Spatiotemporal chaotic localized state in liquid crystal light valve experiments with optical feedback

Localized spatiotemporal dynamics in active fluids

From cytoskeletal networks to tissues, many biological systems behave as active materials. Their composition and stress generation is affected by chemical reaction networks. In such systems, the coupling between mechanics and chemistry enables self-organization, for example, into waves. Recently, contractile mechanochemical systems have been shown to be able to spontaneously develop localized spatial patterns. Here, we show that these localized patterns can present intrinsic spatiotemporal dynamics, including oscillations and chaotic-like dynamics. We discuss their physical origin and bifurcation structure.

Vortex chaoticons in thermal nonlocal nonlinear media

This paper numerically investigates the propagation of Laguerre-Gaussian vortex beams launched in nonlocal nonlinear media, such as lead glass. Our results show that the propagation properties depend on the selection of beam parameters m and p, which represent the azimuthal and radial mode numbers. When p=0, these profiles can be stable solitons for m≤2, or break up and then form a set of single-hump profiles for m≥3, which are unbounded states with scattered remnants of the energy. However, for p≥1, the broken beams can evolve into vortex chaoticons, which exhibit both chaotic and solitonlike properties. The chaotic properties are determined by the positive Lyapunov exponents and spatial decoherence, while the solitonlike properties are demonstrated by the invariance of beam width and the interaction of beams in the form of quasielastic collisions. In addition, the power and orbital angular momentum of unbounded beam states both decay in propagation, while those of the chaoticons maintain their values well.

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 report 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.

Light-Induced Ring Pattern in a Dye-Doped Nematic Liquid Crystal

The use of dye-doped liquid crystals allows the amplification of the coupling of light and liquid crystals. Light can induce the self-organization of the molecular order. The appearance of ring patterns has been observed, which has been associated with phase modulation. However, the morphology and dynamics of the ring patterns are not consistent with self-modulation. Based on an experimental setup with two parallel coherence beams orthogonal to a liquid crystal cell, one of which induces photo-isomerization and the other causes illumination, the formation of ring patterns is studied. To use these two coherent beams, we synthesize methylred methyl ester as a dye-dopant, which is photosensitive only to one of the light beams, and a commercial E7 liquid crystal as a matrix. Based on a mathematical model that accounts for the coupling between the concentration of the cis-state and the order parameter, we elucidate the emergence of the rings as forming patterns in an inhomogeneous medium. The bifurcation diagram is analytically characterized. The emergence, propagation of the rings, and the establishment of the ring patterns are in fair agreement with the experimental observations.

Localized chaos of elliptically polarized cavity solitons in broad-area VCSEL with a saturable absorber

We introduce a spin-flip model for a broad-area vertical-cavity surface-emitting laser (VCSEL) with a saturable absorber. We demonstrate simultaneous existence of orthogonally linearly polarized and elliptically polarized cavity solitons. We show that polarization degree of freedom leads to a period-doubling route to spatially localized chaos of the elliptically polarized cavity solitons.

Dissipative solitons for bistable delayed-feedback systems

We study how nonlinear delayed-feedback in the Ikeda model can induce solitary impulses, i.e., dissipative solitons. The states are clearly identified in a virtual space-time representation of the equations with delay, and we find that conditions for their appearance are bistability of a nonlinear function and negative character of the delayed feedback. Both dark and bright solitons are identified in numerical simulations and physical electronic experiment, showing an excellent qualitative correspondence and proving thereby the robustness of the phenomenon. Along with single spiking solitons, a variety of compound soliton-based structures is obtained in a wide parameter region on the route from the regular dynamics (two quiescent states) to developed spatiotemporal chaos. The number of coexisting soliton-based states is fast growing with delay, which can open new perspectives in the context of information storage.

Two-dimensional spatiotemporal complexity in dual-delayed nonlinear feedback systems: Chimeras and dissipative solitons

We demonstrate for a photonic nonlinear system that two highly asymmetric feedback delays can induce a variety of emergent patterns which are highly robust during the system's global evolution. Explicitly, two-dimensional chimeras and dissipative solitons become visible upon a space-time transformation. Switching between chimeras and dissipative solitons requires only adjusting two system parameters, demonstrating self-organization exclusively based on the system's dynamical properties. Experiments were performed using a tunable semiconductor laser's transmission through a Fabry-Pérot resonator resulting in an Airy function as nonlinearity. Resulting dynamics were bandpass filtered and propagated along two feedback paths whose time delays differ by two orders of magnitude. An excellent agreement between experimental results and the theoretical model given by modified Ikeda equations was achieved.

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.

Two-dimensional localized chaotic patterns in parametrically driven systems

We study two-dimensional localized patterns in weakly dissipative systems that are driven parametrically. As a generic model for many different physical situations we use a generalized nonlinear Schrödinger equation that contains parametric forcing, damping, and spatial coupling. The latter allows for the existence of localized pattern states, where a finite-amplitude uniform state coexists with an inhomogeneous one. In particular, we study numerically two-dimensional patterns. Increasing the driving forces, first the localized pattern dynamics is regular, becomes chaotic for stronger driving, and finally extends in area to cover almost the whole system. In parallel, the spatial structure of the localized states becomes more and more irregular, ending up as a full spatiotemporal chaotic structure.

Splitting in the pinning-depinning transition of fronts in long-delayed bistable systems

We investigate the formation of localized domains through front pinning in a periodically forced, bistable semiconductor laser with long-delayed optoelectronic feedback. At difference with 1D spatially extended systems, the transition from the pinning to the propagation regime occurs via two separated bifurcations, each corresponding to the unpinning of one of the fronts surrounding the localized domain. The bifurcation splitting is systematically explored, unveiling the crucial role played by the forcing frequency. The experimental results are reproduced and interpreted by means of a prototypical model of our system.

Chaoticons described by nonlocal nonlinear Schrödinger equation

It is shown that the unstable evolutions of the Hermite-Gauss-type stationary solutions for the nonlocal nonlinear Schrödinger equation with the exponential-decay response function can evolve into chaotic states. This new kind of entities are referred to as chaoticons because they exhibit not only chaotic properties (with positive Lyapunov exponents and spatial decoherence) but also soliton-like properties (with invariant statistic width and interaction of quasi-elastic collisions).

Optical textures: characterizing spatiotemporal chaos

Macroscopic systems subjected to injection and dissipation of energy can exhibit complex spatiotemporal behaviors as result of dissipative self-organization. Here, we report a one- and two-dimensional pattern forming setup, which exhibits a transition from stationary patterns to spatiotemporal chaotic textures, based on a nematic liquid crystal layer with spatially modulated input beam and optical feedback. Using an adequate projection of spatiotemporal diagrams, we determine the largest Lyapunov exponent. Jointly, this exponent and Fourier transform allow us to distinguish between spatiotemporal chaos and amplitude turbulence concepts, which are usually merged.

Anomalous Diffusion of Dissipative Solitons in the Cubic-Quintic Complex Ginzburg-Landau Equation in Two Spatial Dimensions

We demonstrate the occurrence of anomalous diffusion of dissipative solitons in a "simple" and deterministic prototype model: the cubic-quintic complex Ginzburg-Landau equation in two spatial dimensions. The main features of their dynamics, induced by symmetric-asymmetric explosions, can be modeled by a subdiffusive continuous-time random walk, while in the case dominated by only asymmetric explosions, it becomes characterized by normal diffusion.

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. Here we investigate 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 identify 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.

Time-delayed feedback control of breathing localized structures in a three-component reaction-diffusion system

The dynamics of a single breathing localized structure in a three-component reaction-diffusion system subjected to time-delayed feedback is investigated. It is shown that variation of the delay time and the feedback strength can lead either to stabilization of the breathing or to delay-induced periodic or quasi-periodic oscillations of the localized structure. A bifurcation analysis of the system in question is provided and an order parameter equation is derived that describes the dynamics of the localized structure in the vicinity of the Andronov-Hopf bifurcation. With the aid of this equation, the boundaries of the stabilization domains as well as the dependence of the oscillation radius on delay parameters can be explicitly derived, providing a robust mechanism to control the behaviour of the breathing localized structure in a straightforward manner.

Periodic and chaotic solitons in a semiconductor laser with saturable absorber

In a semiconductor laser with saturable absorber, solitons may spontaneously drift and/or oscillate. We study three different regimes characterized by strong intensity oscillations, both periodic and chaotic. We show that (i) soliton dynamics may be similar to that of passively Q-switched lasers, (ii) solitons may drift and oscillate simultaneously, and (iii) chaotic solitons may coexist with stationary ones and with the laser off solution.

Chaotic behavior of cavity solitons induced by time delay feedback

We investigate spatiotemporal dynamics of cavity solitons in a broad area vertical-cavity surface-emitting laser with saturable absorption subject to time-delayed optical feedback. We show that the inclusion of feedback leads to a period doubling route to chaos of spatially localized light structures.

Experimental observation of localized structures in medium size VCSELs

We report experimental evidence of spontaneous formation of localized structures in a 80 μm diameter Vertical-Cavity Surface-Emitting Laser (VCSEL) biased above the lasing threshold and under optical injection. Such localized structures are bistable with the injected beam power and the VCSEL current. We experimentally investigate the formation of localized structures for different detunings between the injected beam and the VCSEL, and different injection beam waists.

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 quasiperiodicity 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. The Lyapunov spectrum enables us to identify the complex behavior observed as spatiotemporal chaos, and also from the larger Lyapunov exponents characterize the above instabilities.