Strong interaction between plants induces
circular barren patches: fairy circles.
Fairy circles consist of isolated or randomly
distributed circular areas devoid of any
vegetation.They are observed in vast territories
in southern Angola, Namibia and South Africa. We
report on the formation of fairy circles, and we
interpret them as localized structures with a
varying plateau size as a function of the aridity.
Their stabilization cccccmechanism is attributed
to a combined influence of the bistability between
the bare state and the uniformly vegetation state,
and Lorentzian-like nonlocal coupling that models
the competition between plants. We show how a
circular shape is formed,and how the aridity level
influences the size of fairy circles. Finally, we
show that the proposed mechanismis
model-independent . Morover, We propose to
interpret and model peculiar plant morphologies
(cushions and tussocks) observed in the Andean
Altiplano as localized structures.Such structures
resulting in a patchy, aperiodic aspect of the
vegetation cover are hypothesized to self-organize
thanks to the interplay between facilitation and
competition processes occurring at the scale of
basic plant components biologically referred to as
'ramets' (Participants:M.
Tlidi, C.
Fernandez and D. Escaff).
Self-Replication of Localized Vegetation Patches
in Scarce Environments: Desertification due to climate
change and increasing drought periods is a worldwide problem
for both ecology and economy. Our ability to understand how
vegetation manages to survive and propagate through arid and
semiarid ecosystems may be useful in the development of future
strategies to prevent desertification, preserve flora–—and
fauna within—or even make use of scarce resources soils.
We study a robust phenomena observed in semi-arid ecosystems,
by which localized vegetation patches split in a process called
self-replication. Localized patches of vegetation are visible
in nature at various spatial scales. Even though they have
been described in literature, their growth mechanisms remain
largely unexplored. We develop an innovative statistical analysis
based on real field observations to show that patches may
exhibit deformation and splitting. This growth mechanism is
opposite to the desertification since it allows to repopulate
territories devoid of vegetation. We investigate these aspects
by characterizing quantitatively, with a simple mathematical
model, a new class of instabilities that lead to the self-replication
phenomenon observed (Participants:M.
Tlidi, I. Bordeu, P. Couteron, R. Lefever).
Spatiotemporal chaos induces extreme
events in an extended microcavity laser:
Extreme events such as rogue or freak waves in
optics and fluids are often associated with the
merging dynamics of coherent structures.
We show experimental and numerical results on the
physics of extreme event appearance in a spatially
extended semiconductor microcavity laser with an
intracavity saturable absorber. This system can
display deterministic irregular dynamics only,
thanks to spatial coupling through diffraction of
light. Identifying parameter regions where extreme
events are encountered and establishing the origin
of this dynamics in the emergence of deterministic
spatiotemporal chaos, through the correspondence
between the proportion of extreme events and the
dimension of the strange attractor (Participants:
S. Coulibaly, and S.
Barbay).
Extreme events induced by spatiotemporal
chaos in experimental optical patterns.
We report on experimental results in the physics
of extreme events emerging in a liquid-crystal
light valve subjected to optical feedback, and we
establish the relation of this phenomenon with the
appearance of spatiotemporal chaos. This system,
under particular conditions, exhibits stationary
roll patterns that can be destabilized into
quasi-periodic and chaotic textures when control
parameters are properly modified. We have
identified the parameter regions where extreme
fluctuations of the amplitude can emerge and
established their origin through its direct
relation with the experimental largest Lyapunov
exponents, the proportion of extreme events,
and the normed kurtosis (Participants: G.
Gonzalez-Cortes, and M. Wilson).
Transition to Spatiotemporal Intermittency and
Defect Turbulence in Systems under Translational
Coupling: Out
of equilibrium systems under the influence of
enough energy injection exhibit complex
spatiotemporal behaviors. Based on a liquid
crystal light valve experiment with translational
optical feedback, we observe propagation,
spatiotemporal intermittency, and defect
turbulence of striped waves. A prototype model of
pattern formation with translational coupling
shows the same phenomenology. Close to the spatial
instability, a local amplitude equation is
derived. This amplitude equation allows us to
reveal the origin and bifurcation diagram of the
observed complex spatiotemporal dynamics.
Experimental observations have a qualitative
agreement with theoretical findings.
Spatiotemporal
chaos and two-dimensional dissipative rogue waves
in the parametrically driven nonlinear optical
cavity. Driven nonlinear optical
cavities can exhibit complex spatiotemporal
dynamics such as spatiotemporal chaotic dynamics
in one-dimension. We identify a route to
spatiotemporal chaos through an extended
quasiperiodicity. We have estimated the
Kaplan-Yorke dimension that provides a measure
of the strange attractor complexity. Likewise,
we show that the Lugiato-Lefever equation
supports rogues waves in two-dimensional
settings. We characterize rogue-wave formation
by computing the probability distribution of the
pulse height (Participants:M.
Tlidi and K.
Panajotov).
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).
Chaoticon:a spatiotemporal chaotic
localized state. We study the existence,
stability properties, and dynamical evolution of
localized spatiotemporal chaos. We provide
evidence of spatiotemporal chaotic localized
structures in a liquid crystal light valve
experiment with optical feedback. The
observations are supported by numerical
simulations of the Lifshitz model describing the
system. This model exhibits coexistence between
a uniform state and a spatiotemporal chaotic
pattern, which emerge as the necessary
ingredients to obtain localized spatiotemporal
chaos. In addition, we derive a simplified model
that allows us to unveil the front interaction
mechanism at the origin of the localized
spatiotemporal chaotic structures (Participant:N.
Verschueren, U.
Bortolozzo, and S.
Residori).
.
Quasiperiodicity route to spatiotemporal chaos
in one-dimensional pattern-forming systems: we
propose a route to spatiotemporal chaos for
one-dimensional stationary patterns, which is a
natural extension of the quasi-periodicity route for
low-dimensional chaos to extended systems. This
route is studied through a universal model of
pattern formation. The model exhibits a scenario
where stationary patterns become spatiotemporally
chaotic through two successive bifurcations. First,
the pattern undergoes a subcritical Andronov-Hopf
bifurcation leading to an oscillatory pattern.
Subsequently, a secondary bifurcation gives rise to
an oscillation with an incommensurable frequency
with respect to the former one. This last
bifurcation is responsible for the spatiotemporally
chaotic behavior (Participant:N.
Verschueren)
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.
Lorenz
Pendulum
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).
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 ).
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.
Latent bifurcation
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).
LCLV: Experimental setup, bistability
and front propagation.
Bouncing localized structures in a
Liquid-Crystal-Light-Valve experiment
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).
Interface 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)
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).
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).
Van der Waals transition
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).
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).
Vortex Induction via Anisotropy Stabilized
Light-Matter Interaction: By sending circularly polarized light beams
onto a homeotropic nematic liquid crystal cell
with a photosensitive wall, we are able to locally
induce spontaneous matter vortices that remain,
each, stable and trapped at the chosen location.
We study the dual light-matter nature of the
created vortices and demonstrate the ability of
the system to create optical vortices with
opposite topological charges that, consistent with
angular momentum conservation, both derive from
the same defect created in the liquid crystal
texture. Theoretically, we identify a
self-stabilizing mechanism for the matter vortex,
which is provided by the concurrency of
light-induced gradients and anisotropy of the
elastic constants that characterize the
deformation of the liquid crystal medium (Participants:
E.
Vidal, R. Barboza, G. Assanto, U.Bortolozzo,
and
S. Residori).
Harnessing Optical Vortex Lattices in Nematic Liquid Crystals:
By creating self-induced vortexlike defects in the nematic
liquid crystal layer of a light valve, we demonstrate the
realization of programable lattices of optical vortices with
arbitrary distribution in space. On each lattice site, every
matter vortex acts as a photonic spin-to-orbital momentum
coupler and an array of circularly polarized input beams is
converted into an output array of vortex beams with topological
charges consistent with the matter lattice. The vortex arrangements
are explained on the basis of lightinduced matter defects
of both signs and consistent topological rules.
Symmetry breaking of nematic umbilical defects through
an amplitude equation. The existence, stability properties,
and bifurcation diagram of the nematic umbilical defects is
studied by considering a modified Ginzburg-Landau equation.
This model allows to reveal the mechanism of symmetry breaking
of nematic umbilical defects (Participants: E.
Vidal, JD. Davila and M. Kowalczyk).
Umbilical
defect dynamics: Electrically driven nematic liquid crystals
layers are ideal contexts for studying the interactions of local
topological defects. Experimentally, we characterize the coarsening
dynamics in samples containing glass beads as spacers and show
that the inclusion of such imperfections changes the exponent
of the coarsening law. Moreover, we demonstrate that beads that
are slightly deformed alter the surrounding molecular distribution
and attract vortices of both topological charges, thus, presenting
a mainly quadrupolar behavior. Theoretically, based on a model
of vortices diluted in a dipolar medium, a 2/3 exponent is inferred,
which is consistent with the experimental observations.
Exotic states of matter, such as Bose–Einstein
condensates, superfluidity, chiral magnets, superconductivity,
and liquid crystalline blue phases are observed in thermodynamic
equilibrium. Rather than being a result of an aggregation of
matter, their emergence is due to a change of a topological
state of the system. We investigate topological states of matter
in a system with injection and dissipation of energy by means
of oscillatory forcing. In an experiment involving a liquid
crystal cell under the influence of a low-frequency oscillatory
electric field, we observe a transition from a non-vortex state
to a state in which vortices persist, topological transition.
Depending on the period and the type of the forcing, the vortices
self-organise, forming square lattices, glassy states, and disordered
vortex structures. The bifurcation diagram is characterised
experimentally. A continuous topological transition is observed
for the sawtooth and square forcings. The scenario changes dramatically
for sinusoidal forcing where the topological transition is discontinuous,
which is accompanied by serial transitions between square and
glassy vortex lattices. Based on a stochastic amplitude equation,
we recognise the origin of the transition as the balance between
stochastic creation and deterministic annihilation of vortices.
Numerical simulations show topological transitions and the emergence
of square vortex lattice. Our results show that the matter maintained
out of equilibrium by means of the temporal modulation of parameters
can exhibit exotic states.
Vortices are particle-type solutions with topological
charges that can steer the dynamics in various physical systems.
By the application of electromagnetic fields onto a homeotropic
nematic liquid crystal cell, we are able to induce a vortex
triplet that remains stable and trapped at a given location.
For a low frequency of the driven voltage, we observe that the
vortex triplet is unstable and gives rise to the appearance
of a topological lattice. Based on an amplitude equation valid
close to reorientational instability, it allows us to reveal
the origin of the vortex triplet and vortex lattice. Numerical
simulations show a quite fair agreement with theoretical findings
and experimental observations.
Localized dissipative vortices
in chiral nematic liquid crystal cells: Solitary waves and
solitons have played a fundamental role in understanding nonlinear
phenomena and emergent particle-type behaviors in out-of-equilibrium
systems. This type of dynamic phenomenon has not only been essential
to comprehend the behavior of fundamental particles but also
to establish the possibilities of novel technologies based on
optical elements. Dissipative vortices are topological particle-type
solutions in vectorial field out-of-equilibrium systems. These
states can be extended or localized in space. The topological
properties of these states determine the existence, stability
properties, and dynamic evolution. Under homeotropic anchoring,
chiral nematic liquid crystal cells are a natural habitat for
localized vortices or spherulites. However, chiral bubble creation
and destruction mechanisms and their respective bifurcation
diagrams are unknown.We propose a minimal two-dimensional model
based on experimental observations of a temperature-triggered
first-order winding/unwinding transition of a cholesteric liquid
crystal cell and symmetry arguments, and investigate this system
experimentally. This model reveals the main ingredients for
the emergence of chiral bubbles and their instabilities. Experimental
observations have a quite fair agreement with the theoretical
results. Our findings are a starting point to understand the
existence, stability, and dynamical behaviors of dissipative
particles with topological properties.
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)
.
Pinning of Drifting Monostable
Patterns: Under drift forces, a monostable
pattern propagates. However, examples of
nonpropagative dynamics have been observed. The
origin of this pinning effect comes from the
coupling between the slow scale of the envelope to
the fast scale of the modulation of the underlying
pattern. This effect stems from spatial
inhomogeneities in the system. Experiments and
numerics on drifting pattern-forming systems
subjected to inhomogeneous spatial pumping or
boundary conditions confirm this origin of pinning
dynamics (Participants:
C.
Fernandez-Oto, M.A.
Garcia-Nustes and, E.
Louvergneaux).
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
Coulibaly,N.
Mujica, R.Navarro, and T. Sauma).
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).
Phase Shielding Soliton: A
novel type of parametrically excited dissipative
solitons is unveiled. It differs from the
well-known solitons with constant phase by an
intrinsically dynamical evolving shell-type phase
front. Analytical and numerical characterizations
are proposed, displaying quite a good agreement.
In one spatial dimension, the system shows three
types of stationary solitons with shell-like
structure whereas in two spatial dimensions it
displays only one, characterized by a Pi-phase
jump far from the soliton position (Participants:
S
Coulibaly,M.A.
Garcia-Nustes, and Y.
Zarate).
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).
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).
Homoclinic Snaking of Localized Patterns in a
Spatially Forced System: Dissipative
localized structures exhibit intricate bifurcation
diagrams. An adequate theory has been developed in
one space dimension; however, discrepancies arise
with the experiments. Based on an optical feedback
with spatially modulated input beam, we set up a
1D forced configuration in a nematic liquid
crystal layer. We characterize experimentally and
theoretically the homoclinic snaking diagram of
localized patterns, providing a reconciliation
between theory and experiments (Participants:
F.Haudin, R.G.Rojas,U.Bortolozzo,
and
S. Residori).
Asymmetric
counterpropagating front without flow:
Out-of-equilibrium systems exhibit domain walls
between different states. These walls, depending
on the type of connected states, can display rich
spatiotemporal dynamics. We investigate the
asymmetrical counterpropagation of fronts in an
in-plane-switching cell filled with a nematic
liquid crystal. Experimentally, we characterize
the different front shapes and propagation speeds.
These fronts present dissimilar elastic
deformations that are responsible for their
asymmetric speeds. Theoretically, using a
phenomenological model, we describe the observed
dynamics with fair agreement. (Participants:
I.
Andrade-Silva and V. Odent)
Traveling
pulse on a periodic background in parametrically
driven systems: Macroscopic systems with
dissipation and time-modulated injection of
energy, parametrically driven systems, can
self-organize into localized states and/or
patterns. We investigate a pulse that travels over
a one-dimensional
pattern in parametrically driven systems. Based on
a minimal prototype model, we show that the pulses
emerge through a subcritical Andronov-Hopf
bifurcation of the underlying pattern.We describe
a simple physical system, a magnetic wire forced
with a transverse oscillatory magnetic field,
which displays these traveling pulses.(Participants:
A. Leon and
S Coulibaly)
.
From localized spot to the
formation of invaginated labyrinth structures:
The stability of a circular localized spot with
respect to azimuthal perturbations is studied in a
prototype variational model, namely, a
Swift-Hohenberg type equation. The conditions under
which the circular shape of the spot undergoes an
elliptical deformation which transforms it into a
rod-shaped structure are analyzed. As it elongates,
the rod structure exhibits a transversal
instability, generating an invaginated labyrinthine
structure which invades all the space available (Participants:
I.
Bordeu, M.
Tlidi, and R. Lefever).
Rodlike localized structure in
isotropic pattern-forming systems: Stationary
two-dimensional
localized structures have been observed in a wide
variety of dissipative systems. The existence,
stability properties, dynamical evolution, and
bifurcation diagram of an azimuthal symmetry
breaking, rodlike localized structure in the
isotropic prototype model of pattern formation, the
Swift-Hohenberg model, is studied. These rodlike
structures persist under the presence of nongradient
perturbations. Interaction properties of the rodlike
structures are studied. This allows us to envisage
the possibility of different crystal-like
configurations (Participants: I.
Bordeu).
Chimera-type states induced
by local coupling. Coupled oscillators
can exhibit complex self-organization behavior
such as phase turbulence, spatiotemporal
intermittency, and chimera states. The latter
corresponds to a coexistence of coherent and
incoherent states apparently promoted by nonlocal
or global coupling. We have investigated the
existence, stability properties, and bifurcation
diagram of chimera-type states in a system with
local coupling without different time scales.
Based on a model of a chain of nonlinear
oscillators coupled to adjacent neighbors, we have
identified the required attributes to observe
these states: local coupling and bistability
between a stationary and an oscillatory state
close to a homoclinic bifurcation. The local
coupling prevents the incoherent state from
invading the coherent one, allowing concurrently
the existence of a family of chimera states, which
are organized by a homoclinic snaking bifurcation
diagram (Participants: M.A.
Ferre, S
Coulibaly,R.G.Rojas,
and M.A.
Garcia-Nustes).
Chimera-like states in an array of
coupled-waveguide resonators. We
consider coupled-waveguide resonators subject to
optical injection. The dynamics of this simple
device are described by the discrete
Lugiato–Lefever equation. We show that
chimera-like states can be stabilized, thanks to
the discrete nature of the coupled-waveguide
resonators. Such chaotic localized structures are
unstable in the continuous Lugiato–Lefever model;
this is because of dispersive radiation from the
tails of localized structures in the form of two
counter-propagating fronts between the homogeneous
and the complex spatiotemporal state. We
characterize the formation of chimera-like states
by computing the Lyapunov spectra. We show that
localized states have an intermittent
spatiotemporal chaotic dynamical nature. These
states are generated in a parameter regime
characterized by a coexistence between a uniform
steady state and a spatiotemporal intermittency
state (Participants: M.A.
Ferre, S
Coulibaly,R.G.Rojas,
and M.
Tlidi).
Time-delayed nonlocal response inducing traveling
temporal localized structures: time-delayed
nonlocal response induces traveling localized states in bistable
systems. These states result from the interaction of fronts
between homogeneous steady states. We illustrate this mechanism
by considering an experimentally relevant system—the fiber
cavity with the noninstantaneous Raman response. Close to
the nascent bistability, we performed a derivation of a generic
bistable model with a nonlocal delayed response. Analytical
expressions of the width and the speed of traveling localized
states are derived. Without a time-delayed nonlocal response,
traveling localized states are excluded. In addition, we propose
realistic parameters and perform numerical simulations of
the governing model equation.s (Participants: S Coulibaly and M.
Tlidi).
Nonlinear Localization of Dissipative Modulation Instability:
Modulation instability (MI) in the presence of noise typically
leads to an irreversible and complete disintegration of a
plane wave background. Here we study on experiments performed
in a coherently driven nonlinear optical resonator that demonstrate
nonlinear localization of dissipative MI: formation of persisting
domains of MI-driven spatiotemporal chaos surrounded by a
stable quasi-plane-wave background. The persisting localization
ensues from a combination of bistability and complex spatiotemporal
nonlinear dynamics that together permit a locally induced
domain of MI to be pinned by a shallow modulation on the plane
wave background. We further show that the localized domains
of spatiotemporal chaos can be individually addressed—turned
on and off at will—and we explore their transport behavior
as the strength of the pinning is controlled. Our results
reveal new fundamental dynamics at the interface of front
dynamics and MI, and offer a route for tailored patterns of
noiselike bursts of light.
Front
propagation into an unstable state in a forced medium: Spatially
forced systems can exhibit coexistence and a rich interface
dynamics between manipulable states. We study how the propagation
speed of a front into an unstable state can be modified through
periodic space forcing. Based on optical feedback, we set up
a quasi-one-dimensional forced experiment in a liquid-crystal
cell. When changing the forcing parameters, fronts exhibit a
ratchet motion. Unexpectedly, the average speed of fronts decreases
when the strength of the forcing increases. Close to molecular
reorientation transition, an amplitude
equation allows characterizing analytically and numerically
the observed dynamics (Participants: A.
J. Alvarez-Socorro, G.
González-Cortés, and M. Wilson). Nonvariational mechanism of
front propagation: In one-dimensional scalar gradient
systems, the spread of the fronts is proportional to the energy
difference between equilibria. Fronts spreading proportionally
to the energetic difference between equilibria is a characteristic
of one-dimensional scalar gradient systems. Based on a simple
nonvariational bistable model, we show analytically and numerically
that the direction and speed of front propagation is led by
nonvariational dynamics. We provide experimental evidence of
nonvariational front propagation between different molecular
orientations in a quasi-one-dimensional liquid- crystal light
valve subjected to optical feedback. Free diffraction length
allows us to control the variational or nonvariational nature
of this system. Numerical simulations of the phenomenological
model have quite good agreement with experimental observations. (Participants: K. Alfaro-Bittner,
C. Castillo-Pinto, G.
González-Cortés, R.
G. Rojas, and M. Wilson).
.
Labyrinthine patterns transitions: Macroscopic systems
with injection and dissipation of energy exhibit intricate
dissipative structures. Labyrinthine patterns are disordered
spatial structures arising into homogeneous media that show
a short-range order. Here, we investigate the stability properties,
classification, and transitions of labyrinthine patterns.
Based on a prototype pattern forming model, we characterize
the existence of three types of labyrinthine patterns—fingerprint
type, glassy, and scurfy—and reveal the bifurcation diagram.
The defects density, free energy, structure factor, and correlation
length are used as order parameters (Participants:
Echeverría-Alar ).
Localised labyrinthine patterns in ecosystems: Self-organisation
is a ubiquitous phenomenon in ecosystems. These systems can
experience transitions from a uniform cover towards the formation
of vegetation patterns as a result of symmetry-breaking instability.
They can be either periodic or localised in space. Localised
vegetation patterns consist of more or less circular spots
or patches that can be either isolated or randomly distributed
in space. We study a patterning phenomenon consisting of localised
vegetation labyrinths. This intriguing pattern is visible
in satellite photographs taken in many territories of Africa
and Australia. They consist of labyrinths which is spatially
irregular pattern surrounded by either a homogeneous cover
or a bare soil. The phenomenon is not specific to particular
plants or soils. They are observed on strictly homogenous
environmental conditions on flat landscapes, but they are
also visible on hills. The spatial size of localized labyrinth
ranges typically from a few hundred meters to ten kilometres.
A simple modelling approach based on the interplay between
short-range and longrange interactions governing plant communities
or on the water dynamics explains the observations reported
here (Participants: Echeverría-Alar
and M.
Tlidi)..
Nonreciprocal Coupling Induced Self-Assembled
Localized Structures: Chains of coupled oscillators exhibit
energy propagation by means of waves, pulses, and fronts.
Nonreciprocal coupling radically modifies the wave dynamics
of chains. Based on a prototype model of nonlinear chains
with nonreciprocal coupling to nearest neighbors, we study
nonlinear wave dynamics. Nonreciprocal coupling induces a
convective instability between unstable and stable equilibrium.
Increasing the coupling level, the chain presents a propagative
pattern, a traveling wave. This emergent phenomenon corresponds
to the self-assembly of localized structures. The pattern
wavelength is characterized as a function of the coupling.
Analytically, the phase diagram is determined and agrees with
numerical simulations.
