Spatiotemporal Chaotic Behavior Research Articles

Order and Disorder in a Cyclically Competitive Ecological Community

We consider a deterministic model of a rock-paper-scissors cyclic ecological system. The model accounts for species mobility via Fickian diffusion, as well as interspecies interactions describing the cyclic competition scheme. The system admits both single-species equilibria (saddles) and a three-species coexistence equilibrium state. We focus on the regime where coexistence is unstable. When there is no spatial dependence, the solution of the corresponding ordinary differential equation system admits an attracting heteroclinic cycle consisting of orbits connecting the three single-species saddles in the sequence, to which most solutions with physical initial conditions asymptote. When there is spatial dependence (partial differential equation system), we consider mixed initial conditions (all species cohabit the same spatial region), as well as patch initial conditions, where each species is isolated from the other two, and show that, when interspecies competition is not strong, spatiotemporal chaotic behavior generally occurs. We propose a mechanism for the development of chaos—patch splitting—whereby sufficiently large patches are repeatedly split by the diffusion of competitors into the patch. However, when there is strong interspecies competition, ordered spatiotemporal patterns can occur. We consider both 1D patterns, corresponding to a community confined to a thin, circular annulus, and 2D geometries, simulating a petri dish. We show that, in 1 dimension, traveling arrays of single-species patches, as well as modulated traveling waves, consisting of patches which periodically expand and contract (breather modes), exist. In 2 dimensions, spirals, as well as localized patches chasing each other in the sequence, can occur. We consider length and temporal scales appropriate for a bacterial community in a laboratory setting, qualitatively modeling observed Escherichia coli cyclic bacterial systems.

Prediction error growth in a more realistic atmospheric toy model with three spatiotemporal scales

Abstract. This article studies the growth of the prediction error over lead time in a schematic model of atmospheric transport. Inspired by the Lorenz (2005) system, we mimic an atmospheric variable in one dimension, which can be decomposed into three spatiotemporal scales. We identify parameter values that provide spatiotemporal scaling and chaotic behavior. Instead of exponential growth of the forecast error over time, we observe a more complex behavior. We test a power law and the quadratic hypothesis for the scale-dependent error growth. The power law is valid for the first days of the growth, and with an included saturation effect, we extend its validity to the entire period of growth. The theory explaining the parameters of the power law is confirmed. Although the quadratic hypothesis cannot be completely rejected and could serve as a first guess, the hypothesis's parameters are not theoretically justifiable in the model. In addition, we study the initial error growth for the ECMWF forecast system (500 hPa geopotential height) over the 1986 to 2011 period. For these data, it is impossible to assess which of the error growth descriptions is more appropriate, but the extended power law, which is theoretically substantiated and valid for the Lorenz system, provides an excellent fit to the average initial error growth of the ECMWF forecast system. Fitting the parameters, we conclude that there is an intrinsic limit of predictability after 22 d.

On a generalized doubly parabolic Keller–Segel system in one spatial dimension

We study a doubly parabolic Keller–Segel system in one spatial dimension, with diffusions given by fractional Laplacians. We obtain several local and global well-posedness results for the subcritical and critical cases (for the latter we need certain smallness assumptions). We also study dynamical properties of the system with added logistic term. Then, this model exhibits a spatio-temporal chaotic behavior, where a number of peaks emerge. In particular, we prove the existence of an attractor and provide an upper bound on the number of peaks that the solution may develop. Finally, we perform a numerical analysis suggesting that there is a finite time blowup if the diffusion is weak enough, even in presence of a damping logistic term. Our results generalize on one hand the results for local diffusions, on the other the results for the parabolic–elliptic fractional case.

Spatiotemporal chaos synchronization between uncertain complex networks with diverse structures

Spatiotemporal chaos synchronization between uncertain complex networks with diverse structures is investigated. The identification law of unknown parameters and the adaptive law of the configuration matrix element in state equations of network nodes are determined based on stability theory, and the conditions of realizing spatiotemporal chaos synchronization between uncertain complex networks with different structures are discussed and obtained. Further, the Fisher–Kolmogorov system with spatiotemporal chaotic behavior is taken as the nodes of drive and response networks to imitate the experiment. It is found that the synchronization performance between two networks is very stable.

On the performance of anisotropic mesh adaptation for scroll wave turbulence dynamics in reaction–diffusion systems

Nonlinear reaction–diffusion systems are widely employed to study the spatio-temporal chaotic behavior that occurs in excitable media such as cardiac tissue where sufficiently strong perturbations can excite nonlinear propagating waves which can form spiral waves in two dimensions or scroll waves in three dimensions. The numerical simulation of these waves calls for grids that are extremely fine over the whole computational domain to accurately predict the trajectories and multiplication of wave fronts and therefore leads to huge computational challenges. Mesh adaptation methods can reduce the number of degrees of freedom required for a given accuracy but they also have a cost and it is not clear if they are competitive with respect to very fine uniform meshes. Previous mesh adaptation techniques applied to spatio-temporal chaotic behavior have been mostly limited to the two-dimensional case. The purpose of this paper is to explore the efficiency of a three-dimensional anisotropic finite element mesh adaptivity for simulating scroll wave turbulence. The computational efficiency of the proposed method is assessed using reference solutions obtained on a uniform refined mesh with more than 44 millions degrees of freedom. The proposed method reduces significantly the number of elements leading to huge saving in memory as well as in computational time. Examples of the dynamics of ventricular fibrillation in cardiac tissue will be presented illustrating the performance of the overall methodology.

Projective synchronization of the small-world delayed network with uncertainty

Spatiotemporal chaotic systems are taken as nodes to constitute the small-world delayed network with uncertainty, and projective synchronization of the network is researched. The control input of the network and the adaptive law of adjustment parameters are designed based on Lyapunov theorem. Concretely, the Burgers systems with spatiotemporal chaotic behavior in physics domain are taken as nodes to constitute the complex network, and the Fisher–Kolmogorov system is taken as the target system. The simulation results show that the synchronization performance of the network is very stable.

Projective synchronization of a class of complex network based on high-order sliding mode control

Projective synchronization of a class of complex networks is investigated using second-order sliding mode control. The sliding surface and the control input are designed based on stability theory. The Burgers system with spatiotemporal chaotic behavior in the physics domain is taken as nodes to constitute the complex network, and the Fisher–Kolmogorov system is taken as the tracking target. The artificial simulation results show that the synchronization technique is effective.

Chaos for cardiac arrhythmias through a one-dimensional modulation equation for alternans

Instabilities in cardiac dynamics have been widely investigated in recent years. One facet of this work has studied chaotic behavior, especially possible correlations with fatal arrhythmias. Previously chaotic behavior was observed in various models, specifically in the breakup of spiral and scroll waves. In this paper we study cardiac dynamics and find spatiotemporal chaotic behavior through the Echebarria-Karma modulation equation for alternans in one dimension. Although extreme parameter values are required to produce chaos in this model, it seems significant mathematically that chaos may occur by a different mechanism from previous observations.

Hyperlabyrinth chaos: From chaotic walks to spatiotemporal chaos

In this paper we examine a very simple and elegant example of high-dimensional chaos in a coupled array of flows in ring architecture that is cyclically symmetric and can also be viewed as an N-dimensional spatially infinite labyrinth (a "hyperlabyrinth"). The scaling laws of the largest Lyapunov exponent, the Kaplan-Yorke dimension, and the metric entropy are investigated in the high-dimensional limit (3<N<or=101) together with its routes to chaos. It is shown that by tuning the single bifurcation parameter b that governs the dissipation and the number of coupled systems N, the attractor dimension can span the entire range of 0 to N including Hamiltonian (conservative) hyperchaos in the limit of b=0 and, furthermore, spatiotemporal chaotic behavior. Finally, stability analysis reveals interesting and important changes in the dynamics, whether N is even or odd.

Generalized chaotic synchronization in coupled Ginzburg-Landau equations

Generalized synchronization is analyzed in unidirectionally coupled oscillatory systems exhibiting spatiotemporal chaotic behavior described by Ginzburg-Landau equations. Several types of coupling between the systems are analyzed. The largest spatial Lyapunov exponent is proposed as a new characteristic of the state of a distributed system, and its calculation is described for a distributed oscillatory system. Partial generalized synchronization is introduced as a new type of chaotic synchronization in spatially nonuniform distributed systems. The physical mechanisms responsible for the onset of generalized chaotic synchronization in spatially distributed oscillatory systems are elucidated. It is shown that the onset of generalized chaotic synchronization is described by a modified Ginzburg-Landau equation with additional dissipation irrespective of the type of coupling. The effect of noise on the onset of a generalized synchronization regime in coupled distributed systems is analyzed.

Sliding Dynamics of the Frenkel-Kontorova Model

We investigate the dynamics of the Frenkel-Kontorova model with inertia and damping under a uniform driving force. We consider the dependence of the velocity-force characteristic, and the corresponding sliding states on the damping: In the overdamped limit the asymptotic solution with nonzero sliding velocity is unique and periodic. Periodic solutions and their stability can also be calculated for decreasing damping, where the dynamics is dominated by resonance effects of the underlying potential with the phonon modes of the chain. The asymptotic velocity is not unique, leading to hysteresis in the velocity-force characteristic. In addition to periodic sliding states, which lose stability for decreasing damping, quasiperiodic or spatio-temporal chaotic behavior occurs.

Growth morphologies in diffusion fields

Growth morphologies in diffusion fields are investigated by two types of numerical methods: solution of the integro-differential equation of a step profile and Monte Carlo simulation of a lattice gas model. The former method is applied to study the effect of kinetics on dendritic growth, and the capillarity-controlled scalling behavior is found to hold for a small but non-vanishing kinetic coefficient. The latter methods is suitable for studying the effect of noise at the atomic level, and applied to the solidification by adsorption of atoms at the step. By an irreversible solidification, shot-noise is frozen in the aggregation to produce a fractal structure. Even when the noise is reduced, the step profile is found to show a spatio-temporal chaotic behavior after the straight step becomes unstable.

Spatiotemporal chaotic behavior induced by the multiplicative noise process

The time-dependent Ginzburg-Landau equation is studied under the influence of the multiplicative noise process. By analytically studying the linear stability of the homogeneous state and numerically solving the basic equation, we show that a spatiotemporal chaotic behavior is realized in the present system.