Statistics of Topological Defects and Spatiotemporal Chaos in a Reaction-Diffusion System

Annihilation dynamics during spiral defect chaos revealed by particle models

Pair-annihilation events are ubiquitous in a variety of spatially extended systems and are often studied using computationally expensive simulations. Here, we develop an approach in which we simulate the pair-annihilation of spiral wave tips in cardiac models using a computationally efficient particle model. Spiral wave tips are represented as particles with dynamics governed by diffusive behavior and short-ranged attraction. The parameters for diffusion and attraction are obtained by comparing particle motion to the trajectories of spiral wave tips in cardiac models during spiral defect chaos. The particle model reproduces the annihilation rates of the cardiac models and can determine the statistics of spiral wave dynamics, including its mean termination time. We show that increasing the attraction coefficient sharply decreases the mean termination time, making it a possible target for pharmaceutical intervention.

The physics of heart rhythm disorders

The global burden caused by cardiovascular disease is substantial, with heart disease representing the most common cause of death around the world. There remains a need to develop better mechanistic models of cardiac function in order to combat this health concern. Heart rhythm disorders, or arrhythmias, are one particular type of disease which has been amenable to quantitative investigation. Here we review the application of quantitative methodologies to explore dynamical questions pertaining to arrhythmias. We begin by describing single-cell models of cardiac myocytes, from which two and three dimensional models can be constructed. Special focus is placed on results relating to pattern formation across these spatially-distributed systems, especially the formation of spiral waves of activation. Next, we discuss mechanisms which can lead to the initiation of arrhythmias, focusing on the dynamical state of spatially discordant alternans, and outline proposed mechanisms perpetuating arrhythmias such as fibrillation. We then review experimental and clinical results related to the spatio-temporal mapping of heart rhythm disorders. Finally, we describe treatment options for heart rhythm disorders and demonstrate how statistical physics tools can provide insights into the dynamics of heart rhythm disorders.

On Nonlinear Reaction-Diffusion Model with Time Delay on Hexagonal Lattice

In the work, a nonlinear reaction-diffusion model in a class of delayed differential equations on the hexagonal lattice is considered. The system includes a spatial operator of diffusion between hexagonal pixels. The main results deal with the qualitative investigation of the model. The conditions of global asymptotic stability, which are based on the Lyapunov function construction, are obtained. An estimate of the upper bound of time delay, which enables stability, is presented. The numerical study is executed with the help of the bifurcation diagram, phase trajectories, and hexagonal tile portraits. It shows the changes in qualitative behavior with respect to the growth of time delay; namely, starting from the stable focus at small delay values, then through Hopf bifurcation to limit cycles, and finally, through period doublings to deterministic chaos.

Extinction dynamics of spiral defect chaos

Spatially extended excitable systems can exhibit spiral defect chaos (SDC) during which spiral waves continuously form and disappear. To address how this dynamical state terminates using simulations can be computationally challenging, especially for large systems. To circumvent this limitation, we treat the number of spiral waves as a stochastic population with a corresponding birth-death equation and use techniques from statistical physics to determine the mean episode duration of SDC. Motivated by cardiac fibrillation, during which the heart's electrical activity becomes disorganized and shows fragmenting spiral waves, we use generic models of cardiac electrophysiology. We show that the duration can be computed in minimal computational time and that it depends exponentially on domain size. Therefore, the approach can result in efficient and accurate predictions of mean episode duration which may be extended to more complex geometries and models.

Defects formation and spiral waves in a network of neurons in presence of electromagnetic induction

Complex anatomical and physiological structure of an excitable tissue (e.g., cardiac tissue) in the body can represent different electrical activities through normal or abnormal behavior. Abnormalities of the excitable tissue coming from different biological reasons can lead to formation of some defects. Such defects can cause some successive waves that may end up to some additional reorganizing beating behaviors like spiral waves or target waves. In this study, formation of defects and the resulting emitted waves in an excitable tissue are investigated. We have considered a square array network of neurons with nearest-neighbor connections to describe the excitable tissue. Fundamentally, electrophysiological properties of ion currents in the body are responsible for exhibition of electrical spatiotemporal patterns. More precisely, fluctuation of accumulated ions inside and outside of cell causes variable electrical and magnetic field. Considering undeniable mutual effects of electrical field and magnetic field, we have proposed the new Hindmarsh-Rose (HR) neuronal model for the local dynamics of each individual neuron in the network. In this new neuronal model, the influence of magnetic flow on membrane potential is defined. This improved model holds more bifurcation parameters. Moreover, the dynamical behavior of the tissue is investigated in different states of quiescent, spiking, bursting and even chaotic state. The resulting spatiotemporal patterns are represented and the time series of some sampled neurons are displayed, as well.

Nonlinear physics of electrical wave propagation in the heart: a review

The beating of the heart is a synchronized contraction of muscle cells (myocytes) that is triggered by a periodic sequence of electrical waves (action potentials) originating in the sino-atrial node and propagating over the atria and the ventricles. Cardiac arrhythmias like atrial and ventricular fibrillation (AF,VF) or ventricular tachycardia (VT) are caused by disruptions and instabilities of these electrical excitations, that lead to the emergence of rotating waves (VT) and turbulent wave patterns (AF,VF). Numerous simulation and experimental studies during the last 20 years have addressed these topics. In this review we focus on the nonlinear dynamics of wave propagation in the heart with an emphasis on the theory of pulses, spirals and scroll waves and their instabilities in excitable media with applications to cardiac modeling. After an introduction into electrophysiological models for action potential propagation, the modeling and analysis of spatiotemporal alternans, spiral and scroll meandering, spiral breakup and scroll wave instabilities like negative line tension and sproing are reviewed in depth and discussed with emphasis on their impact for cardiac arrhythmias.

Flow-induced arrest of spatiotemporal chaos and transition to a stationary pattern in the Gray-Scott model

We examine the prototypical Gray-Scott model, which mimics cubic autocatalytic reaction with linear decay of the autocatalyst, to model the kinetics of a reaction-diffusion system subjected to advective streamline flow. For a proper choice of boundary conditions and parameter space, the system admits wave-induced spatiotemporal chaos in the absence of flow. We show that flow above a critical value leads to an arrest of the spatiotemporal chaos due to a change in the instability from absolute to convective type. Furthermore, stationary spatial structures are borne out of a second successive bifurcation for yet another critical flow value. The theoretical formulations are corroborated by extensive numerical simulation of the full reaction-diffusion-advection system in one dimension.

Accelerated dynamics in active media: from Turing patterns to sparkling waves

We report the destabilization of stationary Turing patterns and the subsequent emergence of fast spatiotemporal dynamics due to reactant consumption. The localized hexagonal Turing spots switch from a stationary regime to a dynamics state by exhibiting spatial oscillations with two characteristic wavelengths and one representative temporal period. These oscillatory Turing spots are not temporally stable and evolve into traveling spiral tips that, in addition to the unexpected birth of spots, rapidly transform into target patterns and originate multiple collisions and wave breakups due to their proximity, degenerating into a chaotic scenario.

Influence of the medium's dimensionality on defect-mediated turbulence

Spatiotemporal chaos in oscillatory and excitable media is often characterized by the presence of phase singularities called defects. Understanding such defect-mediated turbulence and its dependence on the dimensionality of a given system is an important challenge in nonlinear dynamics. This is especially true in the context of ventricular fibrillation in the heart, where the importance of the thickness of the ventricular wall is contentious. Here, we study defect-mediated turbulence arising in two different regimes in a conceptual model of excitable media and investigate how the statistical character of the turbulence changes if the thickness of the medium is changed from (quasi-) two- dimensional to three dimensional. We find that the thickness of the medium does not have a significant influence in, far from onset, fully developed turbulence while there is a clear transition if the system is close to a spiral instability. We provide clear evidence that the observed transition and change in the mechanism that drives the turbulent behavior is purely a consequence of the dimensionality of the medium. Using filament tracking, we further show that the statistical properties in the three-dimensional medium are different from those in turbulent regimes arising from filament instabilities like the negative line tension instability. Simulations also show that the presence of this unique three-dimensional turbulent dynamics is not model specific.

Chiral symmetry breaking in a reaction-diffusion system

The emergence of order in far-from-equilibrium systems is often accompanied by the formation of spatially asymmetric patterns. About 30 years ago, a general mechanism to select a chiral solution by coupling a reaction-diffusion system to an external chiral electric field was proposed by Nicolis and Prigogine [Proc. Natl. Acad. Sci. USA 78, 659 (1981)]. However, no experimental or even numerical evidence in reaction-diffusion systems has been reported yet. Here we report a chiral symmetry-breaking phenomenon in a reaction-diffusion system coupled to a circularly polarized electric field (CPEF). Specifically, we show that the CPEF breaks the zero-rotation chiral symmetry between clockwise and counterclockwise spiral defects and that ordered spiral waves with preferred chirality arise from defect-mediated turbulence. The occurrence of such chiral symmetry breaking can be understood by the competition between spiral defects with opposite chirality.

Noise-induced spatiotemporal patterns in Hodgkin-Huxley neuronal network

The effect of noise on the pattern selection in a regular network of Hodgkin-Huxley neurons is investigated, and the transition of pattern in the network is measured from subexcitable to excitable media. Extensive numerical results confirm that kinds of travelling wave such as spiral wave, circle wave and target wave could be developed and kept alive in the subexcitable network due to the noise. In the case of excitable media under noise, the developed spiral wave and target wave could coexist and new target-like wave is induced near to the border of media. The averaged membrane potentials over all neurons in the network are calculated to detect the periodicity of the time series and the generated traveling wave. Furthermore, the firing probabilities of neurons in networks are also calculated to analyze the collective behavior of networks.

The formation mechanism of defects, spiral wave in the network of neurons

A regular network of neurons is constructed by using the Morris-Lecar (ML) neuron with the ion channels being considered, and the potential mechnism of the formation of a spiral wave is investigated in detail. Several spiral waves are initiated by blocking the target wave with artificial defects and/or partial blocking (poisoning) in ion channels. Furthermore, possible conditions for spiral wave formation and the effect of partial channel blocking are discussed completely. Our results are summarized as follows. 1) The emergence of a target wave depends on the transmembrane currents with diversity, which mapped from the external forcing current and this kind of diversity is associated with spatial heterogeneity in the media. 2) Distinct spiral wave could be induced to occupy the network when the target wave is broken by partially blocking the ion channels of a fraction of neurons (local poisoned area), and these generated spiral waves are similar with the spiral waves induced by artificial defects. It is confirmed that partial channel blocking of some neurons in the network could play a similar role in breaking a target wave as do artificial defects; 3) Channel noise and additive Gaussian white noise are also considered, and it is confirmed that spiral waves are also induced in the network in the presence of noise. According to the results mentioned above, we conclude that appropriate poisoning in ion channels of neurons in the network acts as ‘defects’ on the evolution of the spatiotemporal pattern, and accounts for the emergence of a spiral wave in the network of neurons. These results could be helpful to understand the potential cause of the formation and development of spiral waves in the cortex of a neuronal system.

Effects of shear flows on nonlinear waves in excitable media

If an excitable medium is moving with relative shear, the waves of excitation may be broken by the motion. We consider such breaks for the case of a constant linear shear flow. The mechanisms and conditions for the breaking of solitary waves and wavetrains are essentially different: the solitary waves require the velocity gradient to exceed a certain threshold, whilst the breaking of repetitive wavetrains happens for arbitrarily small velocity gradients. Since broken waves evolve into new spiral wave sources, this leads to spatio-temporal irregularity.

Dynamical topology and statistical properties of spatiotemporal chaos

For spatiotemporal chaos described by partial differential equations, there are generally locations where the dynamical variable achieves its local extremum or where the time partial derivative of the variable vanishes instantaneously. To a large extent, the location and movement of these topologically special points determine the qualitative structure of the disordered states. We analyze numerically statistical properties of the topologically special points in one-dimensional spatiotemporal chaos. The probability distribution functions for the number of point, the lifespan, and the distance covered during their lifetime are obtained from numerical simulations. Mathematically, we establish a probabilistic model to describe the dynamics of these topologically special points. In spite of the different definitions in different spatiotemporal chaos, the dynamics of these special points can be described in a uniform approach.

Anisotropic velocity statistics of topological defects under shear flow

We report numerical results on the velocity statistics of topological defects during the dynamics of phase ordering and nonrelaxational evolution assisted by an external shear flow. We propose a numerically efficient tracking method for finding the position and velocity of defects and apply it to vortices in a uniform field and dislocations in anisotropic stripe patterns. During relaxational dynamics, the distribution function of the velocity fluctuations is characterized by a dynamical scaling with a scaling function that has a robust algebraic tail with an inverse cube power law. This is characteristic of defects of codimension 2, e.g., point defects in two dimensions and filaments in three dimensions, regardless of whether the motion is isotropic (as for vortices) or highly anisotropic (as for dislocations). However, the anisotropic dislocation motion leads to anisotropic statistical properties when the interaction between defects and their motion is influenced by the presence of an external shear flow transverse to the stripe orientation.

COLLECTIVE BEHAVIORS OF SPIRAL WAVES IN THE NETWORKS OF HODGKIN-HUXLEY NEURONS IN PRESENCE OF CHANNEL NOISE

Collective behaviors of spiral waves in the networks of Hodgkin-Huxley neuron are investigated. A stable rotating spiral wave can be developed to occupy the quiescent areas in networks of neurons by selecting appropriate initial values for the variables in the networks of neurons. In our numerical studies, most neurons are quiescent and finite (few) numbers of neurons are selected with different values to form a spiral seed. In this way, neurons communicating are carried by propagating spiral wave to break through the quiescent domains (areas) in networks of neurons. The effect of membrane temperature on the formation of spiral wave is investigated by selecting different fixed membrane temperatures in the networks, and it is found that a spiral wave cannot be developed if the membrane temperature is close to a certain threshold. A quantitative factor of synchronization is defined to measure the statistical properties and collective behaviors of the spiral wave. And a distinct phase transition, which indicates the critical condition for spiral survival, is observed in the sudden changing point of the factors of synchronization curve vs. certain bifurcation parameter. Internal noise is introduced into ion channels (channel noise) with the Langevin method. It is found that a stable rotating spiral wave is developed and the spiral wave is robust to weak channel noise (the membrane patch is not small). The spiral wave can not grow up and the stable rotating spiral wave encounters instability in presence of strong channel noise. Coherence resonance-like behavior is observed in calculating the factors of synchronization in presence of channel noise.

Self-replicating holes in a vertically vibrated dense suspension

We find self-replicating holes on the surface of a vertically vibrated potato starch suspension. Above certain acceleration, the finite-amplitude deformation of the surface grows to form a hole that penetrates the fluid layer. The circular shape of the hole is not stable, and the hole begins to replicate just like the self-replicating spots in chemical reaction-diffusion systems. At high acceleration, these holes exhibit spatiotemporal chaos. By assessing the statistical properties in a steady state, we show that fluctuation in the number of holes can be understood by a master equation.

Theoretical analysis of defect-mediated turbulence in a catalytic surface reaction

We present a statistical analysis of defect-mediated turbulence in a kinetic model of catalytic CO oxidation on Pt(110). A probabilistic description based on the gain and loss rates of defects is derived. For low values of the CO partial pressure the statistics of topological defects agree with earlier results for the complex Ginzburg-Landau equation. For high values of the CO partial pressure, we observe an additional autocatalytic reproduction of defects that results in a linear dependence of the defect creation rate on the number of defects in the system. The role of correlations between defects of opposite topological charge was found to be weaker than in the experimental system.

Spontaneously periodic wave generation in coupled excitable media

We studied a modified reaction-diffusion model theoretically by coupling two ideal excitable media systems. In the simulated homogeneous system, we observed the propagation of reaction-diffusion wave trains that required no external force after the initial stimulation. We investigated the dependence of the system's oscillation patterns on model parameters, and we discussed the influence of the different dynamic constants of the individual coupled systems on the dynamics of the coupled systems. Some complex two-dimensional patterns generated by our model are shown. We also found similar phenomena in the models for catalytic CO oxidation on Pt(110), and for cardiac tissue.

Defect-mediated turbulence in the Belousov-Zhabotinsky reaction

Statistical properties of topological defects in defect-mediated turbulence due to the Doppler instability are examined experimentally in the Belousov-Zhabotinsky reaction. By applying the phase space reconstruction approach, processes of defect creation, annihilation, and defect movement are analyzed. The defect dynamics can be well interpreted within the framework of stochastic Markovian process. In contrast to previous studies that made direct measure of the gain and loss rates, which is practically difficult, we demonstrate that the rates can be obtained directly from the analysis of the time series of defects, and the shape of the probability distribution function can be reproduced in a simple way.

Filament-induced surface spiral turbulence in three-dimensional excitable media

Filament-induced surface defect-mediated turbulence in bounded three-dimensional (3D) excitable media is investigated in the regime of negative line tension. In this regime turbulence arises due to unstable filaments associated with scroll waves and is purely a 3D phenomenon. It is shown that the statistical properties of the turbulent defect dynamics can be used to distinguish surface defect-mediated turbulence from its 2D analog. Mechanisms for the creation and annihilation of surface defects are discussed and Markov rate equations are employed to model the results.

Active and passive control of spiral turbulence in excitable media

The influence of a spatially localized heterogeneity defect, defined by failure of the diffusion effect, on spiral turbulence suppression in two-dimensional excitable media is studied numerically, based on the Bär model. It is shown that in certain parameter regions spiral turbulence without the defect can be suppressed by a boundary periodic forcing (called active control) if the forcing frequency is properly chosen. However, with a sufficiently large defect this active control method no longer works due to the wake turbulence following the defect. We suggest an auxiliary method of enclosing the defect with a thin layer of material of high excitability (called passive control) to screen the interaction between the defect and the turbulence and to restore the global control effect of the periodic forcing. The possible application of the method in cardiac defibrillation is discussed.

Spatiotemporal chaos of self-replicating spots in reaction-diffusion systems

The statistical properties of self-replicating spots in the reaction-diffusion Gray-Scott model are analyzed. In the chaotic regime of the system, the spots that dominate the spatiotemporal chaos grow and divide in two or decay into the background randomly and continuously. The rates at which the spots are created and decay are observed to be linearly dependent on the number of spots in the system. We derive a probabilistic description of the spot dynamics based on the statistical independence of spots and thus propose a characterization of the spatiotemporal chaos dominated by replicating spots.

Geometric diagnostics of complex patterns: spiral defect chaos

Motivated by the observation of spiral patterns in a wide range of physical, chemical, and biological systems, we present an automated approach that aims at characterizing quantitatively spiral-like elements in complex stripe-like patterns. The approach provides the location of the spiral tip and the size of the spiral arms in terms of their arc length and their winding number. In addition, it yields the number of pattern components (Betti number of order 1), as well as their size and certain aspects of their shape. We apply the method to spiral defect chaos in thermally driven Rayleigh-Benard convection and find that the arc length of spirals decreases monotonically with decreasing Prandtl number of the fluid and increasing heating. By contrast, the winding number of the spirals is nonmonotonic in the heating. The distribution function for the number of spirals is significantly narrower than a Poisson distribution. The distribution function for the winding number shows approximately an exponential decay. It depends only weakly on the heating, but strongly on the Prandtl number. Large spirals arise only for larger Prandtl numbers (Pr approximately > 1). In this regime the joint distribution for the spiral length and the winding number exhibits a three-peak structure, indicating the dominance of Archimedean spirals of opposite sign and relatively straight sections. For small Prandtl numbers the distribution function reveals a large number of small compact pattern components.

Drift of spiral waves controlled by a polarized electric field

The drift behavior of spiral waves under the influence of a polarized electric field is investigated in the light that both the polarized electric field and the spiral waves possess rotation symmetry. Numerical simulations of a reaction-diffusion model show that the drift velocity of the spiral tip can be controlled by changing the polarization mode of the polarized electric field and some interesting drift phenomena are observed. When the electric field is circularly polarized and its rotation follows that of the spiral, the drift speed of the spiral tip reaches its maximal value. On the contrary, opposite rotation between the spiral and electric field locks the drift of the spiral tip. Analytical results based on the weak deformation approximation are consistent with the numerical results. We hope that our theoretical results will be observed in experiments, such as the Belousov-Zhabotinsky reaction.

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.

Statistics of defect trajectories in spatio-temporal chaos in inclined layer convection and the complex Ginzburg-Landau equation

For spatio-temporal chaos observed in numerical simulations of the complex Ginzburg-Landau equation (CGL) and in experiments on inclined-layer convection (ILC) we report numerical and experimental data on the statistics of defects and of defect loops. These loops consist of defect trajectories in space-time that are connected to each other through the pairwise annihilation or creation of the associated defects. While most such loops are small and contain only a few defects, the loop distribution functions decay only slowly with the quantities associated with the loop size, consistent with power-law behavior. For the CGL, two of the three power-law exponents are found to agree, within our computational precision, with those from previous investigations of a simple lattice model. In certain parameter regimes of the CGL and ILC, our results for the single-defect statistics show significant deviations from the previously reported findings that the defect dynamics are consistent with those of random walkers that are created with fixed probability and annihilated through random collisions.

Defect-mediated turbulence in systems with local deterministic chaos

Defect-mediated turbulence is shown to exist in media where the underlying local dynamics is deterministically chaotic. While many of the characteristics of defect-mediated turbulence, such as the exponential decay of correlations and a squared Poissonian distribution for the number of defects, are identical to those seen in oscillatory media, the fluctuations in the number of defects differ significantly. The power spectra suggest the existence of underlying correlations that lead to a different and nonuniversal scaling structure in chaotic media.

Suppression of spiral waves and spatiotemporal chaos by generating target waves in excitable media

A method for suppressing spiral waves and spatiotemporal chaos in excitable media is proposed. Applying suitable periodic force to a single point, we can successfully suppress spiral waves as well as spatiotemporal chaos by generating target waves. After we turn off the external force, target waves finally disappear and the whole system which was in the state of spiral wave or spatiotemporal chaos goes to the spatially homogeneous steady state. It is shown that our control method is not sensitively model dependent. It works for a model for catalytic CO oxidation on platinum as well as for a model for cardiac muscle.

Identification of nonlinear spatiotemporal systems via partitioned filtering

The problem of identifying continuous spatiotemporal nonlinear systems from noisy and indirect observations is determined by its computational complexity. We propose a solution by means of nonlinear state space filtering along with a state partition technique. The method is demonstrated to be computationally feasible for spatiotemporal data with properties that occur typically in experimental recordings. It is applied to one component of the simulated chaotic data of a two-component reaction diffusion system, yielding estimates of both the unobserved state component and the diffusion constant.

Pattern formation on the edge of chaos: mathematical modeling of CO oxidation on a Pt(110) surface under global delayed feedback

Effects of global delayed feedback on diffusion-induced turbulence are studied in a realistic model of catalytic oxidation of carbon monoxide on Pt(110). Spatiotemporal patterns resulting from numerical simulations of this model are identified and analyzed using a transformation into the phase and the amplitude of local oscillations. We find that chemical turbulence can be efficiently controlled by varying the feedback intensity and the delay time in the feedback loop. Near the transition from turbulence to uniform oscillations, various chaotic and regular spatiotemporal patterns-intermittent turbulence, two-phase clusters, cells of hexagonal symmetry, and phase turbulence-are found.

Defibrillation via the elimination of spiral turbulence in a model for ventricular fibrillation

Ventricular fibrillation, the major reason behind sudden cardiac death, is turbulent cardiac electrical activity in which rapid, irregular disturbances in the spatiotemporal electrical activation of the heart make it incapable of any concerted pumping action. Methods of controlling ventricular fibrillation include electrical defibrillation as well as injected medication. Electrical defibrillation, though widely used, involves subjecting the whole heart to massive, and often counterproductive, electrical shocks. We propose a defibrillation method that uses a very low-amplitude shock (of order mV) applied for a brief duration (of order 100 ms) and over a coarse mesh of lines on our model ventricle.

Spatiotemporal chaos and nonequilibrium transitions in a model excitable medium

We present a detailed study of the statistical steady states of a model for CO oxidation on Pt(110) proposed by Bar and co-workers. We show that the stability diagram of this model depends sensitively on the boundary conditions. We elucidate several novel properties of a state with meandering spirals (M) briefly mentioned by Bar and co-workers. (1) We show that, with periodic boundary conditions, M is the state MP, a binary mixture displaying a coexistence of quasiperiodically rotating spirals and chaotically moving pointlike defects. We show that the transition from MP to the turbulent state T1 is continuous; the transition line marks the locus where the two phases cease to be distinct. (2) With Neumann boundary conditions M is the state MN, a single quasiperiodically rotating spiral. We show that the MN-T1 transition is discontinuous or first order. We also characterize the transitions from MP and MN to the state S, which has quasiperiodically rotating spirals. We also propose qualitative mechanisms for these transitions.

Voronoi analysis of the breakdown of order in spontaneous optical spot patterns

In a nonlinear optical single feedback system, patterns of bright spots evolve spontaneously. Making use of the Voronoi construction, the breakdown of spatial order in experimental spot patterns under increase of the pump laser intensity is analyzed. The spot density is found to increase with the pump intensity. Our analysis shows that maximum order is maintained for a finite range of spot densities and starts to break down at a clearly detectable threshold value. The enhanced disorder with increasing density contrasts the behavior of simple hard-core systems. A separate analysis of the metrical properties of the Voronoi cells in ordered and disordered regions yields remarkably smaller nearest-neighbor distances in disordered regions. This suggests that the creation of new spots is a governing mechanism for the observed breakdown of order.

Coherent structure analysis of spatiotemporal chaos

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 Pt(100), 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.