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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).


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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).

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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

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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). 


 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)
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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)
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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 and
S 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). rod

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).