Chaotic Spatiotemporal Patterns Research Articles

Optical feedback-induced spatiotemporal patterns with power law spectra in a liquid crystal light valve.

Pattern formation can be induced by coupling electromagnetic fields to a polarizable and lossy medium. Increasing energy injection patterns can exhibit aperiodic behaviors. We investigate the self-organization of unidimensional aperiodic patterns. Based on a liquid crystal light valve (LCLV) with optical feedback, we observed aperiodic one-dimensional patterns with power laws in the temporal and spatial spectrum density of the light intensity, and their pseudo envelope and phase characteristic of spatiotemporal complexity. Theoretically, a local model describes the system close to nascent bistability and spatial instability. Numerical simulations of this model show chaotic spatiotemporal patterns whose temporal and spatial spectra have exponents similar to those observed experimentally.

Convolutional LSTM-Attention Based Encoder–Decoder Neural Network for Prediction of Chaotic Vibrations of Multi-Dimensional Dynamic Systems

The present research proposes a convolutional long-short term memory (ConvLSTM) with an attention mechanism (AM) model, termed as ConvLSTM-AM, to conduct prediction of chaotic vibrations of multi-dimensional dynamic systems. The proposed data-driven model is based on an encoder–decoder architecture where the lengths of inputs and outputs are variable. Different from other conventional benchmarks which consider the temporal correlation solely to deal with the chaotic sequences, this research work takes the spatial information into account. In this sense, the ConvLSTM is adopted as an encoder to acquire useful chaotic spatiotemporal patterns and retain long-term successive dependencies. LSTM and AM in this research are taken as the main structures of the procedure in decoding, in which the LSTM is used as the further temporal processor and AM is stacked on the top to exploit more salient information of the historical data. Among that, a residual connection between the outputs of LSTM and the information of attention is considered in AM to prevent gradient vanishment. Two datasets of chaotic vibrations of multi-dimensional systems are employed to adequately illustrate the effectiveness and feasibility of the proposed model. Besides, five conventional benchmarks are built to demonstrate the advantages of the proposed model in terms of both training and generalization performance. As found in the research, the training time is reduced with lower testing loss in comparing with the other five counterparts, as the spatial information introduced expedites the training convergence. The present research provides a useful guidance for predicting and analysing chaotic vibrations of multi-dimensional dynamic systems.

Columnar vortices induced by dielectrophoretic force in a stationary cylindrical annulus filled with a dielectric liquid

Abstract

Global dynamics and spatio-temporal patterns of predator–prey systems with density-dependent motion

In this paper, we investigate the global boundedness, asymptotic stability and pattern formation of predator–prey systems with density-dependent prey-taxis in a two-dimensional bounded domain with Neumann boundary conditions, where the coefficients of motility (diffusiq‘dfdon) and mobility (prey-taxis) of the predator are correlated through a prey density-dependent motility function. We establish the existence of classical solutions with uniform-in time bound and the global stability of the spatially homogeneous prey-only steady states and coexistence steady states under certain conditions on parameters by constructing Lyapunov functionals. With numerical simulations, we further demonstrate that spatially homogeneous time-periodic patterns, stationary spatially inhomogeneous patterns and chaotic spatio-temporal patterns are all possible for the parameters outside the stability regime. We also find from numerical simulations that the temporal dynamics between linearised system and nonlinear systems are quite different, and the prey density-dependent motility function can trigger the pattern formation.

Pattern formation in a diffusive intraguild predation model with nonlocal interaction effects

In this paper, we investigate the spatiotemporal pattern formation in a diffusive intraguild predation (IGP) model with a nonlocal interaction term in the growth of the shared resource, which extends previous studies of local reaction-diffusion IGP model. We first perform the stability and Hopf bifurcation analyses for the unique positive equilibrium of the corresponding non-spatial system, and give analytical formulas to determine the direction and stability of the bifurcating periodic solutions. Then the linear stability analysis for the nonlocal model shows that the nonlocal interaction is a key mechanism for the formation of Turing patterns. Numerical simulations show that low conversion rate from resource to IG predator can induce stationary Turing patterns, intermediate conversion rate can induce regular oscillatory patterns, and high conversion rate can induce irregular spatiotemporal chaotic patterns for certain diffusive rate. The impact of nonlocal interaction on the resulting patterns with certain diffusive rate is further explored by numerical simulations, which show that nonlocal interaction can induce pattern transition from stationary Turing patterns to non-stationary oscillatory patterns, and even to spatiotemporal chaotic patterns with the increase of the nonlocal interaction tensity. In addition, spatiotemporal chaotic patterns are found in the Turing-Hopf parametric space, which enrich pattern dynamics for diffusive IGP models with nonlocal interactions.

Cross-Diffusion Induced Turing and non-Turing Patterns in Rosenzweig–MacArthur Model

Pattern formation is widely studied in spatio-temporal prey– predator models with only self-diffusion terms. Models with cross-diffusion, besides self-diffusion, take care of the situation in which presence, absence, abundance or scarcity of one species affect the movement of a population of another species in a given domain. Here, we consider cross-diffusion induced pattern formation in a prey–predator model with Rosenzweig–MacArthur type reaction kinetics in a one-dimensional spatial domain. Spatio-temporal prey–predator model with Rosenzweig–MacArthur type reaction kinetics and self-diffusion is unable to generate Turing patterns, rather it produces travelling wave, periodic travelling wave, modulated periodic travelling wave and spatio-temporal chaotic patterns. However, addition of density dependent cross-diffusion leads to satisfaction of Turing instability conditions and generation of stationary Turing patterns. Also, the dynamics of the patterns formed in the self-diffusion model are preserved. Furthermore, cross-diffusion affects the speed of travelling waves produced in the self-diffusion model. Our focus in this work is to investigate the bifurcation of travelling wave solution into Turing patterns and transition of one pattern into another in the presence of cross-diffusion.

Analysis of bifurcation, chaos and pattern formation in a discrete time and space Gierer Meinhardt system

This paper is concerned with the spatiotemporal behaviors of a Gierer–Meinhardt system in discrete time and space form. Through the linear stability analysis, the parametric conditions are gained to ensure the stability of the homogeneous steady state of the system. Based on the bifurcation theory, as well as center manifold theorem, we derive the critical parameter values of the flip, Neimark–Sacker and Turing bifurcation respectively. Besides, the specific parameter expression to form patterns are also determined. In order to identify chaos among regular behaviors, we calculate the Maximum Lyapunov exponents. The results obtained in this paper are illustrated by numerical simulations. From the simulations, we can see some complex dynamics, such as period doubling cascade, invariant cycles, periodic windows, chaotic behaviors, and some striking Turing patterns, e.g. circle, mosaic, spiral, spatiotemporal chaotic patterns and so on, which can be produced by flip-Turing instability, Neimark–Sacker–Turing instability and chaos.

Stationary, non-stationary and invasive patterns for a prey-predator system with additive Allee effect in prey growth

This study deals with the pattern formation scenarios of a reaction-diffusion system of prey-predator interactions with Allee effect in prey growth, Holling type-II functional response and density dependent death rate for predator. The necessary conditions, such as the presence of Hopf and Turing bifurcations, to get pattern formation are rendered and exhaustive numerical simulations are performed to validate the analytical results. Numerical simulations affirm the existence of stationary patterns such as hot spot, cold spot, labyrinthine and mixture of spot and stripe patterns along with the non-stationary patterns such as spatiotemporal chaotic and quasi-periodic patterns in presence of both weak and strong Allee effects in prey growth. Numerical simulations also reveal the coexistence of both species (either stationary or chaotic in nature depending on the values for diffusion coefficients) in contrast to the extinction scenario for the corresponding temporal model. Further, we investigate the invasion of exotic species for the considered model for strong Allee effect. In this case, we find out the patchy invasion scenario apart from the invasion through propagation of continuous population fronts which is a common invasion scenario for deterministic prey-predator models. In the patchy invasion regime, we observe a striking patchy spatial distribution of species in the wake of an expanding population front as well as complete patchy invasion.

Two-dimensional nonlinear modes and frequency combs in bottle microresonators.

We theoretically investigate frequency comb generation in a bottle microresonator accounting for the azimuthal and axial degrees of freedom. We first identify a discrete set of the axial nonlinear modes of a bottle microresonator that appear as tilted resonances bifurcating from the spectrum of linear axial modes. We then study azimuthal modulational instability of these modes and show that families of two-dimensional (2D) soliton states localized both azimuthally and axially bifurcate from them at critical pump frequencies. Depending on detuning, 2D solitons can be stable, form persistent breathers or chaotic spatio-temporal patterns, or exhibit collapse-like evolution.

Spatio-temporal pattern formation in predator-prey systems with fitness taxis

We pose a spatial predator–prey model in which the movement of animals is not purely diffusive, but also contains a drift term in the direction of higher specific growth rates. We refer to this as fitness taxis. We conduct a linear stability analysis of the resulting coupled reaction–advection–diffusion equations and derive conditions under which spatial patterns form. We find that for some parameters the problem is ill posed and short waves grow with unbounded speeds. To eliminate this, we introduce spatial kernels in the model, yielding coupled integro-differential equations, and conduct a similar stability analysis for this system. Through numerical simulation, we find that a variety of patterns can emerge, including stationary spatial patterns, standing and travelling waves, and seemingly chaotic spatio-temporal patterns. We argue that fitness taxis represents a simple and generic extension of diffusive motion, is ecologically plausible, and provides an alternative mechanism for formation of patterns in spatially explicit ecosystem models, with emphasis on non-stationary spatio-temporal dynamics.

A period doubling route to spatiotemporal chaos in a system of Ginzburg-Landau equations for nematic electroconvection

In this paper we investigate the transition from periodic solutions to spatiotemporal chaos in a system of four globally coupled Ginzburg Landau equations describing the dynamics of instabilities in the electroconvection of nematic liquid crystals, in the weakly nonlinear regime. If spatial variations are ignored, these equations reduce to the normal form for a Hopf bifurcation with \begin{document}$O(2) × O(2)$\end{document} symmetry. Both the amplitude system and the normal form are studied theoretically and numerically for values of the parameters including experimentally measured values of the nematic liquid crystal Merck I52. Coexistence of low dimensional and extensive spatiotemporal chaotic patterns, as well as a temporal period doubling route to spatiotemporal chaos, corresponding to a period doubling cascade towards a chaotic attractor in the normal form, and a kind of spatiotemporal intermittency that is characteristic for anisotropic systems are identified and characterized. A low-dimensional model for the intermittent dynamics is obtained by perturbing the eight-dimensional normal form by imperfection terms that break a continuous translation symmetry.

Spatio-temporal pattern formation in Rosenzweig–MacArthur model: Effect of nonlocal interactions

Spatio-temporal pattern formation in reaction–diffusion models of interacting populations is an active area of research due to various ecological aspects. Instability of homogeneous steady-states can lead to various types of patterns, which can be classified as stationary, periodic, quasi-periodic, chaotic, etc. The reaction–diffusion model with Rosenzweig–MacArthur type reaction kinetics for prey–predator type interaction is unable to produce Turing patterns but some non-Turing patterns can be observed for it. This scenario changes if we incorporate non-local interactions in the model. The main objective of the present work is to reveal possible patterns generated by the reaction–diffusion model with Rosenzweig–MacArthur type prey–predator interaction and non-local consumption of resources by the prey species. We are interested in the existence of Turing patterns in this model and in the effect of the non-local interaction on the periodic travelling wave and spatio-temporal chaotic patterns. Global bifurcation diagrams are constructed to describe the transition from one pattern to another one.

Bifurcation, chaos and pattern formation in a space- and time-discrete predator–prey system

The spatiotemporal dynamics of a space- and time-discrete predator–prey system is investigated in this research. The conditions for stable homogeneous stationary state of the system are derived via stability analysis. By using center manifold theorem and bifurcation theory, critical parameter values for flip bifurcation, Hopf bifurcation and Turing bifurcation are determined, respectively. Based on the bifurcation analysis, pattern formation conditions are also provided. Numerical simulations are performed not only to illustrate the theoretical results, but also to exhibit new and complex dynamical behaviors, including period-doubling cascade, invariant circles, periodic windows, chaotic dynamics, and pattern formation. Maximum Lyapunov exponents are calculated to distinguish chaos from regular behaviors. In the routes from bifurcation to chaos, flip-Turing instability and Hopf-Turing instability emerge, capturing the formation of diverse complex patterns, such as mosaic, circle, spiral, spatiotemporal chaotic patterns, and so on. The analysis and results in this research contribute to a new understanding on the relationship among bifurcation, chaos and pattern formation.

Patterns, Bifurcations, Multistability and Hysteresis in an Inhomogeneous Coupled Map Lattice

We investigate local interaction between logistic maps in an inhomogeneous lattice numerically with respect to an inhomogeneity parameter [Formula: see text] and the coupling constant [Formula: see text]. In our model, the inhomogeneity appears in the form of different values of the map parameter at different sites. The phase diagram of the model in the [Formula: see text]–[Formula: see text] plane gives seven qualitatively different patterns. These are: synchronized patterns, steady (fixed in time) patterns with spatial period-two, spatially chaotic together with temporally periodic patterns, spatially coherent accompanied with temporally quasi-periodic patterns, spatial intermittency with temporal period-two patterns, spatiotemporal intermittency patterns or spatiotemporal chaotic patterns. Our system exhibits, tangent bifurcations in the transition from synchronized patterns to steady patterns, period-doubling bifurcation from steady patterns to temporal periodic patterns and Neimark–Sacker (Hopf) bifurcation from steady patterns to temporal quasi-periodic patterns. The system also shows the possibility of multistable attractors and the phenomena of hysteresis for some parameter values. We identify our results using techniques such as time series, space-time plot, Fourier transform, bifurcation diagram, stability analysis.

Time delay can enhance spatio-temporal chaos in a prey–predator model

In this paper we explore how the time delay induced Hopf-bifurcation interacts with Turing instability to determine the resulting spatial patterns. For this study, we consider a delayed prey–predator model with Holling type-II functional response and intra-specific competition among the predators. Analytical criteria for the delay induced Hopf-bifurcation and for the delayed spatio-temporal model are provided with numerical example to validate the analytical results. Exhaustive numerical simulation reveals the appearance of three types of stationary patterns, cold spot, labyrinthine, mixture of stripe-spot and two non-stationary patterns, quasi-periodic and spatio-temporal chaotic patterns. The qualitative features of the patterns for the non-delayed and the delayed spatio-temporal model are the same but their occurrence is solely controlled by the temporal parameters, rate of diffusivity and magnitude of the time delay. It is evident that the magnitude of time delay parameter beyond the Hopf-bifurcation threshold mostly produces spatio-temporal chaotic patterns.

A Simple Feedback Control Approach for Output Modulation of Spatiotemporal Patterns in a Class of Tubular Reactors

Tubular chemical reactors can support spatiotemporal patterns. Pattern modulation in reactive systems may be of major importance for adequate operation of some industrial applications. In this work, we introduce a robust control approach for the output regulation and tracking of spatiotemporal patterns in a class of tubular reactors at a desired position. The control design includes a state estimator of lumped uncertain terms, which is coupled with an inverse model-based feedback control that assigns a desired output closed-loop behavior. The result is an efficient controller that is robust against feed disturbances while introducing good output regulation and tracking properties. The controller is implemented via numerical simulations in three benchmark tubular reactors displaying spatiotemporal patterns: (i) spatiotemporal periodic oscillation in a tubular reactor with axial dispersion, (ii) chaotic spatiotemporal patterns in a tubular reactor with recycle, and (iii) spatiotemporal periodic oscillations in the Belousov–Zhabotinseky reaction–diffusion system.

Intermittency, quasiperiodicity and chaos in probe-induced ferroelectric domain switching

Ferroelectric domain switching on the surface of a lithium niobate thin film can be induced by the tip of a scanning probe microscope, and gives rise to both regular and chaotic spatiotemporal patterns. Moreover, the long-range interactions that govern these phenomena can be tuned by varying temperature, humidity, domain spacing and tip bias.

Chimera states as chaotic spatiotemporal patterns

Chimera states are a recently new discovered dynamical phenomenon that appears in arrays of nonlocally coupled oscillators and displays a spatial pattern of coherent and incoherent regions. We report here an additional feature of this dynamical regime: an irregular motion of the position of the coherent and incoherent regions, i.e., we reveal the nature of the chimera as a spatiotemporal pattern with a regular macroscopic pattern in space, and an irregular motion in time. This motion is a finite-size effect that is not observed in the thermodynamic limit. We show that on a large time scale, it can be described as a Brownian motion. We provide a detailed study of its dependence on the number of oscillators N and the parameters of the system.

Transition to spatiotemporal chaos can resolve the paradox of enrichment

The dynamics and stability of interacting populations in connection to spatial phenomena such as pattern formation and spatiotemporal chaos have recently become a focus of intensive research in theoretical ecology. In this paper, we demonstrate a surprising relation between the long-standing enigma known as Rosenzweig’s paradox of enrichment and the formation of chaotic spatiotemporal patterns in an ecological community. Using two different spatially explicit models (a standard diffusion–reaction system and a diffusion–reaction system with cutoff at low population densities), we show by means of computer simulations that transition to spatiotemporal chaos can prevent species extinction in a situation when it would be expected in the case of regular dynamics. The patterns arising in our models are self-organized, and not induced directly by pre-existing spatial heterogeneity of the environment. We also show that the type of the system’s response to enrichment essentially depends on the system size and on the rates of eutrophication.

Stationary spatially complex solutions in cross‐flow reactors with two reactions

AbstractFormation of stationary spatially multiperiodic or even chaotic patterns is analyzed for a simple model of a cross‐flow reactor with two consecutive reactions and realistically high Le and Pe. Spatial patterns emerge much like dynamic temporal patterns in a mixed system of the same kinetics. The sequence of period doubling bifurcations is determined for the corresponding ODE system and is completely confirmed by direct numerical simulations of the full PDE model. The incorporation of a slow nondiffusing inhibitor led to chaotic spatiotemporal patterns.