Spatiotemporal chaos in an electric current driven ionic reaction-diffusion system

Two types of transitions from the time-periodic spatiotemporal patterns to chaotic ones in the spatially one-dimensional ionic reaction-diffusion system forced either with direct or alternating electric field are described and analyzed by numerical techniques. An ionic version of the Brusselator kinetic scheme is considered. The Karhunen-Loeve decomposition technique is shown to be a possible tool for the global representation of dynamic behavior, but fails as a tool in the identification of the route of transition to chaos in the case of direct current forcing. Higher dimensional chaos with two positive Lyapunov exponents has been identified for the case of alternating current forcing. Results of the Karhunen-Loeve analysis are compared to results of classical analysis of local time series (attractor dimensions, Lyapunov exponents).     

Spatiotemporal chaos control in two-wave driven systems

Spatiotemporal chaos control is considered by taking a one-dimensional driven/damped nonlinear drift-wave equation as a model. We apply an additional sinusoidal wave to suppress spatiotemporal chaos, and the system becomes a two-sinusoidal-wave driven system (the original driving wave with frequency ω and an additional controlling wave with frequency Ω). Numerical simulations show that when the frequency of the controlling wave is in the proper range, spatiotemporal chaos can be modified into a regular state where the amplitudes of all modes vary periodically with frequency Ω-ω while the phases of all modes evolve quasi-periodically with a running frequency Ω overlapped by a small modulation of frequency Ω-ω. The physical reason for this peculiar phenomenon is attributed to a frequency entrainment in the competition of the two external waves.     

Spatiotemporal chaos and noise

Low-dimensional chaotic dynamical systems can exhibit many characteristic properties of stochastic systems, such as broad Fourier spectra. They are distinguishable from stochastic processes through finite values for their dimension, Lyapunov exponents, and Kolmogorov-Sinai entropy. We discuss how these characteristic observables are modified in spatiotemporal chaotic systems like. coupled map lattices. We analyze with the help of Lyapunov concepts how the stochastic limit is approached and how these properties can be observed directly through local dimension measurements from reconstructed time series. Finally, we discuss the interaction of spatiotemporal attractors with external noise and possible connections to problems of pattern selection and stability.     

Segregation in reaction-diffusion systems

Segregation of reactants in reaction-diffusion systems is a spatial structure that can be formed either as a result of a dynamical process or as an initially prepared system. In this paper we review our recent results on both such systems. First we study the dynamic segregation at a single trap, in particular in the presence of fields and disorder. Then we study properties of the dynamic reaction front produced due to initial segregation of the reactants in the A + B→C system. Both systems are shown to exhibit anomalous kinetic properties.     

Nonlocal boundary dynamics of traveling spots in a reaction-diffusion system

The boundary integral method is extended to derive a closed integro-differential equation applicable to computation of the shape and propagation speed of a steadily moving spot and to the analysis of dynamic instabilities in the sharp boundary limit. Expansion of the boundary integral near the locus of traveling instability in a standard reaction-diffusion model proves that the bifurcation is supercritical whenever the spot is stable to splitting. Thus, stable propagating spots do already exist in the basic activator-inhibitor model, without additional long-range variables.     

Effect of noise on defect chaos in a reaction-diffusion model

The influence of noise on defect chaos due to breakup of spiral waves through Doppler and Eckhaus instabilities is investigated numerically with a modified Fitzhugh-Nagumo model. By numerical simulations we show that the noise can drastically enhance the creation and annihilation rates of topological defects. The noise-free probability distribution function for defects in this model is found not to fit with the previously reported squared-Poisson distribution. Under the influence of noise, the distributions are flattened, and can fit with the squared-Poisson or the modified-Poisson distribution. The defect lifetime and diffusive property of defects under the influence of noise are also checked in this model.     

Karhunen-Loeve local characterization of spatiotemporal chaos in a reaction-diffusion system

By computing the Karhunen-Loeve decomposition (KLD) correlation length xi(KLD) of a reaction-diffusion system in the extensive chaos regime, we show that it is a sensitive measure of spatial dynamical inhomogeneities. It reveals substantial spatial nonuniformity of the dynamics at the boundaries and can also detect slow spatial variations in system parameters. The intensive length xi(KLD) can be easily computed from small local subsystems and is found to have a similar parametric dependence as the two-point correlation length computed over the full system size.     

Localized patterns in reaction-diffusion systems

We discuss a variety of experimental and theoretical studies of localized stationary spots, oscillons, and localized oscillatory clusters, moving and breathing spots, and localized waves in reaction-diffusion systems. We also suggest some promising directions for future research in this area.     

Anomalous dynamics in reaction-diffusion systems

Summary We review recent developments in the study of the diffusion reaction systems of the type A+B→C in which the reactants are initially separated. We consider the case where the A and B particles are initially placed uniformly in Euclidean space atx>0 andx<0, respectively. We find that whereas ford≥2 the mean-field exponent characterizes the width of the reaction zone, fluctuations are relevant in the one-dimensional system. We also presented analytical and numerical results for the reaction rate on fractals and percolation systems. Paper presented at the I International Conference on Scaling Concepts and Complex Fluids, Copanello, Italy, July 4–8, 1994.     

时空混沌系统的主动-间隙耦合同步

提出了离散系统中的主动-间隙耦合同步方法。该方法由同步相和自治相组成,在同步相,同步方案使得混沌系统趋于同步,而在自治相,两系统间的误差将迅速放大,导致同步失去。但只要同步相足够大,最终可实现系统的完全同步。从理论上讨论了同步条件,并在数值实验上讨论了同步相与耦合强度的关系。     

Hysteresis in one-dimensional reaction-diffusion systems

We introduce a simple nonequilibrium model for a driven diffusive system with nonconservative reaction kinetics in one dimension. The steady state exhibits a phase with broken ergodicity and hysteresis which has no analog in systems investigated previously. We identify the main dynamical mode, viz., the random motion of a shock in an effective potential, which provides a unified framework for understanding phase coexistence as well as ergodicity breaking. This picture also leads to the exact phase diagram of the system.     

Spatiotemporal chaos: the microscopic perspective

Extended nonequilibrium systems can be studied in the framework of field theory or from the dynamical systems perspective. Here we report numerical evidence that the sum of a well-defined number of instantaneous Lyapunov exponents for the complex Ginzburg-Landau equation is given by a simple function of the space average of the square of the macroscopic field. This relationship follows from an explicit formula for the time-dependent values of almost all the exponents.     

Spatiotemporal communication with synchronized optical chaos

We propose a model system that allows communication of spatiotemporal information using an optical chaotic carrier waveform. The system is based on broad-area nonlinear optical ring cavities, which exhibit spatiotemporal chaos in a wide parameter range. Message recovery is possible through chaotic synchronization between transmitter and receiver. Numerical simulations demonstrate the feasibility of the proposed scheme, and the benefit of the parallelism of information transfer with optical wave fronts.     

Analysis of domain-solutions in reaction-diffusion systems

We investigate a standard model for bistable reaction-diffusion-systems, which shares characteristic properties with the van-der-Pol oscillator for distributed generators and the FitzHugh-Nagumo system. In this system we study the effect of a long ranging inhibitor. As a main result we show the existence of two inhomogeneous stationary solutions—the smaller one is always a saddle which corresponds to a critical nucleus, while the larger one arises as astable solution. In carrying out the linear stability analysis for these solutions, we have to treat the Schrödinger-equation for a double-well potential. This is done approximately by a supersymmetric approach which yields the eigenvalues and eigenfunctions of the Schrödinger-equation. Furthermore we compare our analytical findings with numerical results-especially the occurrence of oscillating solutions is shown.     

Quantum statistics of reaction-diffusion systems

A strictly quantum-statistical theory of inhomogeneous reactions is presented. The treatment is based on the theory of multi-channel reactive scattering. For the configuration probabilities of the reactants a system of reaction-diffusion equations is obtained. The expressions for the diffusion tensor and the reaction-rate coefficients are established in terms of microscopic parameters characteristic of the reacting species.