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).

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.

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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).

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).

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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).

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)

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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 andS
Coulibaly)

.

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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).