We develop a hierarchical structure (HS) analysis for quantitative description of statistical states of spatially extended systems. Examples discussed here include an experimental reaction-diffusion system with Belousov-Zhabotinsky kinetics, the two-dimensional complex Ginzburg-Landau equation and the modified FitzHugh-Nagumon equation, which all show complex dynamics of spirals and defects. We demonstrate that the spatial-temporal fluctuation fields in the above mentioned systems all display the HS similarity property originally proposed for the study of fully developed turbulence [Z.-S. She and E. Leveque, Phys. Rev. Lett. {\bf 72}, 336 (1994)]. The derived values of a HS parameter $β$ from experimental and numerical data in various physical regimes exhibit consistent trends and characterize the degree of turbulence in the systems near the transition, and the degree of heterogeneity of multiple disorders far from the transition. It is suggested that the HS analysis offers a useful quantitative description for the complex dynamics of two-dimensional spatiotemporal patterns.
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 set up, 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 spatiotempo…
We demonstrate targeting and control over spatiotemporal chaos in an optical feedback loop experiment. Different stationary target patterns are stabilized in real-time by means of a two dimensional space extended perturbation field driven by an interfaced computer and applied in real-space to a liquid crystal display device inserted within a control optical loop. The flexibility of the system in switching between different target patterns is also demonstrated.
We report the first experimental evidence of a successful synchronization of spatiotemporal chaos. The experiments were performed on two unidirectionally coupled, nonlinear-optical systems of single-feedback type. The synchronization was investigated for different degrees of complexity of the spontaneous structures. In all cases, the cross-correlation between the two system states was found to increase with the strength of the coupling. Numerical simulations yield comparable results and throw a light on the role of spatial inhomogeneities, which hamper a perfect synchronization.
We report experiments on defect-tracking in the state of undulation chaos observed in thermal convection of an inclined fluid layer. We characterize the ensemble of defect trajectories according to their velocities, relative positions, diffusion, and gain and loss rates. In particular, the defects exhibit incidents of rapid transverse motion which result in power law distributions for a number of quantitative measures. We examine connections between this behavior and Lévy flights and anomalous diffusion. In addition, we describe time-reversal and system size invariance for defect creation and …
Using large-scale parallel numerical simulations we explore spatiotemporal chaos in Rayleigh-Bénard convection in a cylindrical domain with experimentally relevant boundary conditions. We use the variation of the spectrum of Lyapunov exponents and the leading order Lyapunov vector with system parameters to quantify states of high-dimensional chaos in fluid convection. We explore the relationship between the time dynamics of the spectrum of Lyapunov exponents and the pattern dynamics. For chaotic dynamics we find that all of the Lyapunov exponents are positively correlated with the leading orde…
Diffusion-induced turbulence in spatially extended oscillatory media near a supercritical Hopf bifurcation can be controlled by applying global time-delay autosynchronization. We consider the complex Ginzburg-Landau equation in the Benjamin-Feir unstable regime and analytically investigate the stability of uniform oscillations depending on the feedback parameters. We show that a noninvasive stabilization of uniform oscillations is not possible in this type of systems. The synchronization diagram in the plane spanned by the feedback parameters is derived. Numerical simulations confirm the analy…
We report experimental evidence of the route to spatiotemporal chaos in a large 1D-array of hotspots in a thermoconvective system. Increasing the driving force, a stationary cellular pattern becomes unstable towards a mixed pattern of irregular clusters which consist of time-dependent localized patterns of variable spatiotemporal coherence. These irregular clusters coexist with the basic cellular pattern. The Fourier spectra corresponding to this synchronization transition reveals the weak coupling of a resonant triad. This pattern saturates with the formation of a unique domain of great spati…
Numerical simulations of two dimensional pattern formation in an anisotropic bistable reaction-diffusion medium reveal a new dynamical state, stratified spatiotemporal chaos, characterized by strong correlations along one of the principal axes. Equations that describe the dependence of front motion on the angle illustrate the mechanism leading to stratified chaos.
We generalize the concept of geometrical resonance to perturbed sine-Gordon, Nonlinear Schrödinger and Complex Ginzburg-Landau equations. Using this theory we can control different dynamical patterns. For instance, we can stabilize breathers and oscillatory patterns of large amplitudes successfully avoiding chaos. On the other hand, this method can be used to suppress spatiotemporal chaos and turbulence in systems where these phenomena are already present. This method can be generalized to even more general spatiotemporal systems.
Alexander B. Medvinsky \emph{et al} [A. B. Medvinsky, I. A. Tikhonova, R. R. Aliev, B.-L. Li, Z.-S. Lin, and H. Malchow, Phys. Rev. E \textbf{64}, 021915 (2001)] and Marcus R. Garvie \emph{et al} [M. R. Garvie and C. Trenchea, SIAM J. Control. Optim. \textbf{46}, 775-791 (2007)] shown that the minimal spatially extended reaction-diffusion model of phytoplankton-zooplankton can exhibit both regular, chaotic behavior, and spatiotemporal patterns in a patchy environment. Based on that, the spatial plankton model is furtherly investigated by means of computer simulations and theoretical analysis i…
We present the first large scale numerical simulation of three-dimensional Rayleigh-Bénard convection near onset, under free-free boundary conditions for a fluid of Prandtl number $σ=0.5$. We find that a spatiotemporally chaotic state emerges immediately above onset, which we investigate as a function of the reduced control parameter $ε$. We conclude that the transition from conduction to spatiotemporal chaos is second order and of ``mean field'' character. We also present a simple theory for the time-averaged convective current. Finally, we show that the time-averaged structure factor satisfi…
In coupled reaction-diffusion systems, modes with two different length scales can interact to produce a wide variety of spatiotemporal patterns. Three-wave interactions between these modes can explain the occurrence of spatially complex steady patterns and time-varying states including spatiotemporal chaos. The interactions can take the form of two short waves with different orientations interacting with one long wave, or vice-versa. We investigate the role of such three-wave interactions in a coupled Brusselator system. As well as finding simple steady patterns when the waves reinforce each o…
For Rayleigh-Benard convection in a square cell with a fluid of Prandtl number one, we report experimental results on the transition between a stationary pattern of ideal straight rolls (ISR) and the spatiotemporal chaotic state of spiral defect chaos (SDC). In contrast to experiments in circular geometries, we found ISR states below a particular value of the control parameter and SDC states above this value. By characterizing the pattern with a global measure, the pattern entropy, we found that the transition from SDC to ISR showed similarities to phase transitions in equilibrium finite-size …
Experimental and theoretical investigations of undulation patterns in high-pressure, inclined layer gas convection at a Prandtl number near unity are reported. Particular focus is given to the competition between the spatiotemporal chaotic state of undulation chaos and stationary patterns of ordered undulations. In experiments a competition and bistability between the two states is observed, with ordered undulations most prevalent at higher Rayleigh number. The spectral pattern entropy, spatial correlation lengths, and defect statistics are used to characterize the competing states. The experi…
In a quasi-1D thermal convective system consisting of a large array of nonlinearly coupled oscillators, clustering is the way to achieve a regime of mostly antiphase synchronized oscillators. This regime is characterized by a spatiotemporal doubling of traveling modes. As the dynamics is explored beyond a spatiotemporal chaos regime with weak coupling, new interacting modes emerge through a supercritical bifurcation. In this new regime, the system exhibits coherent subsystems of antiphase synchronized oscillators, which are stationary clusters following a spatiotemporal beating phenomena. This…
Arrhythmias are potentially fatal disruptions to the normal heart rhythm, but their underlying dynamics is still poorly understood. Theoretical modeling is an important tool to fill this gap. Typical studies often employ detailed multidimensional conductance-based models. We describe the cardiac muscle with a three-dimensional map-based membrane potential model in lattices. Although maps retain the biophysical behavior of cells and generate computationally efficient tissue models, few studies have used them to understand cardiac dynamics. Our study captures healthy and pathological behaviors w…
Small-sized systems exhibit a finite number of routes to chaos. However, in extended systems, not all routes to complex spatiotemporal behavior have been fully explored. Starting from the sine-Gordon model of parametrically driven chain of damped nonlinear oscillators, we investigate a route to spatiotemporal chaos emerging from standing waves. The route from the stationary to the chaotic state proceeds through quasiperiodic dynamics. The standing wave undergoes the onset of oscillatory instability, which subsequently exhibits a different critical frequency, from which the complexity originate…
We present a phenomenological theory for spatiotemporal chaos (STC) in Rayleigh-Benard convection, based on the generalized Swift-Hohenberg model. We apply a random phase approximation to STC and conjecture a scaling form for the structure factor $S(k)$ with respect to the correlation length $ξ_2$. We hence obtain analytical results for the time-averaged convective current $J$ and the time-averaged vorticity current $Ω$. We also define power-law behaviors such as $J \sim ε^μ$, $Ω\sim ε^λ$ and $ξ_2 \sim ε^{-ν}$, where $ε$ is the control parameter. We find from our theory that $μ= 1$, $ν\ge 1/2$…
We present an analytical and numerical study of the Klein-Gordon kink-soliton dynamics in inhomogeneous media. In particular, we study an external field that is almost constant for the whole system but that changes its sign at the center of coordinates and a localized impurity with finite-width. The soliton solution of the Klein-Gordon-like equations is usually treated as a structureless point-like particle. A richer dynamics is unveiled when the extended character of the soliton is taken into account. We show that interesting spatiotemporal phenomena appear when the structure of the soliton i…
Nonlinear reaction-diffusion systems admit a wide variety of spatiotemporal patterns or structures. In this lecture, we point out that there is certain advantage in studying discrete arrays, namely cellular neural/nonlinear networks (CNNs), over continuous systems. Then, to illustrate these ideas, the dynamics of diffusively coupled one and two dimensional cellular nonlinear networks (CNNs), involving Murali-Lakshmanan-Chua circuit as the basic element, is considered. Propagation failure in the case of uniform diffusion and propagation blocking in the case of defects are pointed out. The mecha…
Laboratory experiments point out the existence of patterns made of alternately laminar and turbulent oblique bands in plane Couette flow in its way to/from turbulence as the Reynolds number R is varied. Many previous theoretical and numerical works on the problem have considered small aspect-ratio systems subjected to periodic boundary conditions, while experiments correspond to the opposite limit of large aspect-ratio. Here, by means of fully resolved direct numerical simulations of the Navier-Stokes equations at decreasing R, we scrutinize the transition from temporal to spatiotemporal behav…
