Inwardly rotating spiral waves in a reaction-diffusion system

Self-organization of active colloids mediated by chemical interactions

Self-propelled colloidal particles exhibit rich non-equilibrium phenomena and have promising applications in fields such as drug delivery and self-assembled active materials. Previous experimental and theoretical studies have shown that chemically active colloids that consume or produce a chemical can self-organize into clusters with diverse characteristics depending on the effective phoretic interactions. In this paper, we investigate self-organization in systems with multiple chemical species that undergo a network of reactions and multiple colloidal species that participate in different reactions. Active colloids propelled by complex chemical reactions with potentially nonlinear kinetics can be realized using enzymatic reactions that occur on the surface of enzyme-coated particles. To demonstrate how the self-organizing behavior depends on the chemical reactions active colloids catalyze and their chemical environment, we consider first a single type of colloid undergoing a simple catalytic reaction, and compare this often-studied case with self-organization in binary mixtures of colloids with sequential reactions, and binary mixtures with nonlinear autocatalytic reactions. Our results show that in general active colloids at low particle densities can form localized clusters in the presence of bulk chemical reactions and phoretic attractions. The characteristics of the clusters, however, depend on the reaction kinetics in the bulk and on the particles and phoretic coefficients. With one or two chemical species that only undergo surface reactions, the space for possible self-organizations are limited. By considering the additional system parameters that enter the chemical reaction network involving reactions on the colloids and in the fluid, the design space of colloidal self-organization can be enlarged, leading to a variety of non-equilibrium structures.

Time-delay-induced spiral chimeras on a spherical surface of globally coupled oscillators

We consider globally coupled networks of identical oscillators, located on the surface of a sphere with interaction time delays, and show that the distance-dependent time delays play a key role for the spiral chimeras to occur as a generic state in different systems of coupled oscillators. For the phase oscillator system, we analyze the existence and stability of stationary solutions along the Ott-Antonsen invariant manifold to find the bifurcation structure of the spiral chimera state. We demonstrate via an extensive numerical experiment that the time-delay-induced spiral chimeras are also present for coupled networks of the Stuart-Landau and Van der Pol oscillators in the same parameter regime as that of phase oscillators, with a series of evenly spaced band-type regions. It is found that the spiral chimera state occurs as a consequence of a resonant-type interplay between the intrinsic period of an individual oscillator and the interaction time delay as a topological structure property.

Pattern formation by bacteria-phage interactions

The interactions between bacteria and phages-viruses that infect bacteria-play critical roles in agriculture, ecology, and medicine; however, how these interactions influence the spatial organization of both bacteria and phages remain largely unexplored. Here, we address this gap in knowledge by developing a theoretical model of motile, proliferating bacteria that aggregate via motility-induced phase separation (MIPS) and encounter phage that infect and lyse the cells. We find that the non-reciprocal predator-prey interactions between phage and bacteria strongly alter spatial organization, in some cases giving rise to a rich array of finite-scale stationary and dynamic patterns in which bacteria and phage coexist. We establish principles describing the onset and characteristics of these diverse behaviors, thereby helping to provide a biophysical basis for understanding pattern formation in bacteria-phage systems, as well as in a broader range of active and living systems with similar predator-prey or other non-reciprocal interactions.

Effects of noise on the critical points of Turing instability in complex ecosystems

Noise is ubiquitous in natural and artificial systems. In a noisy environment, the interactions among nodes may fluctuate randomly, leading to more complicated interactions. In this paper we focus on the effects of noise and network topology on the Turing pattern of ecological networks with activator-inhibitor structure, which may be interpreted as prey-predator interactions. Based on the stability theory of stochastic differential equations, a sufficient condition for the uniform state is derived. The analytical results indicate that noise is beneficial for the uniform state. When the ratio between the diffusion coefficients of the predator and prey increases, the ecosystems can exhibit a transition from a uniform stable state to a Turing pattern, while when the ratio decreases, the ecosystems transit from a Turing pattern to a uniform stable state. There are two crucial critical points in Turing patterns, forward and backward. We find that both forward and backward critical points increase as the noise intensity increases. This means that noise favors a stable homogeneous state compared to a state with a heterogeneous pattern, which is consistent with the analytical results. In addition, noise can weaken the hysteresis phenomenon and even eliminate it in some cases. Furthermore, we report that network topology plays an important role in modulating the uniform state of ecosystems, such as the size of prey-predator systems, the network connectivity, and the strength of interaction.

Finding type and location of the source of cardiac arrhythmias from the averaged flow velocity field using the determinant-trace method

Life threatening cardiac arrhythmias result from abnormal propagation of nonlinear electrical excitation waves in the heart. Finding the locations of the sources of these waves remains a challenging problem. This is mainly due to the low spatial resolution of electrode recordings of these waves. Also, these recordings are subjected to noise. In this paper, we develop a different approach: the AFV-DT method based on an averaged flow velocity (AFV) technique adopted from the analysis of optical flows and the determinant-trace (DT) method used for vector field analysis of dynamical systems. This method can find the location and determine all important types of sources found in excitable media such as focal activity, spiral waves, and waves rotating around obstacles. We test this method on in silico data of various wave excitation patterns obtained using the Luo-Rudy model for cardiac tissue. We show that the method works well for data with low spatial resolutions (up to 8×8) and is stable against noise. Finally, we apply it to two clinical cases and show that it can correctly identify the arrhythmia type and location. We discuss further steps on the development and improvement of this approach.

Chemical micro-oscillators based on the Belousov–Zhabotinsky reaction

The results of studies on the development of micro-oscillators (MOs) based on the Belousov –Zhabotinsky (BZ) oscillatory chemical reaction are integrated and systematized. The mechanisms of the BZ reaction and the methods of immobilization of the catalyst of the BZ reaction in micro-volumes are briefly discussed. Methods for creating BZ MOs based on water microdroplets in the oil phase and organic and inorganic polymer microspheres are considered. Methods of control and management of the dynamics of BZ MO networks are described, including methods of MO synchronization. The prospects for the design of neural networks of MOs with intelligent-like behaviour are outlined. Such networks present a new area of nonlinear chemistry, including, in particular, the creation of a chemical ‘computer’.

The bibliography includes 250 references.

Ill-matched timescales in coupled systems can induce oscillation suppression

We explore the behavior of two coupled oscillators, considering combinations of similar and dissimilar oscillators, with their intrinsic dynamics ranging from periodic to chaotic. We first investigate the coupling of two different real-world systems, namely, the chemical mercury beating heart oscillator and the electronic Chua oscillator, with the disparity in the timescales of the constituent oscillators. Here, we are considering a physical situation that is not commonly addressed: the coupling of sub-systems whose characteristic timescales are very different. Our findings indicate that the oscillations in coupled systems are quenched to oscillation death (OD) state, at sufficiently high coupling strength, when there is a large timescale mismatch. In contrast, phase synchronization occurs when their timescales are comparable. In order to further strengthen the concept, we demonstrate this timescale-induced oscillation suppression and phase synchrony through numerical simulations, with the disparity in the timescales serving as a tuning or control parameter. Importantly, oscillation suppression (OD) occurs for a significantly smaller timescale mismatch when the coupled oscillators are chaotic. This suggests that the inherent broad spectrum of timescales underlying chaos aids oscillation suppression, as the temporal complexity of chaotic dynamics lends a natural heterogeneity to the timescales. The diversity of the experimental systems and numerical models we have chosen as a test-bed for the proposed concept lends support to the broad generality of our findings. Last, these results indicate the potential prevention of system failure by small changes in the timescales of the constituent dynamics, suggesting a potent control strategy to stabilize coupled systems to steady states.

Nonequilibrium thermodynamics of glycolytic traveling wave: Benjamin-Feir instability

Evolution of the nonequilibrium thermodynamic entities corresponding to dynamics of the Hopf instabilities and traveling waves at a nonequilibrium steady state of a spatially extended glycolysis model is assessed here by implementing an analytically tractable scheme incorporating a complex Ginzburg-Landau equation (CGLE). In the presence of self and cross diffusion, a more general amplitude equation exploiting the multiscale Krylov-Bogoliubov averaging method serves as an essential tool to reveal the various dynamical instability criteria, especially Benjamin-Feir (BF) instability, to estimate the corresponding nonlinear dispersion relation of the traveling wave pattern. The critical control parameter, wave-number selection criteria, and magnitude of the complex amplitude for traveling waves are modified by self- and cross-diffusion coefficients within the oscillatory regime, and their variabilities are exhibited against the amplitude equation. Unlike the traveling waves, a low-amplitude broad region appears for the Hopf instability in the concentration dynamics as the system phase passes through minima during its variation with the control parameter. The total entropy production rate of the uniform Hopf oscillation and glycolysis wave not only qualitatively reflects the global dynamics of concentrations of intermediate species but almost quantitatively. Despite the crucial role of diffusion in generating and shaping the traveling waves, the diffusive part of the entropy production rate has a negligible contribution to the system's total entropy production rate. The Hopf instability shows a more complex and colossal change in the energy profile of the open nonlinear system than in the traveling waves. A detailed analysis of BF instability shows a contrary nature of the semigrand Gibbs free energy with discrete and continuous wave numbers for the traveling wave. We hope the Hopf and traveling wave pattern around the BF instability in terms of energetics and dissipation will open up new applications of such dynamical phenomena.

Programmed Locomotion of an Active Gel Driven by Spiral Waves

Active media that host spiral waves can display complex modes of locomotion driven by the dynamics of those waves. We use a model of a photosensitive stimulus-responsive gel that supports the propagation of spiral chemical waves to study locomotive transition and programmed locomotion. The mode transition between circular and toroidal locomotion results from the onset of spiral tip meandering that arises via a secondary Hopf bifurcation as the level of illumination is increased. This dynamic instability of the system introduces a second circular locomotion with a small diameter caused by tip meandering. The original circular locomotion with large diameter is driven by the push-pull asymmetry of the wavefront and waveback of the simple spiral waves initiated at one corner of gel. By harnessing this mode transition of the gel locomotion via coded illumination, we design programmable pathways of nature-inspired angular locomotion of the gel.

Nucleation of spatiotemporal structures from defect turbulence in the two-dimensional complex Ginzburg-Landau equation

We numerically investigate nucleation processes in the transient dynamics of the two-dimensional complex Ginzburg-Landau equation toward its "frozen" state with quasistationary spiral structures. We study the transition kinetics from either the defect turbulence regime or random initial configurations to the frozen state with a well-defined low density of quasistationary topological defects. Nucleation events of spiral structures are monitored using the characteristic length between the emerging shock fronts. We study two distinct situations, namely when the system is quenched either far from the transition limit or near it. In the former deeply quenched case, the average nucleation time for different system sizes is measured over many independent realizations. We employ an extrapolation method as well as a phenomenological formula to account for and eliminate finite-size effects. The nonzero (dimensionless) barrier for the nucleation of single spiral droplets in the extrapolated infinite system size limit suggests that the transition to the frozen state is discontinuous. We also investigate the nucleation of spirals for systems that are quenched close to but beyond the crossover limit and of target waves which emerge if a specific spatial inhomogeneity is introduced. In either of these cases, we observe long, "fat" tails in the distribution of nucleation times, which also supports a discontinuous transition scenario.

Pattern Formation in Heterostructured Gel by the Ferrocyanide-Iodate-Sulfite Reaction

Pattern formation in the reaction-diffusion systems for the ferrocyanide-iodate-sulfite reaction has been investigated. Previous studies have been conducted in a uniform medium. However, in this study, we reported the pattern formation in heterostructured gels with different network densities. The chemical states of the gel depend on the diffusivity, which in turn depends on the network density of the gel. Consequently, a pH pattern reflecting the heterostructured gel emerged. Furthermore, adjusting the condition produces novel patterns in the heterostructured gel.

Selection of spatiotemporal patterns in arrays of spatially distributed oscillators indirectly coupled via a diffusive environment

Emergence of self-organized behaviors in diverse living systems often depends on population density. In these systems, cell-cell communications are usually mediated by the surrounding environment. Collective behaviors (e.g., synchrony and dynamical quorum sensing) of such systems with stirred environment have been extensively studied, but the spatiotemporal dynamics of the oscillators coupled via a diffusive environment (without stirring) is rather understudied. We here perform a computational study on the selection and competition of wave patterns in arrays of spatially distributed oscillators immersed in a diffusive medium. We find that population density plays a crucial role in the selection of wave patterns: (i) for a single spiral in the system, its rotation either inward or outward could be controlled by population density, and (ii) for spiral and target waves coexisting initially in the system, wave competition happens and population density decides which type of wave will finally survive. The latter phenomenon is further confirmed in a system whose individual element is excitable rather than self-sustained oscillatory. The mechanism underlying all these observations is attributed to the frequency competition. Our results in the excitable case may have implications on the experimental results.

Mechanism and Phenomenology of an Oscillating Chemical Reaction

Chemical reactions, which are far from equilibrium, are capable of displaying oscillations in species concentrations and hence in colour, electrode potential, pH and/or temperature. The oscillations arise from the interplay between positive and negative kinetic feedback. Mechanisms for such reactions are presented, along with the rich phenomenology that these systems exhibit, from complex oscillations and chemical waves, to stationary concentration patterns. This review will focus on the Belousov-Zhabotinksy reaction but reference to other reactions will be made where appropriate.

The Tris(2,2'-Bipyridyl)Ruthenium-Catalysed Belousov–Zhabotinsky Reaction

The Belousov – Zhabotinsky (BZ) reaction is the prototypical oscillating chemical reaction. The tris(2,2'-bipyridine)ruthenium-catalysed BZ reaction (often simply referred to as the ruthenium-catalysed BZ reaction) displays photosensitivity and has been widely exploited for examination of the effects of illumination on nonlinear reaction kinetics. In this review, we investigate the behaviour of the ruthenium-catalysed BZ reaction. The mechanism of the reaction is analysed and we examine how light sensitivity is incorporated into kinetic models of the reaction. The temporal dynamics of the photosensitive reaction is presented and, finally, we discuss the extraordinary wealth of behaviour that has been observed in the spatially-distributed system when perturbed by visible light.

Inwardly Rotating Spirals in a Nonoscillatory Medium

We report the spontaneous formation of spiral patterns observed at a downward-facing free surface of a horizontal liquid film. The surface is unstable to the Rayleigh-Taylor instability and the resulting liquid discharge from the film can occur in the form of propagating liquid curtains. They are born at the film circular periphery and exhibit patterns of inwardly rotating spiral arms. With the help of a phenomenologically constructed cellular automaton, we show that the patterns arise from the phase locking leading to periodic liquid discharge at constant flow rate over the whole film surface.

Nonlinear Concave Spiral Waves in Active Media Transferring Energy

Spiral concave autowaves are widely implemented in physics, chemistry, hydrodynamics, meteorology and other fields. A mathematical model of spiral concave autowaves based on the Fitzhugh-Nagumo equation and modified axiomatic models are presented. The existence of spiral concave autowaves transferring energy was predicted via computational experiments. Applications of spiral concave autowaves carrying energy in hydrodynamics, generation of tornadoes, breaking waves, and tsunamis and examples of such autowaves in biology and medicine are reviewed and the importance of concave spiral autowaves transferring energy is emphasized.

Under Diffusion Control: from Structuring Matter to Directional Motion

Self-organization in synthetic chemical systems is quickly developing into a powerful strategy for designing new functional materials. As self-organization requires the system to exist far from thermodynamic equilibrium, chemists have begun to go beyond the classical equilibrium self-assembly that is often applied in bottom-up supramolecular synthesis, and to learn about the surprising and unpredicted emergent properties of chemical systems that are characterized by a higher level of complexity and extended reactivity networks. The present review focuses on self-organization in reaction-diffusion systems. Selected examples show how the emergence of complex morphogenesis is feasible in synthetic systems leading to hierarchically and nanostructured matter. Starting from well-investigated oscillating reactions, recent developments extend diffusion-limited reactivity to supramolecular systems. The concept of dynamic instability is introduced and illustrated as an additional tool for the design of smart materials and actuators, with emphasis on the realization of motion even at the macroscopic scale. The formation of spatio-temporal patterns along diffusive chemical gradients is exploited as the main channel to realize symmetry breaking and therefore anisotropic and directional mechanical transformations. Finally, the interaction between external perturbations and chemical gradients is explored to give mechanistic insights in the design of materials responsive to external stimuli.

From dynamic self-assembly to networked chemical systems

Although dynamic self-assembly, DySA, is a relatively new area of research, the past decade has brought numerous demonstrations of how various types of components - on scales from (macro)molecular to macroscopic - can be arranged into ordered structures thriving in non-equilibrium, steady states. At the same time, none of these dynamic assemblies has so far proven practically relevant, prompting questions about the field's prospects and ultimate objectives. The main thesis of this Review is that formation of dynamic assemblies cannot be an end in itself - instead, we should think more ambitiously of using such assemblies as control elements (reconfigurable catalysts, nanomachines, etc.) of larger, networked systems directing sequences of chemical reactions or assembly tasks. Such networked systems would be inspired by biology but intended to operate in environments and conditions incompatible with living matter (e.g., in organic solvents, elevated temperatures, etc.). To realize this vision, we need to start considering not only the interactions mediating dynamic self-assembly of individual components, but also how components of different types could coexist and communicate within larger, multicomponent ensembles. Along these lines, the review starts with the discussion of the conceptual foundations of self-assembly in equilibrium and non-equilibrium regimes. It discusses key examples of interactions and phenomena that can provide the basis for various DySA modalities (e.g., those driven by light, magnetic fields, flows, etc.). It then focuses on the recent examples where organization of components in steady states is coupled to other processes taking place in the system (catalysis, formation of dynamic supramolecular materials, control of chirality, etc.). With these examples of functional DySA, we then look forward and consider conditions that must be fulfilled to allow components of multiple types to coexist, function, and communicate with one another within the networked DySA systems of the future. As the closing examples show, such systems are already appearing heralding new opportunities - and, to be sure, new challenges - for DySA research.

Nonlinear behavior and fluctuation-induced dynamics in the photosensitive Belousov-Zhabotinsky reaction

The photosensitive Belousov-Zhabotinsky (pBZ) reaction has been used extensively to study the properties of chemical oscillators. In particular, recent experiments revealed the existence of complex spatiotemporal dynamics for systems consisting of coupled micelles (V < 10^-21 L) or droplets (V ≈ [10^-8-10^-11] L) in which the pBZ reaction takes place. These results have been mostly understood in terms of reaction-diffusion models. However, in view of the small size of the droplets and micelles, large fluctuations of concentrations are to be expected. In this work, we investigate the role of fluctuations on the dynamics of a single droplet with stochastic simulations of an extension of the Field-Körös-Noyes (FKN) model taking into account the photosensitivity. The birhythmicity and chaotic behaviors predicted by the FKN model in the absence of fluctuations become transient or intermittent regimes whose lifetime decreases with the size of the droplet. Simple oscillations are more robust and can be observed even in small systems (V > 10^-12 L), which justifies the use of deterministic models in microfluidic systems of coupled oscillators. The simulations also reveal that fluctuations strongly affect the efficiency of inhibition by light, which is often used to control the kinetics of these systems: oscillations are found for parameter values for which they are supposed to be quenched according to deterministic predictions.

Dynamics of spiral waves rotating around an obstacle and the existence of a minimal obstacle

Pinning of vortices by obstacles plays an important role in various systems. In the heart, anatomical reentry is created when a vortex, also known as the spiral wave, is pinned to an anatomical obstacle, leading to a class of physiologically very important arrhythmias. Previous analyses of its dynamics and instability provide fine estimates in some special circumstances, such as large obstacles or weak excitabilities. Here, to expand theoretical analyses to all circumstances, we propose a general theory whose results quantitatively agree with direct numerical simulations. In particular, when obstacles are small and pinned spiral waves are destabilized, an accurate explanation of the instability in two-dimensional media is provided by the usage of a mapping rule and dimension reduction. The implications of our results are to better understand the mechanism of arrhythmia and thus improve its early prevention.

Pattern formation in a reaction-diffusion system of Fitzhugh-Nagumo type before the onset of subcritical Turing bifurcation

We investigate numerically the behavior of a two-component reaction-diffusion system of Fitzhugh-Nagumo type before the onset of subcritical Turing bifurcation in response to local rigid perturbation. In a large region of parameters, the initial perturbation evolves into a localized structure. In a part of that region, closer to the bifurcation line, this structure turns out to be unstable and covers all the available space over the course of time in a process of self-completion. Depending on the parameter values in two-dimensional (2D) space, this process happens either through generation and evolution of new peaks on oscillatory tails of the initial pattern, or through the elongation, deformation, and rupture of initial structure, leading to space-filling nonbranching snakelike patterns. Transient regimes are also possible. Comparison of these results with 1D simulations shows that the prebifurcation region of parameters where the self-completion process is observed is much larger in the 2D case.

Dynamical regimes of four oscillators with excitatory pulse coupling

Dynamics of four almost identical chemical oscillators pulse coupled via excitatory coupling with time delays are systematically studied.

Materials learning from life: concepts for active, adaptive and autonomous molecular systems

A broad overview of functional aspects in biological and synthetic out-of-equilibrium systems.

Waves spontaneously generated by heterogeneity in oscillatory media

Wave propagation is an important characteristic for pattern formation and pattern dynamics. To date, various waves in homogeneous media have been investigated extensively and have been understood to a great extent. However, the wave behaviors in heterogeneous media have been studied and understood much less. In this work, we investigate waves that are spontaneously generated in one-dimensional heterogeneous oscillatory media governed by complex Ginzburg-Landau equations; the heterogeneity is modeled by multiple interacting homogeneous media with different system control parameters. Rich behaviors can be observed by varying the control parameters of the systems, whereas the behavior is incomparably simple in the homogeneous cases. These diverse behaviors can be fully understood and physically explained well based on three aspects: dispersion relation curves, driving-response relations, and wave competition rules in homogeneous systems. Possible applications of heterogeneity-generated waves are anticipated.

Reaction-diffusion processes at the nano- and microscales

The bottom-up fabrication of nano- and microscale structures from primary building blocks (molecules, colloidal particles) has made remarkable progress over the past two decades, but most research has focused on structural aspects, leaving our understanding of the dynamic and spatiotemporal aspects at a relatively primitive stage. In this Review, we draw inspiration from living cells to argue that it is now time to move beyond the generation of structures and explore dynamic processes at the nanoscale. We first introduce nanoscale self-assembly, self-organization and reaction-diffusion processes as essential features of cells. Then, we highlight recent progress towards designing and controlling these fundamental features of life in abiological systems. Specifically, we discuss examples of reaction-diffusion processes that lead to such outcomes as self-assembly, self-organization, unique nanostructures, chemical waves and dynamic order to illustrate their ubiquity within a unifying context of dynamic oscillations and energy dissipation. Finally, we suggest future directions for research on reaction-diffusion processes at the nano- and microscales that we find hold particular promise for a new understanding of science at the nanoscale and the development of new kinds of nanotechnologies for chemical transport, chemical communication and integration with living systems.

Spiral wave chimeras in locally coupled oscillator systems

The recently discovered chimera state involves the coexistence of synchronized and desynchronized states for a group of identical oscillators. In this work, we show the existence of (inwardly) rotating spiral wave chimeras in the three-component reaction-diffusion systems where each element is locally coupled by diffusion. A transition from spiral waves with the smooth core to spiral wave chimeras is found as we change the local dynamics of the system or as we gradually increase the diffusion coefficient of the activator. Our findings on the spiral wave chimera in the reaction-diffusion systems suggest that spiral chimera states may be found in chemical and biological systems that can be modeled by a large population of oscillators indirectly coupled via a diffusive environment.

Cross-diffusion-driven hydrodynamic instabilities in a double-layer system: General classification and nonlinear simulations

Cross diffusion, whereby a flux of a given species entrains the diffusive transport of another species, can trigger buoyancy-driven hydrodynamic instabilities at the interface of initially stable stratifications. Starting from a simple three-component case, we introduce a theoretical framework to classify cross-diffusion-induced hydrodynamic phenomena in two-layer stratifications under the action of the gravitational field. A cross-diffusion-convection (CDC) model is derived by coupling the fickian diffusion formalism to Stokes equations. In order to isolate the effect of cross-diffusion in the convective destabilization of a double-layer system, we impose a starting concentration jump of one species in the bottom layer while the other one is homogeneously distributed over the spatial domain. This initial configuration avoids the concurrence of classic Rayleigh-Taylor or differential-diffusion convective instabilities, and it also allows us to activate selectively the cross-diffusion feedback by which the heterogeneously distributed species influences the diffusive transport of the other species. We identify two types of hydrodynamic modes [the negative cross-diffusion-driven convection (NCC) and the positive cross-diffusion-driven convection (PCC)], corresponding to the sign of this operational cross-diffusion term. By studying the space-time density profiles along the gravitational axis we obtain analytical conditions for the onset of convection in terms of two important parameters only: the operational cross-diffusivity and the buoyancy ratio, giving the relative contribution of the two species to the global density. The general classification of the NCC and PCC scenarios in such parameter space is supported by numerical simulations of the fully nonlinear CDC problem. The resulting convective patterns compare favorably with recent experimental results found in microemulsion systems.

From chemical systems to systems chemistry: Patterns in space and time

We present a brief, idiosyncratic overview of the past quarter century of progress in nonlinear chemical dynamics and discuss what we view as the most exciting recent developments and some challenges and likely areas of progress in the next 25 years.

Self-organized target and spiral patterns through the "coffee ring" effect

We studied the precipitation pattern of fullerene C60 nanocrystals generated through the evaporation of a confined liquid bridge. In contrast to the usual "coffee ring" pattern, both target and spiral patterns were observed. The characteristics of the pattern critically depended on the concentration of the solution, the temperature, and the level of vacuum. In addition, the morphology of the microscopic precipitates varied greatly as a function of these experimental parameters. This pattern formation can be interpreted as a two-step rhythmic nucleation/precipitation of fullerene crystals during receding motion of the contact line. Symmetric motion of the contact line produces a target pattern, and the propagation of distortion of the liquid interface caused by a disturbance generates a spiral pattern.

Scroll wave drift along steps, troughs, and corners

Three-dimensional excitable systems can create nonlinear scroll waves that rotate around one-dimensional phase singularities. Recent theoretical work predicts that these filaments drift along step-like height variations. Here, we test this prediction using experiments with thin layers of the Belousov-Zhabotinsky reaction. We observe that over short distances scroll waves are attracted towards the step and then rapidly commence a steady drift along the step line. The translating filaments always reside on the shallow side of the step near the edge. Accordingly, filaments in the deep domain initially collide with and shorten at the step wall. The drift speeds obey the predicted proportional dependence on the logarithm of the height ratio and the direction depends on the vortex chirality. We also observe drift along the perimeter of rectangular plateaus and find that the filaments perform sharp turns at the corners. In addition, we investigate rectangular troughs for which vortices of equal chirality can drift in different directions. The latter two effects are reproduced in numerical simulations with the Barkley model. The simulations show that narrow troughs instigate scroll wave encounters that induce repulsive interaction and symmetry breaking. Similar phenomena could exist in the geometrically complicated ventricles of the human heart where reentrant vortex waves cause tachycardia and fibrillation.

Interfacial hydrodynamic instabilities driven by cross-diffusion in reverse microemulsions

When two microemulsions are put in contact in the gravity field along a horizontal contact line, cross-diffusion can trigger the transport of one species in the presence of a gradient in concentration of another species. We show here theoretically that such cross-diffusion effects can induce buoyancy-driven convective instabilities at the interface between two solutions of different compositions even when initially the less dense solution lies on top of the denser one. Two different sources of convective modes are identified depending whether positive or negative cross-diffusion is involved. We evidence the two predicted cross-diffusion driven instabilities experimentally using a two-layer stratification of Aerosol-OT (AOT) water-in-oil microemulsions solutions with different water or AOT composition.

Entoptic perceptions of spiral waves and rare inward spirals

This report concerns Entoptic Rotating Spiral Waves as observed and documented by the author over a period of 46 years (1962-2008). The manifestations of these state-dependent, elusive rotating spiral entities were brief, emerging only during sleep-to-waking arousal epochs (in limbo). The images were seen only with closed lids in favorable ambient lighting-here, termed the umbral view. The clusters of rotating spiral entities emerge briefly to conscious view; their angular subtenses are estimated to be between 1° and 4°, and the rotations at ten-turns per second. Epochs of these activities commonly continued for about 20 s, with longevity of each visible entity up to 4 s. 90% of all observed entities were circular and outwardly levorotary; 5% were elliptical, appearing only as horizontal (prolate) entities. Overlapping units were rare, and were chiefly elliptical. Observations of twin spirals were also rare, seen in counter rotations, each twin inwardly rotating.

Chiral selection and frequency response of spiral waves in reaction-diffusion systems under a chiral electric field

Chirality is one of the most fundamental properties of many physical, chemical, and biological systems. However, the mechanisms underlying the onset and control of chiral symmetry are largely understudied. We investigate possibility of chirality control in a chemical excitable system (the Belousov-Zhabotinsky reaction) by application of a chiral (rotating) electric field using the Oregonator model. We find that unlike previous findings, we can achieve the chirality control not only in the field rotation direction, but also opposite to it, depending on the field rotation frequency. To unravel the mechanism, we further develop a comprehensive theory of frequency synchronization based on the response function approach. We find that this problem can be described by the Adler equation and show phase-locking phenomena, known as the Arnold tongue. Our theoretical predictions are in good quantitative agreement with the numerical simulations and provide a solid basis for chirality control in excitable media.

Spatially distributed current oscillations with electrochemical reactions in microfluidic flow cells

The formation of spatiotemporal patterns is investigated by using a chemical reaction on the surface of a high-aspect-ratio metal electrode positioned in a flow channel. A partial differential equation model is formulated for nickel dissolution in sulfuric acid in a microfluidic flow channel. The model simulations predict oscillatory patterns that are spatially distributed on the electrode surface; the downstream portion of the metal surface exhibits large-amplitude, nonlinear oscillations of dissolution rates, whereas the upstream portion displays small-amplitude, harmonic oscillations with a phase delay. The features of the dynamical response can be interpreted by the dependence of local dynamics on the widely varying surface conditions and the presence of strong coupling. The patterns can be observed for both contiguous and segmented metal surfaces. The existence of spatially distributed current oscillations is confirmed in experiments with Ni electrodissolution in a microfluidic device. The results show the impact of a widely heterogeneous environment on the types of patterns of chemical reaction rates.

Synthesis of programmable reaction-diffusion fronts using DNA catalyzers

We introduce a DNA-based reaction-diffusion (RD) system in which reaction and diffusion terms can be precisely and independently controlled. The effective diffusion coefficient of an individual reaction component, as we demonstrate on a traveling wave, can be reduced up to 2.7-fold using a self-assembled hydrodynamic drag. The intrinsic programmability of this RD system allows us to engineer, for the first time, orthogonal autocatalysts that counterpropagate with minimal interaction. Our results are in excellent quantitative agreement with predictions of the Fisher-Kolmogorov-Petrovskii-Piscunov model. These advances open the way for the rational engineering of pattern formation in pure chemical RD systems.

Influences of periodic mechanical deformation on pinned spiral waves

In a generic model of excitable media, we study the behavior of spiral waves interacting with obstacles and their dynamics under the influences of simple periodic mechanical deformation (PMD). Depending on the characteristics of the obstacles, i.e., size and excitability, the rotation of a pinned spiral wave shows different scenarios, e.g., embedding into or anchoring on an obstacle. Three different drift phenomena induced by PMD are observed: scattering on small partial-excitable obstacles, meander-induced unpinning on big partial-excitable obstacles, and drifting around small unexcitable obstacles. Their underlying mechanisms are discussed. The dependence of the threshold amplitude of PMD on the characteristics of the obstacles to successfully remove pinned spiral waves on big partial-excitable obstacles is studied.

Superdiffusive cusp-like waves in the mercuric iodide precipitate system and their transition to regular reaction bands

We report a two-dimensional (2D) reaction-diffusion system that exhibits a superdiffusive propagating wave with anomalous cusp-like contours. This wave results from a leading precipitation reaction (wavefront) and a trailing redissolution (waveback) between initially separated mercuric chloride and potassium iodide to produce mercuric iodide precipitate (HgI2) in a thin sheet of a solid hydrogel (agar) medium. The propagation dynamics is accompanied by continuous polymorphic transformations between the metastable yellow crystals and the stable red crystals of HgI2. We study the dynamics of wavefront and waveback propagation that reveals interesting anomalous superdiffusive behavior without the influence of external enhancement. We find that a transition from superdiffusive to subdiffusive dynamics occurs as a function of outer iodide concentration. Inner mercuric concentrations lead to the transition from the anomalous cusp-like to cusp-free regular bands. While gel concentration affects the speed of propagation of the wave, it has no effect on its shape or on its superdiffusive dynamics. Microscopically, we show that the macroscopic wave propagation and polymorphic transformations are accompanied by an Ostwald ripening mechanism in which larger red HgI2 crystals are formed at the expense of smaller yellow HgI2 crystals.

Unpinning of rotating spiral waves in cardiac tissues by circularly polarized electric fields

Spiral waves anchored to obstacles in cardiac tissues may cause lethal arrhythmia. To unpin these anchored spirals, comparing to high-voltage side-effect traditional therapies, wave emission from heterogeneities (WEH) induced by the uniform electric field (UEF) has provided a low-voltage alternative. Here we provide a new approach using WEH induced by the circularly polarized electric field (CPEF), which has higher success rate and larger application scope than UEF, even with a lower voltage. And we also study the distribution of the membrane potential near an obstacle induced by CPEF to analyze its mechanism of unpinning. We hope this promising approach may provide a better alternative to terminate arrhythmia.

Core solutions of rigidly rotating spiral waves in highly excitable media

Analytical spiral wave solutions for reaction-diffusion equations play an important role in studying spiral wave dynamics. In this paper, we focus on such analytical solutions in the case of highly excitable media. We present numerical evidence that, for rigidly rotating spiral waves in highly excitable media, the species values in the spiral core region do harmonic oscillations but not relaxation ones, and their amplitudes grow linearly with the distance from the rotation center. An analytical solution is proposed to describe such spiral wave dynamics, and the quantitative comparisons between the numerical results and the analytical solutions show that the proposed spiral core solution works well in highly excitable media.

Three-dimensional superdiffusive chemical waves in a precipitation system

We report and characterize new three-dimensional precipitation circular targets and spiral waves with superdiffusive dynamics.

Coupled chemical oscillators and emergent system properties

We review recent work on a variety of systems, from the nanometre to the centimetre scale, including microemulsions, microfluidic droplet arrays, gels and flow reactors, in which chemical oscillators interact to generate novel spatiotemporal patterns and/or mechanical motion.