coexistence,
although the system does not satisfy the hypotheses used
to understand atomic systems. The dynamic behaviour that
we observe—coalescence, coagulation and wave propagation—is
common to a wide class of phase transitions.We have combined
experimental, numerical and theoretical studies to build
a theoretical framework for this transition (Participants:
P. Cordero,
N. Mujica,
& D. Risso).
Van der Waals-like transition in fluidized granular matter:
We have studied the phase separation of fluidized granular
matter. Molecular dynamics simulations of grain system,
in two spatial dimensions, with a vibrating wall and without
gravity exhibit appearance, coagulation and evaporation
of bubbles. By identifying the mechanism responsible of
phase separation, we have shown that the phenomenon is analogous
to the spinodal decomposition of the gas-liquid transition
of the Van der Waals model. In the onset of phase separation,
we have deduced a macroscopic model that agrees quite well
with molecular dynamics simulations. Furthermore, an hydrodynamic
description of granular media confirms the proposed mechanism
(Participants:
A. Argentina
and R. Soto).
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Van der Waals transition
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Additive
noise induces Front propagation:
The effect of additive noise on a static front that connects
a stable homogeneous state with an also stable but spatially
periodic state is studied. Numerical simulations show that
noise induces front propagation. The conversion of random
fluctuations into direct motion of front's core is responsible
of the propagation; noise prefers to create or remove a bump,
because the necessary perturbations to nucleate or destroy
a bump are different. From a prototype model with noise, we
deduce an adequate equation for the front's core. Analytical
expression for the front velocity is deduced, which is in
good agreement with numerical simulations (Participants:
C. Falcon and E. Tirapegui
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Localized patterns and hole solutions in one-dimensional extended
systems: We have studied the existence, stability properties,
dynamical evolution and bifurcation diagram of localized patterns
and hole solutions in one-dimensional extended systems from
the point of view of front interactions. An adequate envelope
equation is derived from a prototype model, amended
amplitude equation, that exhibits these particle-like
solutions. This equation allow us to obtain an analytical
expression for the front interaction, which is in good agreement
with numerical simulations (Participant: C. Falcon).
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Noisy spatial bifurcation:
A universal behaviors for the generic bifurcations of one-dimensional
systems in the presence of additive noise is studied. In particular,
an analytical expression for the supercritical bifurcation shape
of transverse one-dimensional 1D is given. From this universal
expression, the shape of the bifurcation, its location, and
its evolution with the noise level are completely defined. Experimental
results obtained for a 1D transverse Kerr-type slice subjected
to optical feedback are in excellent agreement
(Participants: G. Agez, E.
Louvergneaux, and R. Rojas) |
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| Soliton pair interaction law in parametrically
driven Newtonian fluid: An experimental and theoretical
study of the motion and interaction of the localized excitations
in a vertically driven small rectangular water container
is realized. Close to the Faraday instability, the parametrically
driven damped nonlinear Schrödinger equation models
this system. This model allows one to characterize the
pair interaction law between localized excitations. Experimentally
we have a good agreement with the pair interaction law
(Participants:S.
Coulibally, N.
Mujica, R.Navarro, and T. Sauma). |
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Solitons interaction
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Parametrically driven instability in quasi-reversal
system: A family of localized states which connect asymptotically
a uniform oscillatory state with itself, in the magnetization
of an easy-plane ferromagnetic spin chain when an oscillatory
magnetic field is applied and in a parametrically driven damped
pendula chain is studied. The conventional approach to these
systems, the parametrically driven damped nonlinear Schrodinger
equation, does not account for these states. Adding higher order
terms to this model we were able to obtain these localized structures
(Participants:
S Coulibaly and D. Laroze). |
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| Transversal interface dynamics of a front:
Interfaces in two-dimensional systems exhibit unexpected complex
dynamical behaviors; the dynamics of a border connecting a stripe
pattern and a uniform state is studied. Numerical simulations
of a prototype isotropic model—the subcritical Swift-Hohenberg
equation—show that this interface has transversal spatial
periodic structures, zigzag dynamics and complex coarsening
process. Close to a spatial bifurcation, an amended amplitude
equation and a one-dimensional interface model allow us to characterize
the dynamics exhibited by this interface (Participants:
G. Elias D. Escaff and R. Rojas). |
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Driven Front Propagation in 1-D Spatially Periodic
Media: front propagation in one-dimensional spatially periodic
media exhibits complex dynamics. Based on an optical feedback
with a spatially amplitude modulated beam, we set up a one-dimensional
forced experiment in a nematic liquid crystal cell. By changing
the forcing parameters, the front exhibits a pinning effect
and oscillatory motion, which are confirmed by numerical simulations
for the average liquid crystal tilt angle. A spatially forced
dissipative phi-4 model, derived at the onset of bistability,
accounts qualitatively for the observed dynamics.(Participants:
F.Haudin, R.G.Elias, R.G.Rojas, U.Bortolozzo,
and S. Residori). |
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