Finding influential nodes in networks using pinning control: Centrality measures confirmed with electrochemical oscillators.

The spatiotemporal organization of networks of dynamical units can break down resulting in diseases (e.g., in the brain) or large-scale malfunctions (e.g., power grid blackouts). Re-establishment of function then requires identification of the optimal intervention site from which the network behavior is most efficiently re-stabilized. Here, we consider one such scenario with a network of units with oscillatory dynamics, which can be suppressed by sufficiently strong coupling and stabilizing a single unit, i.e., pinning control. We analyze the stability of the network with hyperbolas in the control gain vs coupling strength state space and identify the most influential node (MIN) as the node that requires the weakest coupling to stabilize the network in the limit of very strong control gain. A computationally efficient method, based on the Moore-Penrose pseudoinverse of the network Laplacian matrix, was found to be efficient in identifying the MIN. In addition, we have found that in some networks, the MIN relocates when the control gain is changed, and thus, different nodes are the most influential ones for weakly and strongly coupled networks. A control theoretic measure is proposed to identify networks with unique or relocating MINs. We have identified real-world networks with relocating MINs, such as social and power grid networks. The results were confirmed in experiments with networks of chemical reactions, where oscillations in the networks were effectively suppressed through the pinning of a single reaction site determined by the computational method.

A numerical investigation of the effect of external resistance and applied potential on the distribution of periodicity and chaos in the anodic dissolution of nickel.

The oscillatory electrodissolution of nickel is one among several reactions utilized as a model-system to study the emergence of oscillations and pattern formation in electrochemical interfaces, in addition to frequently providing experimental proofs for theoretical predictions in synchronization engineering. The reaction was modeled in 1992 by Haim and co-workers [J. Phys. Chem. 1992, 96, 2676] and since then the model has been used with great success. Although some numerical studies have been done in this regard, there is apparently no detailed investigation of the effect of control parameters on the complex dynamics of nickel dissolution. Here, we provide a well-detailed and rigorous analysis of the effect of the external resistance and applied potential by simulating high-resolution phase diagrams based on the calculation of Lyapunov exponents and isospike diagrams. Our findings clearly indicate a strong dependence of the self-similar periodic islands, the so-called shrimps (i.e., periodic islands within chaotic domains in the parameter space), with the control parameters. Overall, we have observed a low density of periodic structures in the phase diagrams, being completely suppressed for large values of resistance and potential. The shrimp-like structures become gradually elongated with an increase of the control parameters to the point where only diagonally aligned periodic bands intertwined with chaotic domains are present. Interestingly, period-doubling cascades were observed not only on the shrimps but also on the periodic bands. The detailed distribution of chaos and periodicity of oscillatory electrodissolution reactions in resistance-potential phase diagrams can bring, for instance, important information to experimentalists to set a desired dynamic behavior and, therefore, to create novel nanostructured self-organized materials.

Quasiperiodic behavior in the electrodeposition of Cu/Sn multilayers: extraction of activation energies and wavelet analysis.

Living systems are one of the many examples in which self-organizing systems yield more intricate structures than those that can be achieved using a "step-by-step" approach. This phenomenon can be observed in electrochemical organic synthesis, oscillating metal deposition and in chemical clocks. There is a plenitude of temporal instabilities during self-organization, from ordinary period-one oscillations, going through quasiperiodicity, to the onset of chaos. Here, we describe the emergence of quasiperiodic behavior during the oscillatory electro-deposition of Cu/Sn. The time-series were characterized using a continuous wavelet transform in order to extract the oscillation frequency of the process with changes in temperature, and to calculate the apparent activation energies. Two different energy ranges are presented, and these are attributed to an activation barrier (∼50 kJ mol-1) which is closely related to a fast time-scale of the feedback loops responsible for the current oscillations and a diffusional process (∼20 kJ mol-1), connected to the slow modulations of the oscillation amplitude, giving rise to quasiperiodic dynamics. This kinetic information might provide information on the self-organized synthesis of Cu/Sn metallic multilayers which are kept far from the thermodynamic equilibrium. As self-organization in chemical systems is rapidly developing into a powerful strategy for designing new functional materials, the availability of kinetic parameters is of major interest in rational design.

Multivariate statistical analysis of chemical and electrochemical oscillators for an accurate frequency selection.

The effect of experimental parameters on the frequency of chemical oscillators has been systematically studied since the first observations of clock reactions. The approach is mainly based on univariate changes in one specific parameter while others are kept constant. The frequency is then monitored and the effect of each parameter is discussed separately. This type of analysis, however, does not take into account the multiple interactions among the controllable parameters and the synergic responses on the oscillation frequency. We have carried out a multivariate statistical analysis of chemical (BZ-ferroin catalyzed reaction) and electrochemical (Cu/Cu2O cathodic deposition) oscillators and identified the contributions of the experimental parameters on frequency variations. The BZ reaction presented a strong dependence on the initial concentration of sodium bromate and temperature, resulting in a frequency increase. The concentration of malonic acid, the organic substrate, affects the system but with lower intensity compared with the combination of sodium bromate and temperature. On the other hand, the Cu/Cu2O electrochemical oscillator was shown to be less sensitive to changes in the temperature. The applied current density and pH were the two parameters which most perturbed the system. Interestingly, the frequency behaved nonmonotonically with a quadratic dependence. The multivariate analysis of both oscillators exhibited significant differences - while the homogenous oscillator displayed a linear dependence with the factors, the heterogeneous one revealed a more complex dependence with quadratic terms. Our results may contribute, for instance, in the synthesis of self-organized materials in which an accurate frequency selection is required and, depending on its value, different physicochemical properties are obtained.

Modulation of Turing Patterns in the CDIMA Reaction by Ultraviolet and Visible Light.

We have carried out the first systematic study of the effects of ultraviolet light, both alone and in combination with visible white light, on Turing patterns in the chlorine dioxide-iodine-malonic acid (CDIMA) reaction. The ultraviolet light used has a sharp peak at 368 nm and can perturb the system selectively. It primarily decomposes chlorine dioxide in a zeroth-order reaction, and when it is used to illuminate Turing patterns, shrunken spots are formed with an imperfect hexagonal arrangement. The ultraviolet light competes directly with the visible white light via the photoreaction with dissolved chlorine dioxide, which prevents the total suppression of patterns at intermediate intensities of white light. These results suggest that specific wavelengths of light in the ultraviolet spectrum selectively modify the chemistry behind the pattern formation and can be utilized to generate novel self-organized structures under forcing conditions. We propose a modified Lengyel-Epstein model to incorporate the effect of ultraviolet illumination and obtain good qualitative agreement between simulations and experiments. These results support the idea that chlorine dioxide photoreaction is a key step in modulating CDIMA patterns under ultraviolet illumination.

Restoring oscillatory behavior from amplitude death with anti-phase synchronization patterns in networks of electrochemical oscillations.

The dynamical behavior of delay-coupled networks of electrochemical reactions is investigated to explore the formation of amplitude death (AD) and the synchronization states in a parameter region around the amplitude death region. It is shown that difference coupling with odd and even numbered ring and random networks can produce the AD phenomenon. Furthermore, this AD can be restored by changing the coupling type from difference to direct coupling. The restored oscillations tend to create synchronization patterns in which neighboring elements are in nearly anti-phase configuration. The ring networks produce frozen and rotating phase waves, while the random network exhibits a complex synchronization pattern with interwoven frozen and propagating phase waves. The experimental results are interpreted with a coupled Stuart-Landau oscillator model. The experimental and theoretical results reveal that AD behavior is a robust feature of delayed coupled networks of chemical units; if an oscillatory behavior is required again, even a small amount of direct coupling could be sufficient to restore the oscillations. The restored nearly anti-phase oscillatory patterns, which, to a certain extent, reflect the symmetry of the network, represent an effective means to overcome the AD phenomenon.

Fronts and patterns in a spatially forced CDIMA reaction.

We use the CDIMA chemical reaction and the Lengyel-Epstein model of this reaction to study resonant responses of a pattern-forming system to time-independent spatial periodic forcing. We focus on the 2 : 1 resonance, where the wavenumber of a one-dimensional periodic forcing is about twice the wavenumber of the natural stripe pattern that the unforced system tends to form. Within this resonance, we study transverse fronts that shift the phase of resonant stripe patterns by π. We identify phase fronts that shift the phase discontinuously, and pairs of phase fronts that shift the phase continuously, clockwise and anti-clockwise. We further identify a front bifurcation that destabilizes the discontinuous front and leads to a pair of continuous fronts. This bifurcation is the spatial counterpart of the nonequilibrium Ising-Bloch (NIB) bifurcation in temporally forced oscillatory systems. The spatial NIB bifurcation that we find occurs as the forcing strength is increased, unlike earlier studies of the NIB bifurcation. Furthermore, the bifurcation is subcritical, implying a range of forcing strength where both discontinuous Ising fronts and continuous Bloch fronts are stable. Finally, we find that both Ising fronts and Bloch fronts can form discrete families of bound pairs, and we relate arrays of these front pairs to extended rectangular and oblique patterns.

Coupled slow and fast surface dynamics in an electrocatalytic oscillator: model and simulations.

The co-existence of disparate time scales is pervasive in many systems. In particular for surface reactions, it has been shown that the long-term evolution of the core oscillator is decisively influenced by slow surface changes, such as progressing deactivation. Here we present an in-depth numerical investigation of the coupled slow and fast surface dynamics in an electrocatalytic oscillator. The model consists of four nonlinear coupled ordinary differential equations, investigated over a wide parameter range. Besides the conventional bifurcation analysis, the system was studied by means of high-resolution period and Lyapunov diagrams. It was observed that the bifurcation diagram changes considerably as the irreversible surface poisoning evolves, and the oscillatory region shrinks. The qualitative dynamics changes accordingly and the chaotic oscillations are dramatically suppressed. Nevertheless, periodic cascades are preserved in a confined region of the resistance vs. voltage diagram. Numerical results are compared to experiments published earlier and the latter reinterpreted. Finally, the comprehensive description of the time-evolution in the period and Lyapunov diagrams suggests further experimental studies correlating the evolution of the system's dynamics with changes of the catalyst structure.

Role of Structural and Compositional Changes of Cu2O Nanocubes in Nitrate Electroreduction to Ammonia.

Nitrate electroreduction reaction (NO3RR) to ammonia (NH3) still faces fundamental and technological challenges. While Cu-based catalysts have been widely explored, their activity and stability relationship are still not fully understood. Here, we systematically monitored the dynamic alterations in the chemical and morphological characteristics of Cu2O nanocubes (NCs) during NO3RR in an alkaline electrolyte. In 1 h of electrolysis from -0.10 to -0.60 V vs RHE, the electrocatalyst achieved the maximum NH3 faradaic efficiency (FE) and yield rate at -0.3 V (94% and 149 µmol h-1 cm-2, respectively). Similar efficiency could be found at a lower overpotential (-0.20 V vs RHE) in long-term electrolysis. At -0.20 V vs RHE, the catalyst FE increased from 73% in the first 2 h to ∼90% in 10 h of electrolysis. Electron microscopy revealed the loss of the cubic shape with the formation of sintered domains. In situ Raman, X-ray diffraction (XRD), and in situ Cu K-edge X-ray absorption near-edge spectroscopy (XANES) indicated the reduction of Cu2O to oxide-derived Cu0 (OD-Cu). Nevertheless, a remaining Cu2O phase was noticed after 1 h of electrolysis at -0.3 V vs RHE. This observation indicates that the activity and selectivity of the initially well-defined Cu2O NCs are not solely dependent on the initial structure. Instead, it underscores the emergence of an OD-Cu-rich surface, evolving from near-surface to underlying layers over time and playing a crucial role in the reaction pathways. By employing online differential electrochemical mass spectrometry (DEMS) and in situ Fourier transform infrared spectroscopy (FTIR), we experimentally probed the presence of key intermediates (NO and NH2OH) and byproducts of NO3RR (N2 and N2H x ) for NH3 formation. These results show a complex relationship between activity and stability of the nanostructured Cu2O oxide catalyst for NO3RR.

Mechanistic aspects of the linear stabilization of non-stationary electrochemical oscillations.

The problem of non-stationarity in experimentally recorded time-series is common in many (electro)chemical systems. Underlying this non-stationarity is the slow drift in some uncontrollable parameter, and it occurs in spite of the fact that all controllable parameters are kept constant. Particularly for electrochemical systems, some of us have recently suggested [J. Phys. Chem. C, 144, (2010), 22262-22268] an empirical method to stabilize experimental time-series. The method was exemplified for the electro-oxidation of methanol and different patterns were satisfactorily stabilized. In this paper we further elaborate some mechanistic aspects of this method and test it for the electro-oxidation of formaldehyde, a system that has some resemblance with the electro-oxidation of methanol, but produces a richer dynamics. In terms of the reaction mechanism, we were able to describe the coupling and to separate the surface processes of the two sub-systems: the fast one (or the core-oscillator) and the slow one, responsible for the drift.

Forcing of Turing patterns in the chlorine dioxide-iodine-malonic acid reaction with strong visible light.

We investigate the sensitivity of Turing patterns in the chlorine dioxide-iodine-malonic acid reaction to illumination by strong white light. Intense illumination results in an increase of [I(-)], in contrast to previous studies, which found only decreased [I(-)] for weak and intermediate intensities of illumination. We propose an expanded mechanism to explain the experimental observations. Both experimental and numerical results suggest that [ClO2] is the key parameter that determines whether the high iodide state is obtained under strong illumination. When strong illumination is applied through a spatially periodic mask with black and white stripes, a dark state with high [I(-)] is produced in the illuminated domain and a light state with low [I(-)] forms in the nonilluminated domain. Depending on the black:white ratio of the mask and its wavelength, Turing patterns can coexist with either the light or the dark state in the nonilluminated domain.

The dual pathway in action: decoupling parallel routes for CO2 production during the oscillatory electro-oxidation of methanol.

As in the case of most small organic molecules, the electro-oxidation of methanol to CO(2) is believed to proceed through a so-called dual pathway mechanism. The direct pathway proceeds via reactive intermediates such as formaldehyde or formic acid, whereas the indirect pathway occurs in parallel, and proceeds via the formation of adsorbed carbon monoxide (CO(ad)). Despite the extensive literature on the electro-oxidation of methanol, no study to date distinguished the production of CO(2) from direct and indirect pathways. Working under, far-from-equilibrium, oscillatory conditions, we were able to decouple, for the first time, the direct and indirect pathways that lead to CO(2) during the oscillatory electro-oxidation of methanol on platinum. The CO(2) production was followed by differential electrochemical mass spectrometry and the individual contributions of parallel pathways were identified by a combination of experiments and numerical simulations. We believe that our report opens some perspectives, particularly as a methodology to be used to identify the role played by surface modifiers in the relative weight of both pathways-a key issue to the effective development of catalysts for low temperature fuel cells.

Turing patterns in the chlorine dioxide-iodine-malonic acid reaction with square spatial periodic forcing.

We use the photosensitive chlorine dioxide-iodine-malonic acid reaction-diffusion system to study wavenumber locking of Turing patterns to two-dimensional "square" spatial forcing, implemented as orthogonal sets of bright bands projected onto the reaction medium. Various resonant structures emerge in a broad range of forcing wavelengths and amplitudes, including square lattices and superlattices, one-dimensional stripe patterns and oblique rectangular patterns. Numerical simulations using a model that incorporates additive two-dimensional spatially periodic forcing reproduce well the experimental observations.

Temperature (over)compensation in an oscillatory surface reaction.

Biological rhythms are regulated by homeostatic mechanisms that assure that physiological clocks function reliably independent of temperature changes in the environment. Temperature compensation, the independence of the oscillatory period on temperature, is known to play a central role in many biological rhythms, but it is rather rare in chemical oscillators. We study the influence of temperature on the oscillatory dynamics during the catalytic oxidation of formic acid on a polycrystalline platinum electrode. The experiments are performed at five temperatures from 5 to 25 °C, and the oscillations are studied under galvanostatic control. Under oscillatory conditions, only non-Arrhenius behavior is observed. Overcompensation with temperature coefficient (q(10), defined as the ratio between the rate constants at temperature T + 10 °C and at T) < 1 is found in most cases, except that temperature compensation with q(10) ≈ 1 predominates at high applied currents. The behavior of the period and the amplitude result from a complex interplay between temperature and applied current or, equivalently, the distance from thermodynamic equilibrium. High, positive apparent activation energies were obtained under voltammetric, nonoscillatory conditions, which implies that the non-Arrhenius behavior observed under oscillatory conditions results from the interplay among reaction steps rather than from a weak temperature dependence of the individual steps.

Tuning Electrochemical Bistability by Surface Area Blocking in the Cathodic Deposition of Copper.

We report herein a precise control of the electrochemical bistability induced by surface area changes during the cathodic deposition of copper. Small additions of 1,10-phenanthroline (Phen) in the reaction media present an inhibiting effect on the global rate mainly due to the adsorption of protonated Phen. The increase of its concentration favors a shrinkage of the bifurcation (saddle-node) diagram and shifts it to less negative potentials. The dynamic instability is verified by impedance measurements, and a negative impedance is clearly found. We calculated the apparent molar mass of the adsorbents using in situ gravimetric monitoring in the electrochemical experiments, and the results indicate that mass changes occur mainly due to the reduction of copper from bivalent ions dissolved in the reaction media. Importantly, the adsorption of protonated Phen molecules does not show a considerable contribution in mass variations but prevents the formation of a copper course grained morphology over the surface. Imaging analysis indicates finer nodulations at the lower branch compared to the upper branch in the bistability domain. On the basis of these observations, a kinetic mechanism is proposed and a good agreement is obtained between the apparent molar mass extracted from experiments and the theoretical values. Altogether, our results contribute to a detailed physical chemical description of the nonlinear behavior, bringing new insights about this reaction and pointing out the possibility to design switchable surface electrodes by taking advantage of the bistable behavior.

Elucidation of Reaction Mechanisms Far from Thermodynamic Equilibrium.

Far from equilibrium: This thesis provides a deep mechanistic analysis of the electrooxidation of methanol when the system is kept far from the thermodynamic equilibrium. Under an oscillatory regime, interesting characteristics between the elementary reaction steps were observed. We were able to elucidate the effect of the intrinsic drift in a potential time-series responsible for spontaneous transition of temporal patterns and the carbon dioxide decoupling from direct and indirect pathways.

Phase-selective entrainment of nonlinear oscillator ensembles.

The ability to organize and finely manipulate the hierarchy and timing of dynamic processes is important for understanding and influencing brain functions, sleep and metabolic cycles, and many other natural phenomena. However, establishing spatiotemporal structures in biological oscillator ensembles is a challenging task that requires controlling large collections of complex nonlinear dynamical units. In this report, we present a method to design entrainment signals that create stable phase patterns in ensembles of heterogeneous nonlinear oscillators without using state feedback information. We demonstrate the approach using experiments with electrochemical reactions on multielectrode arrays, in which we selectively assign ensemble subgroups into spatiotemporal patterns with multiple phase clusters. The experimentally confirmed mechanism elucidates the connection between the phases and natural frequencies of a collection of dynamical elements, the spatial and temporal information that is encoded within this ensemble, and how external signals can be used to retrieve this information.

Restoration of rhythmicity in diffusively coupled dynamical networks.

Oscillatory behaviour is essential for proper functioning of various physical and biological processes. However, diffusive coupling is capable of suppressing intrinsic oscillations due to the manifestation of the phenomena of amplitude and oscillation deaths. Here we present a scheme to revoke these quenching states in diffusively coupled dynamical networks, and demonstrate the approach in experiments with an oscillatory chemical reaction. By introducing a simple feedback factor in the diffusive coupling, we show that the stable (in)homogeneous steady states can be effectively destabilized to restore dynamic behaviours of coupled systems. Even a feeble deviation from the normal diffusive coupling drastically shrinks the death regions in the parameter space. The generality of our method is corroborated in diverse non-linear systems of diffusively coupled paradigmatic models with various death scenarios. Our study provides a general framework to strengthen the robustness of dynamic activity in diffusively coupled dynamical networks.

The electro-oxidation of ethylene glycol on platinum over a wide pH range: oscillations and temperature effects.

We report a comprehensive study of the electro-oxidation of ethylene glycol (EG) on platinum with emphasis on the effects exerted by the electrolyte pH, the EG concentration, and temperature, under both regular and oscillatory conditions. We extracted and discussed parameters such as voltammetric activity, reaction orders (with respect to [EG]), oscillation's amplitude, frequency and waveform, and the evolution of the mean electrode potential at six pH values from 0 to 14. In addition, we obtained the apparent activation energies under several different conditions. Overall, we observed that increasing the electrolyte pH results in a discontinuous transition in most properties studied under both voltammetric and oscillatory regimes. As a relevant result in this direction, we found that the increase in the reaction order with pH is mediated by a minimum (~ 0) at pH = 12. Furthermore, the solution pH strongly affects all features investigated, c.f. the considerable increase in the oscillatory frequency and the decrease in the, oscillatory, activation energy as the pH increase. We suggest that adsorbed CO is probably the main surface-blocking species at low pH, and its absence at high pH is likely to be the main reason behind the differences observed. The size of the parameter region investigated and the amount of comparable parameters and properties presented in this study, as well as the discussion that followed illustrate the strategy of combining investigations under conventional and oscillatory regimes of electrocatalytic systems.

Temperature effects on the oscillatory electro-oxidation of methanol on platinum.

We report in this paper the effect of temperature on the oscillatory electro-oxidation of methanol on polycrystalline platinum in aqueous sulfuric acid media. Potential oscillations were studied under galvanostatic control and at four temperatures ranging from 5 to 35 degrees C. For a given temperature, the departure from thermodynamic equilibrium does not affect the oscillation period and results in a slight increase of the oscillation amplitude. Apparent activation energies were also evaluated in voltammetric and chronoamperometric experiments and were compared to those obtained under oscillatory conditions. In any case, the apparent activation energies values fell into the region between 50 and 70 kJ mol(-1). Specifically under oscillatory conditions an apparent activation energy of 60 +/- 3 kJ mol(-1) and a temperature coefficient q10 of about 2.3 were observed. The present findings extend our recently published report (J. Phys. Chem. A, 2008, 112, 4617) on the temperature effect on the oscillatory electro-oxidation of formic acid. We found that, despite the fact that both studies were carried out under similar conditions, unlike the case of formic acid, only conventional, Arrhenius, dynamics was observed for methanol.