Oscillations in an array of bistable microelectrodes coupled through a globally conserved quantity

The dynamical behavior of an array of microelectrodes is investigated under controlled current conditions during CO electrooxidation, a bistable electrochemical reaction with an S-shaped negative differential resistance (S-NDR) current-potential curve. Under these conditions, the total current constitutes a globally conserved quantity, thus coupling all microelectrodes globally. Upon increasing the total current, the microelectrodes activate one by one, with a single microelectrode being on its intermediate S-NDR current branch and the other ones being either on their passive or their active branches. When a few coupled microelectrodes are activated, the electrochemical system exhibits spontaneous potential oscillations. Mathematical analysis shows that oscillations arise already in a two group approximation of the dynamics, the two groups consisting of 1 electrode and n - 1 electrodes with n ≥ 3, respectively, with each group being described by a single evolution equation. In this minimal representation, oscillations occur when the single electrode is on the intermediate branch and the larger group is on the active branch.

1/f 2 noise in bistable electrocatalytic reactions on mesoscale electrodes

The formation of a self-organized spatial domain during current-controlled CO oxidation, a kinetically bistable reaction, is investigated experimentally and by deterministic simulations as a function of the electrode size and of the supporting electrolyte concentration. Decreasing the microelectrode size leads to the suppression of the spatial instability at the electrode and thus stabilizes the S-NDR branch of the reaction. The critical microelectrode size capable of supporting sustained domain formation is shown to be strongly affected by the sulfuric acid concentration, the characteristic time of the positive feedback loop increasing with the sulfate concentration. Furthermore, we demonstrate that for microelectrode diameters close to the instability threshold, small amplitude electrochemical potential fluctuations appear in the system. These potential fluctuations cannot be captured by deterministic mathematical models and are attributed to a strong enhancement of molecular fluctuations or intrinsic noise in the vicinity of the spatial instability. Analysis of the electrochemical noise revealed a 1/f ² frequency dependence and several common features with neuronal shot noise.

A classification scheme for chimera states

We present a universal characterization scheme for chimera states applicable to both numerical and experimental data sets. The scheme is based on two correlation measures that enable a meaningful definition of chimera states as well as their classification into three categories: stationary, turbulent, and breathing. In addition, these categories can be further subdivided according to the time-stationarity of these two measures. We demonstrate that this approach is both consistent with previously recognized chimera states and enables us to classify states as chimeras which have not been categorized as such before. Furthermore, the scheme allows for a qualitative and quantitative comparison of experimental chimeras with chimeras obtained through numerical simulations.

Robust autoassociative memory with coupled networks of Kuramoto-type oscillators

Uncertain recognition success, unfavorable scaling of connection complexity, or dependence on complex external input impair the usefulness of current oscillatory neural networks for pattern recognition or restrict technical realizations to small networks. We propose a network architecture of coupled oscillators for pattern recognition which shows none of the mentioned flaws. Furthermore we illustrate the recognition process with simulation results and analyze the dynamics analytically: Possible output patterns are isolated attractors of the system. Additionally, simple criteria for recognition success are derived from a lower bound on the basins of attraction.

Chimeras in globally coupled oscillatory systems: From ensembles of oscillators to spatially continuous media

We study an oscillatory medium with a nonlinear global coupling that gives rise to a harmonic mean-field oscillation with constant amplitude and frequency. Two types of cluster states are found, each undergoing a symmetry-breaking transition towards a related chimera state. We demonstrate that the diffusional coupling is non-essential for these complex dynamics. Furthermore, we investigate localized turbulence and discuss whether it can be categorized as a chimera state.

Clustering as a prerequisite for chimera states in globally coupled systems

The coexistence of coherently and incoherently oscillating parts in a system of identical oscillators with symmetrical coupling, i.e., a chimera state, is even observable with uniform global coupling. We address the question of the prerequisites for these states to occur in globally coupled systems. By analyzing two different types of chimera states found for nonlinear global coupling, we show that a clustering mechanism to split the ensemble into two groups is needed as a first step. In fact, the chimera states inherit properties from the cluster states in which they originate. Remarkably, they can exist in parameter space between cluster and chaotic states, as well as between cluster and synchronized states.

Two-cluster solutions in an ensemble of generic limit-cycle oscillators with periodic self-forcing via the mean-field

We study two-cluster solutions of an ensemble of generic limit-cycle oscillators in the vicinity of a Hopf bifurcation, i.e., Stuart-Landau oscillators, with a nonlinear global coupling. This coupling leads to conserved mean-field oscillations acting back on the individual oscillators as a forcing. A reduction to two effective equations makes a linear stability analysis of the cluster solutions possible. These equations exhibit a π-rotational symmetry, leading to a complex bifurcation structure and a wide variety of solutions. In fact, the principal bifurcation structure resembles that of a 2:1 resonance tongue, while inside the tongue we observe an 1:1 entrainment.

Investigation of Front Propagation in an Electrochemical System: Experiments and Numerical Simulations

We examined the spatio-temporal behavior of an electrochemical system in the bistable regime, in which the system might take on either a high current density state (active) or a low current density state (passive) at one value of the externally applied voltage. The transition from the passive to the active state is accompanied by accelerating fronts with sharp interfaces, whereas the reverse transition from the active to the passive state exhibits much smoother spatial variations. Also the evolution of the total current density displays qualitative differences in the two cases. Both, the differences of the spatial patterns as well as those of the total current densities are reproduced with a mathemati­cal model, which also reveals the origin of the asymmetry of the transitions: a global coupling, intrinsic in all electrochemical systems, in combination with the specific dependence of the reaction current on the electrode potential.

Coexistence of synchrony and incoherence in oscillatory media under nonlinear global coupling

We report a novel mechanism for the formation of chimera states, a peculiar spatiotemporal pattern with coexisting synchronized and incoherent domains found in ensembles of identical oscillators. Considering Stuart-Landau oscillators, we demonstrate that a nonlinear global coupling can induce this symmetry breaking. We find chimera states also in a spatially extended system, a modified complex Ginzburg-Landau equation. This theoretical prediction is validated with an oscillatory electrochemical system, the electro-oxidation of silicon, where the spontaneous formation of chimeras is observed without any external feedback control.

Subharmonic phase clusters in the complex Ginzburg-Landau equation with nonlinear global coupling

A wide variety of subharmonic n -phase cluster patterns was observed in experiments with spatially extended chemical and electrochemical oscillators. These patterns cannot be captured with a phase model. We demonstrate that the introduction of a nonlinear global coupling (NGC) in the complex Ginzburg-Landau equation has subharmonic cluster pattern solutions in wide parameter ranges. The NGC introduces a conservation law for the oscillatory state of the homogeneous mode, which describes the strong oscillations of the mean field in the experiments. We show that the NGC causes a pronounced 2:1 self-resonance on any spatial inhomogeneity, leading to two-phase subharmonic clustering, as well as additional higher resonances. Nonequilibrium Ising-Bloch transitions occur as the coupling strength is varied.

Resonance tongues in a system of globally coupled FitzHugh-Nagumo oscillators with time-periodic coupling strength

We investigate the dynamics of a population of globally coupled FitzHugh-Nagumo oscillators with a time-periodic coupling strength. While for synchronizing global coupling, the in-phase state is always stable, the oscillators split into several cluster states for desynchronizing global coupling, most commonly in two, irrespective of the coupling strength. This confines the ability of the system to form n:m locked states considerably. The prevalence of two and four cluster states leads to large 2:1 and 4:1 subharmonic resonance regions, while at low coupling strength for a harmonic 1:1 or a superharmonic 1:m time-periodic coupling coefficient, any resonances are absent and the system exhibits nonresonant phase drifting cluster states. Furthermore, in the unforced, globally coupled system the frequency of the oscillators in a cluster state is in general lower than that of the uncoupled oscillator and strongly depends on the coupling strength. Periodic variation of the coupling strength at twice the natural frequency causes each oscillator to keep oscillating with its autonomous oscillation period.

Effects of Boundaries on Pattern Formation: Catalytic Oxidation of CO on Platinum

The effect of boundaries on pattern formation was studied for the catalytic oxidation of carbon monoxide on platinum surfaces. Photolithography was used to create microscopic reacting domains on polycrystalline foils and single-crystal platinum (110) surfaces with inert titanium overlayers. Certain domain geometries give rise to patterns that have not been observed on the untreated catalyst and bring to light surface mechanisms that have no analog in homogeneous reaction-diffusion systems.

Confined Spatio-Temporal Chaos During Metal Electrodissolution: Simulations

The electrodissolution of metal electrodes exhibits often sustained oscillations of the current density. We study the spatio-temporal dynamics of a metal disk electrode embedded in an insulator in a cylindrical electrochemical cell with a ring-shaped CE in the oscillatory region. The chosen cell geometry introduces a strong radial parameter dependence into the system. For low conductivity the dynamics is spatio-temporally chaotic. A small aspect ratio of the cell supports 2-dimensional space-time chaos on the entire disk electrode. For larger aspect ratios and thus larger parameter gradients, however, we observe a ‘self-confinement’ of the turbulent dynamics to the central part of the disk electrode, the outer ring of the electrode exhibiting angularly uniform oscillations. It can be expected that the state of confined chaos is not restricted to electrochemical systems but can be established also in other dynamical systems by introducing an appropriate parameter gradient.