Soft-mode turbulence (SMT) is a recently discovered type of spatiotemporal chaos observed in the electrohydrodynamic instability (EHD) of a nematic liquid crystal with homeotropic alignment. Its novelty is that it occurs as the first supercritical bifurcation from a stable stationary state in the weakly nonlinear regime of EHD. A particle, small compared to the characteristic length of the macroscopic flow, injected in SMT travels with random velocity similar to a particle in Brownian motion. We find that the particle trajectory exhibits anomalous Brownian motion such as pumped flight and that the macroscopic diffusion constant due to weak turbulence is about 103 times larger than the diffusion constant associated with the rest state. The stochastic properties of SMT are discussed.

In this paper a parallel Hash algorithm construction based on the Chebyshev Halley methods with variable parameters is proposed and analyzed. The two core characteristics of the recommended algorithm are parallel processing mode and chaotic behaviors. Moreover in this paper, an algorithm for one way hash function construction based on chaos theory is introduced. The proposed algorithm contains variable parameters dynamically obtained from the position index of the corresponding message blocks. Theoretical analysis and computer simulation indicate that the algorithm can assure all performance requirements of hash function in an efficient and flexible style and secure against birthday attacks or meet-in-the-middle attacks, which is good choice for data integrity or authentication.

Mechanisms and scenarios of pattern formation in predator-prey systems have been a focus of many studies recently as they are thought to mimic the processes of ecological patterning in real-world ecosystems. Considerable work has been done with regards to both Turing and non-Turing patterns where the latter often appears to be chaotic. In particular, spatiotemporal chaos remains a controversial issue as it can have important implications for population dynamics. Most of the results, however, were obtained in terms of 'traditional' predator-prey models where the per capita predation rate depends on the prey density only. A relatively new family of ratio-dependent predator-prey models remains less studied and still poorly understood, especially when space is taken into account explicitly, in spite of their apparent ecological relevance. In this paper, we consider spatiotemporal pattern formation in a ratio-dependent predator-prey system. We show that the system can develop patterns both inside and outside of the Turing parameter domain. Contrary to widespread opinion, we show that the interaction between two different type of instability, such as the Turing-Hopf bifurcation, does not necessarily lead to the onset of chaos; on the contrary, the emerging patterns remain stationary and almost regular. Spatiotemporal chaos can only be observed for parameters well inside the Turing-Hopf domain. We then investigate the relative importance of these two instability types on the onset of chaos and show that, in a ratio-dependent predatorprey system, the Hopf bifurcation is indeed essential for the onset of chaos whilst the Turing instability is not.

Different transient-chaos related phenomena of spatiotemporal systems are reviewed. Special attention is paid to cases where spatiotemporal chaos appears in the form of chaotic transients only. The asymptotic state is then spatially regular. In systems of completely different origens, ranging from fluid dynamics to chemistry and biology, the average lifetimes of these spatiotemporal transients are found, however, to grow rapidly with the system size, often in an exponential fashion. For sufficiently large spatial extension, the lifetime might turn out to be larger than any physically realizable time. There is increasing numerical and experimental evidence that in many systems such transients mask the real attractors. Attractors may then not be relevant to certain types of spatiotemporal chaos, or turbulence. The observable dynamics is governed typically by a high-dimensional chaotic saddle. We review the origen of exponential scaling of the transient lifetime with the system size, and compare this with a similar scaling with system parameters known in low-dimensional problems. The effect of weak noise on such supertransients is discussed. Different crisis phenomena of spatiotemporal systems are presented and fractal properties of the chaotic saddles underlying high-dimensional supertransients are discussed. The recent discovery according to which turbulence in pipe flows is a very long lasting transient sheds new light on chaotic transients in other spatially extended systems.

Cardiac arrhythmias such as ventricular tachycardia (VT) or ventricular fibrillation (VF) are the leading cause of death in the industrialised world. There is a growing consensus that these arrhythmias arise because of the formation of spiral waves of electrical activation in cardiac tissue; unbroken spiral waves are associated with VT and broken ones with VF. Several experimental studies have been carried out to determine the effects of inhomogeneities in cardiac tissue on such arrhythmias. We give a brief overview of such experiments, and then an introduction to partialdifferential-equation models for ventricular tissue. We show how different types of inhomogeneities can be included in such models, and then discuss various numerical studies, including our own, of the effects of these inhomogeneities on spiral-wave dynamics. The most remarkable qualitative conclusion of our studies is that the spiral-wave dynamics in such systems depends very sensitively on the positions of these inhomogeneities.

We study the spatiotemporal dynamics, in one and two spatial dimensions, of two complex fields which are the two components of a vector field satisfying a vector form of the complex Ginzburg-Landau equation. We find synchronization and generalized synchronization of the spatiotemporally chaotic dynamics. The two kinds of synchronization can coexist simultaneously in different regions of the space, and they are mediated by localized structures. A quantitative characterization of the degree of synchronization is given in terms mutual information measures.

This paper investigates a globally nonlocal coupled map lattice. A rigorous proof to the existence of chaos in the scene of Li–Yorke in that system is presented in terms of the Marotto theorem. Analytical sufficient conditions under which the system is chaotic, and has synchronous behaviors are determined, respectively. The wider regions associated with chaos and synchronous behaviors are shown by simulations. Spatiotemporal chaos, synchronous chaos and some other synchronous behaviors such as fixed points, 2-cycles and 22-cycles are also shown by simulations for some values of the parameters.► A globally nonlocal coupled map lattice is introduced. ► A sufficient condition for the existence of Li–Yorke chaos is determined. ► A sufficient condition for synchronous behaviors is obtained.

A mathematically simple example of a high-dimensional (many-species) Lotka-Volterra model that exhibits spatiotemporal chaos in one spatial dimension is described. The model consists of a closed ring of identical agents, each competing for fixed finite resources with two of its four nearest neighbors. The model is prototypical of more complicated models in its quasiperiodic route to chaos (including attracting 3-tori), bifurcations, spontaneous symmetry breaking, and spatial pattern formation.

A simple model for nuclear reactor is proposed. With increasing the fuel concentration, our minimal model shows two successive phases; subcritical and supercritical. In subcritical regime, the neutron population grows with increasing the fuel concentration. In the supercritical state, the Lyapunov exponent is positive implying that the neutron diffusion phenomena are spatiotemporal chaos. In the present paper, the infinite multiplication factor curve is qualitatively reproduced for fuel concentration. We have derived the critical fuel concentration based on the Lyapunov exponents. The basic objective of the work is to improve the prediction of the critical neutron population with respect to the fuel concentration.

Properties of the complex Ginzburg-Landau equation with drift are studied focusing on the Benjamin-Feir stable regime. On a finite interval with Neumann boundary conditions the equation exhibits bistability between a spatially uniform time-periodic state and a variety of nonuniform states with complex time dependence. The origen of this behavior is identified and contrasted with the bistable behavior present with periodic boundary conditions and no drift.

Every sixth death in industrialised countries occurs because of cardiac arrhythmias like ventricular tachycardia (VT) and ventricular fibrillation (VF). There is growing consensus that VT is associated with an unbroken spiral wave of electrical activation on cardiac tissue but VF with broken waves, spiral turbulence, spatiotemporal chaos and rapid, irregular activation. Thus spiral-wave activity in cardiac tissue has been studied extensively. Nevertheless many aspects of such spiral dynamics remain elusive because of the intrinsically high-dimensional nature of the cardiac-dynamical system. In particular, the role of tissue heterogeneities in the stability of cardiac spiral waves is still being investigated. Experiments with conduction blocks in cardiac tissue yield a variety of results: some suggest that blocks can eliminate VF partially or completely, leading to VT or quiescence, but others show that VF is unaffected by obstacles. We propose theoretically that this variety of results is a natural manifestation of a fractal boundary that must separate the basins of the attractors associated, respectively, with VF and VT. We substantiate this with extensive numerical studies of Panfilov and Luo-Rudy I models, where we show that the suppression of VF depends sensitively on the position, size, and nature of the inhomogeneity.

In this paper we show that the analysis of the dynamics in localized regions, i.e., sub-systems can be used to characterize the chaotic dynamics and the synchronization ability of the spatiotemporal systems. Using noisy scalar time-series data for driving along with simultaneous self-adaptation of the control parameter representative control goals like suppressing spatiotemporal chaos and synchronization of spatiotemporally chaotic dynamics have been discussed.

We find that the global symbolic dynamics of a diffusively coupled map lattice (CML) is wellapproximated by a very small local model for weak to moderate coupling strengths. A local symbolic model is a truncation of the full symbolic model to one that considers only a single element and a few neighbors. Using interval analysis we give rigorous results for a range of coupling strengths and different local model widths. Examples are presented of extracting a local symbolic model from data and of controlling spatiotemporal chaos.

The experimental study of electroconvection in a homeotropically aligned nematic ͑MBBA͒ is reported. The system undergoes a supercritical bifurcation ''rest state-spatiotemporal chaos.'' The chaos is caused by longwavelength modulation of the orientation of convective rolls. For the first time the observations both below and beyond the Lifshitz point are accompanied by quantitative analysis of temporal autocorrelation functions of turbulent modes. The dependence of the correlation time on the control parameter is obtained. A secondary bifurcation from normal to abnormal rolls is discussed. ͓S1063-651X͑97͒51412-7͔ PACS number͑s͒: 47.27.Cn, 47.52.ϩj, 47.65.ϩa, 61.30.Gd FIG. 2. Inverse correlation time vs below the Lifshitz point ͓͑a͒ f ϭ500 Hz͔ and beyond it ͓͑b͒ f ϭ2000 Hz͔.

Surface wave patterns are investigated experimentally in a system geometry that has become a paradigm of quantum chaos: the stadium billiard. Linear waves in bounded geometries for which classical ray trajectories are chaotic are known to give rise to scarred patterns. Here, we utilize parametrically forced surface waves (Faraday waves), which become progressively nonlinear beyond the wave instability threshold, to investigate the subtle interplay between boundaries and nonlinearity. Only a subset (three main types) of the computed linear modes of the stadium are observed in a systematic scan. These correspond to modes in which the wave amplitudes are strongly enhanced along paths corresponding to certain periodic ray orbits. Many other modes are found to be suppressed, in general agreement with a prediction by Agam and Altshuler based on boundary dissipation and the Lyapunov exponent of the associated orbit. Spatially asymmetric or disordered (but time-independent) patterns are also found even near onset. As the driving acceleration is increased, the time-independent scarred patterns persist, but in some cases transitions between modes are noted. The onset of spatiotemporal chaos at higher forcing amplitude often involves a nonperiodic oscillation between spatially ordered and disordered states. We characterize this phenomenon using the concept of pattern entropy. The rate of change of the patterns is found to be reduced as the state passes temporarily near the ordered configurations of lower entropy. We also report complex but highly symmetric (time-independent) patterns far above onset in the regime that is normally chaotic.

We generalize the concept of geometrical resonance to perturbed sine-Gordon, Nonlinear Schr€ odinger, u 4 , 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. A short report of some of our results has been published in [Europhys. Lett. 64 (2003) 743].

We study the spatiotemporal dynamics, in one and two spatial dimensions, of two complex fields which are the two components of a vector field satisfying a vector form of the complex Ginzburg-Landau equation. We find synchronization and generalized synchronization of the spatiotemporally chaotic dynamics. The two kinds of synchronization can coexist simultaneously in different regions of the space, and they are mediated by localized structures. A quantitative characterization of the degree of synchronization is given in terms of mutual information measures.

We introduce a measure to quantify spatiotemporal turbulence in extended systems. It is based on the statistical analysis of a coherent structure decomposition of the evolving system. Applied to a cellular excitable medium and a reaction-diffusion model describing the oxidation of CO on Pt͑100͒, it reveals power-law scaling of the size distribution of coherent space-time structures for the state of spiral turbulence. The coherent structure decomposition is also used to define an entropy measure, which sharply increases in these systems at the transition to turbulence.

The dynamical behavior of a large one-dimensional system obeying the cubic complex Ginzburg-Landau equation is studied numerically as a function of parameters near a supercritical bifurcation. Two types of chaotic behavior can be distinguished beyond the Benjamin-Feir instability, a phase turbulence regime with a conserved phase winding number and no phase dislocations (space-time defects), and a defect regime with a nonzero density of defects. The transition between the two can either be continuous or discontinuous (hysteretic), depending on parameters. The spatial decay of the phase correlation function is inferred to be exponential in both regimes, with a sharp decrease of the correlation length upon entering the defect phase. The temporal decay of correlations is exponential in the defect regime.

We study spectral properties of three-dimensional photonic crystals formed by a periodic array of air cubes separated by a thin film of optically dense dielectric material. The thickness δ of the dielectric component is assumed to be small, whereas its permittivity ε is large. The spectrum of the photonic crystal is studied as δ → 0 while εδ = η −1 is kept constant. Under the additional condition of a vanishing normal component of the magnetic field on the surface of the optically dense dielectric film, we carry out a thorough analytical and numerical analysis of the spectrum of the photonic crystal. In particular, for small η we obtain the linear and quadratic approximations of the spectral bands and, given δ, we estimate the minimum value of ε for which at least one spectral gap opens up.

The imbalance of the boundary energy flow due to energy injection at one end and a nonlinear van der Pol boundary condition at the other end of the spatial one-dimensional interval can cause chaotic vibration of the linear wave equation [Chen et al., 1998b, 1998c]. However, such chaotic vibration is isotropic with respect to space and time because the two associated families of characteristics both propagate with the same speed and, thus, the "strength of chaos" is the same along both x and t directions. In this paper, we show that by including a mixed partial derivative linear energy transport term in the wave equation, nonlinearity in the van der Pol boundary condition can also cause chaotic vibration (without energy injection from the other end). Two new families of characteristics now travel with different speeds, leading to strong mixing of waves and nonisotropic spatiotemporal chaos. Parameter range for the route to chaos, including period-doubling, homoclinic orbits...

Bred vectors are a type of finite perturbation used in prediction studies of atmospheric models that exhibit spatially extended chaos. We study the structure, spatial correlations, and the growthrates of logarithmic bred vectors (which are constructed by using a given norm). We find that, after a suitable transformation, logarithmic bred vectors are roughly piecewise copies of the leading Lyapunov vector. This fact allows us to deduce a scaling law for the bred vector growth rate as a function of their amplitude. In addition, we relate growth rates with the spectrum of Lyapunov exponents corresponding to the most expanding directions. We illustrate our results with simulations of the Lorenz '96 model.

In this work we perform a detailed study of the scaling properties of Lyapunov vectors (LVs) for two different one-dimensional Hamiltonian lattices: the Fermi-Pasta-Ulam and Φ 4 models. In this case, characteristic (also called covariant) LVs exhibit qualitative similarities with those of dissipative lattices but the scaling exponents are different and seemingly nonuniversal. In contrast, backward LVs (obtained via Gram-Schmidt orthonormalizations) present approximately the same scaling exponent in all cases, suggesting it is an artificial exponent produced by the imposed orthogonality of these vectors. We are able to compute characteristic LVs in large systems thanks to a 'bit reversible' algorithm, which completely obviates computer memory limitations.

The characterization of chaotic spatiotemporal dynamics has been studied for a representative nonlinear autocatalytic reaction mechanism coupled with diffusion. This has been carded out by an analysis of the Lyapunov spectrum in spatially localised regions. The linear scaling relationships observed in the invariant measures as a function of the sub-system size have been utilized to assess the controllability, stability and synchronization properties of the chaotic dynamics. The dynamical synchronization properties of this high-dimensional system has been analyzed using suitable Lyapunov functionals. The possibility of controlling spatiotemporal chaos for relevant objectives using available noisy scalar time-series data with simultaneous selfadaptation of the control parameter(s) has also been discussed.

We study Lyapunov vectors (LVs) corresponding to the largest Lyapunov exponents in systems with spatiotemporal chaos. We focus on characteristic LVs and compare the results with backward LVs obtained via successive Gram-Schmidt orthonormalizations. Systems of a very different nature such as coupled-map lattices and the (continuous-time) Lorenz '96 model exhibit the same features in quantitative and qualitative terms. Additionally we propose a minimal stochastic model that reproduces the results for chaotic systems. Our work supports the claims about universality of our earlier results [I. G. Szendro et al., Phys. Rev. E 76, 025202(R) (2007)] for a specific coupled-map lattice.

We generalize the concept of geometrical resonance to perturbed sine-Gordon, Nonlinear Schr€ odinger, u 4 , 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. A short report of some of our results has been published in [Europhys. Lett. 64 (2003) 743].

The control of chaos ͑suppression and enhancement͒ of a damped pendulum subjected to two perpendicular periodic excitations of its pivot ͑one chaos inducing and the other chaos controlling͒ is investigated. Analytical ͑Melnikov analysis͒ and numerical ͑Lyapunov exponents͒ results show that the initial phase difference between the two excitations plays a fundamental role in the control scenario. We demonstrate the effectiveness of the method in suppressing spatiotemporal chaos of chains of identical chaotic coupled pendula where homogeneous regularization is obtained under localized control on a minimal number of pendula. Additionally, we demonstrate the robustness of the control scenario against changes in the coupling function. In particular, synchronization-induced homogeneous regularization of chaotic chains can be highly enhanced by considering time-varying couplings instead of stationary couplings. Ε 0.1 , Φ Π 10 Ε 0.3 , Φ Π 10 Ε 0.9 , Φ Π 10 Ε 0.7 , Φ 0 Ε 0.7 , Φ Π 10 Ε 0.7 , Φ Π 2 FIG. 1. Trajectories of the pendulum pivot ͓Eqs. ͑2͒ and ͑3͔͒ in the Cartesian plane ͑x , y͒ for A y = 1 and diverse values of the relative amplitude and phase difference .

We reinvestigate the dynamics of the grow and collapse of Bose-Einstein condensates in a system of trapped ultracold atoms with negative scattering lengths, and found a new behavior in the long time scale evolution: the number of atoms can go far beyond the static stability limit. The condensed state is described by the solution of the time-dependent nonlinear Schrödinger equation, in a model that includes atomic feeding and three-body dissipation. Our results for the model show that, by changing the feeding parameter and when a substantial depletion of the ground-state exists, a chaotic behavior is found. We consider a criterion proposed by Deissler and Kaneko ͓Phys. Lett. A 119, 397 ͑1987͔͒ to diagnose spatiotemporal chaos.

Invasions in oscillatory systems generate in their wake spatiotemporal oscillations, consisting of either periodic wavetrains or irregular oscillations that appear to be spatiotemporal chaos. We have shown previously that when a finite domain, with zero-flux boundary conditions, has been fully invaded, the spatiotemporal oscillations persist in the irregular case, but die out in a systematic way for periodic traveling waves. In this paper, we consider the effect of environmental inhomogeneities on this persistence. We use numerical simulations of several predator-prey systems to study the effect of random spatial variation of the kinetic parameters on the die-out of regular oscillations and the long-time persistence of irregular oscillations. We find no effect on the latter, but remarkably, a moderate spatial variation in parameters leads to the persistence of regular oscillations, via the formation of target patterns. In order to study this target pattern production analytically, we turn to λ-ω systems. Numerical simulations confirm analagous behavior in this generic oscillatory system. We then repeat this numerical study using piecewise linear spatial variation of parameters, rather than random variation, which also gives formation of target patterns under certain circumstances, which we discuss. We study this in detail by deriving an analytical approximation to the targets formed when the parameter λ 0 varies in a simple, piecewise linear manner across the domain, using perturbation theory. We end by discussing the applications of our results in ecology and chemistry.

We study the complex behavior in soft-mode turbulence (SMT), a recently discovered type of spatiotemporal chaos observed in electrohydrodynamic instability (EHD) of a nematic liquid crystal with homeotropic alignment. A particle, small compared to the characteristic length of the macroscopic ow, injected in SMT travels with random velocity such as an active Brownian motion. Tracking the particle trajectory induces changes in the ow velocity as a function of space and time (the Lagrange picture). The mean amplitude of the velocity is linearly proportional to the control parameter , normalized voltage in EHD. The probability density distribution of the particle velocity changes from LÃ evy for small to Gaussian distribution for large through intermediate distributions. The di erent regimes of anomalous di usions are also observed. The stochastic properties of SMT are discussed based on these results.

We conjecture that in one-dimensional spatially extended systems the propagation velocity of correlations coincides with a zero of the convective Lyapunov spectrum. This conjecture is successfully tested in three different contexts: (i) a Hamiltonian system (a Fermi-Pasta-Ulam chain of oscillators); (ii) a general model for spatiotemporal chaos (the complex Ginzburg-Landau equation); (iii) experimental data taken from a CO 2 laser with delayed feedback. In the last case, the convective Lyapunov exponent is determined directly from the experimental data. PACS numbers: 05.45.Jn, 42.55.Lt, 42.65.Sf 3616 0031-9007͞00͞85(17)͞3616(4)$15.00

We study the transition from laminar to chaotic behavior in deterministic chaotic coupled map lattices and in an extension of the stochastic Domany-Kinzel cellular automaton [E. Domany and W. Kinzel, Phys. Rev. Lett. 53, 311 (1984)]. For the deterministic coupled map lattices, we find evidence that "solitons" can change the nature of the transition: for short soliton lifetimes it is of second order, while for longer but finite lifetimes, it is more reminiscent of a first-order transition. In the second-order regime, the deterministic model behaves like directed percolation with infinitely many absorbing states; we present evidence obtained from the study of bulk properties and the spreading of chaotic seeds in a laminar background. To study the influence of the solitons more specifically, we introduce a soliton including variant of the stochastic Domany-Kinzel cellular automaton. Similar to the deterministic model, we find a transition from second- to first-order behavior ...

We conjecture that in one-dimensional spatially extended systems the propagation velocity of correlations coincides with a zero of the convective Lyapunov spectrum. This conjecture is successfully tested in three different contexts: (i) a Hamiltonian system (a Fermi-Pasta-Ulam chain of oscillators); (ii) a general model for spatiotemporal chaos (the complex Ginzburg-Landau equation); (iii) experimental data taken from a CO 2 laser with delayed feedback. In the last case, the convective Lyapunov exponent is determined directly from the experimental data. PACS numbers: 05.45.Jn, 42.55.Lt, 42.65.Sf 3616 0031-9007͞00͞85(17)͞3616(4)$15.00

Replication and di erentiation of spots in a class of reaction{di usion equations are studied by extending the Gray{Scott model with self-replicating spots so that it includes many chem-ical species. By examining many possible reaction networks, the behavior of this model is categorized into three types: replication of homogeneous xed spots, replication of oscillatory spots, and di erentiation from \multipotent spots". These multipotent spots either replicate or di erentiate into other types of spots with di erent xed-point dynamics, and as a result, an inhomogeneous pattern of spots is formed. This di erentiation process of spots is analyzed in terms of the loss of chemical diversity and decrease of the local Kolmogorov{Sinai entropy. Initial condition dependence and robustness of a pattern against macroscopic perturbation are also analyzed. Relevance of the results to developmental cell biology is also discussed.

Chaotic pattern dynamics in many experimental systems show structured time averages. We suggest that simple universal boundary effects underly this phenomenon and exemplify them with the Kuramoto-Sivashinsky equation in a finite domain. As in the experiments, averaged patterns in the equation recover global symmetries locally broken in the chaotic field. Plateus in the average pattern wavenumber as a function of the system size are observed and studied and the different behaviors at the central and boundary regions are discussed. Finally, the structure strenght of average patterns is investigated as a function of system size.

Surface wave patterns are investigated experimentally in a system geometry that has become a paradigm of quantum chaos: the stadium billiard. Linear waves in bounded geometries for which classical ray trajectories are chaotic are known to give rise to scarred patterns. Here, we utilize parametrically forced surface waves (Faraday waves), which become progressively nonlinear beyond the wave instability threshold, to investigate the subtle interplay between boundaries and nonlinearity. Only a subset (three main types) of the computed linear modes of the stadium are observed in a systematic scan. These correspond to modes in which the wave amplitudes are strongly enhanced along paths corresponding to certain periodic ray orbits. Many other modes are found to be suppressed, in general agreement with a prediction by Agam and Altshuler based on boundary dissipation and the Lyapunov exponent of the associated orbit. Spatially asymmetric or disordered (but time-independent) patterns are also found even near onset. As the driving acceleration is increased, the time-independent scarred patterns persist, but in some cases transitions between modes are noted. The onset of spatiotemporal chaos at higher forcing amplitude often involves a nonperiodic oscillation between spatially ordered and disordered states. We characterize this phenomenon using the concept of pattern entropy. The rate of change of the patterns is found to be reduced as the state passes temporarily near the ordered configurations of lower entropy. We also report complex but highly symmetric (time-independent) patterns far above onset in the regime that is normally chaotic.

From the analyticity properties of the equation governing infinitesimal perturbations, it is conjectured that all types of Lyapunov exponents introduced in spatially extended 1D systems can be derived from a single function that we call entropy potential. The general consequences of its very existence on the Kolmogorov-Sinai entropy of generic spatiotemporal patterns are discussed. KEY WORDS: Spatiotemporal chaos, coupled map lattices, entropy potential, comoving Lyapunov exponents. PACS numbers: 05.45.+b , 42.50.Lc Submitted to J. Stat. Phys. (1997) Typeset using REVT E X I. INTRODUCTION The first part of this work [1] (hereafter referred to as LPT) was devoted to the definition and discussion of the properties of temporal (TLS) and spatial (SLS) Lyapunov spectra of 1D extended dynamical systems. In this second part we first show how the two approaches are deeply related by proving, in some simple cases, and conjecturing, in general, that all stability properties can be derived fr...

Abstract: We present the results of a wavelet-based approach to the study of the chaotic dynamics of a one dimensional model that shows a direct transition to spatiotemporal chaos. We find that the dynamics of this model in the spatiotemporally chaotic regime may be understood in terms of localized dynamics in both space and scale (wavenumber). A projection onto a Daubechies basis yields a good separation of scales, as shown by an examination of the contribution of different wavelet levels to the power spectrum. At most ...

The experimental study of electroconvection in a homeotropically aligned nematic ͑MBBA͒ is reported. The system undergoes a supercritical bifurcation ''rest state-spatiotemporal chaos.'' The chaos is caused by longwavelength modulation of the orientation of convective rolls. For the first time the observations both below and beyond the Lifshitz point are accompanied by quantitative analysis of temporal autocorrelation functions of turbulent modes. The dependence of the correlation time on the control parameter is obtained. A secondary bifurcation from normal to abnormal rolls is discussed. ͓S1063-651X͑97͒51412-7͔ PACS number͑s͒: 47.27.Cn, 47.52.ϩj, 47.65.ϩa, 61.30.Gd FIG. 2. Inverse correlation time vs below the Lifshitz point ͓͑a͒ f ϭ500 Hz͔ and beyond it ͓͑b͒ f ϭ2000 Hz͔.

Spatiotemporally chaotic dynamics of a Kuramoto-Sivashinsky system is described by means of an infinite hierarchy of its unstable spatiotemporally periodic solutions. An intrinsic parametrization of the corresponding invariant set serves as accurate guide to the high-dimensional dynamics, and the periodic orbit theory yields several global averages characterizing the chaotic dynamics.

Synchronization of spatiotemporally chaotic extended systems is considered in the context of coupled onedimensional Complex Ginzburg-Landau equations (CGLE). A regime of coupled spatiotemporal intermittency (STI) is identified and described in terms of the space-time synchronized chaotic motion of localized structures. A quantitative measure of synchronization as a function of coupling parameter is given through distribution functions and information measures. The coupled STI regime is shown to dissapear into regular dynamics for situations of strong coupling, hence a description in terms of a single CGLE is not appropiate.

Coupled Ginzburg-Landau equations appear in a variety of contexts involving instabilities in oscillatory media. When the relevant unstable mode is of vectorial character (a common situation in nonlinear optics), the pair of coupled equations has special symmetries and can be written as a vector complex Ginzburg-Landau (CGL) equation. Dynamical properties of localized structures of topological character in this vector-field case are considered. Creation and annihilation processes of different kinds of vector defects are described, and some of them interpreted in theoretical terms. A transition between different regimes of spatiotemporal dynamics is described.