Spatially organized partial synchronization through the chimera mechanism in a network of electrochemical reactions.

Detailed experimental and numerical results are presented about the pattern formation mechanism of spatially organized partially synchronized states in a networked chemical system with oscillatory metal dissolution. Numerical simulations of the reaction system are used to identify experimental conditions (heterogeneity, network topology, and coupling time-scale) under which the chemical reactions, which take place in a network, are split into coexisting coherent and incoherent domains through the chimera mechanism. Experiments are carried out with a network of twenty electrodes arranged in a ring with seven nearest neighbor couplings in both directions along the ring. The patterns are characterized by analyzing the oscillation frequencies and entrainments to the mean field of the phases of oscillations. The chimera state forms from two domains of elements: the chimera core in which the elements have identical frequencies and are entrained to their corresponding mean field and the chimera shell where the elements exhibit desynchrony with each other and the mean field. The experiments point out the importance of low level of heterogeneities (e.g., surface conditions) and optimal level of coupling strength and time-scale as necessary components for the realization of the chimera state. For systems with large heterogeneities, a 'remnant' chimera state is identified where the pattern is strongly affected by the presence of frequency clusters. The exploration of dynamical features with networked reactions could open up ways for identification of novel types of patterns that cannot be observed with reaction diffusion systems (with localized interactions) or with reactions under global constraints, coupling, or feedback.

Spatially organized dynamical states in chemical oscillator networks: synchronization, dynamical differentiation, and chimera patterns.

Dynamical processes in many engineered and living systems take place on complex networks of discrete dynamical units. We present laboratory experiments with a networked chemical system of nickel electrodissolution in which synchronization patterns are recorded in systems with smooth periodic, relaxation periodic, and chaotic oscillators organized in networks composed of up to twenty dynamical units and 140 connections. The reaction system formed domains of synchronization patterns that are strongly affected by the architecture of the network. Spatially organized partial synchronization could be observed either due to densely connected network nodes or through the 'chimera' symmetry breaking mechanism. Relaxation periodic and chaotic oscillators formed structures by dynamical differentiation. We have identified effects of network structure on pattern selection (through permutation symmetry and coupling directness) and on formation of hierarchical and 'fuzzy' clusters. With chaotic oscillators we provide experimental evidence that critical coupling strengths at which transition to identical synchronization occurs can be interpreted by experiments with a pair of oscillators and analysis of the eigenvalues of the Laplacian connectivity matrix. The experiments thus provide an insight into the extent of the impact of the architecture of a network on self-organized synchronization patterns.

Synchronization of electrochemical oscillators with differential coupling.

Experiments are presented to describe the effect of capacitive coupling of two electrochemical oscillators during Ni dissolution in sulfuric acid solution. Equivalent circuit analysis shows that the coupling between the oscillators occurs through the difference between the differentials of the electrode potentials. The differential nature of the coupling introduces strong negative nonisochronicity (i.e., phase shear, strong dependence of the period on the amplitude) in the coupling mechanism with smooth oscillators (under conditions just above a Hopf bifurcation point). Because of the negative nonisochronicity, asymmetrically coupled oscillators exhibit anomalous phase synchronization in the form of frequency difference enhancement. At strong coupling bistability is observed between in-phase and antiphase synchronized states. With relaxation oscillators, in contrast to the resistive coupling where antiphase synchronization can occur, the typical system response with weak coupling is out-of-phase synchronization. When the capacitance is applied on the individual resistors attached to the electrodes the oscillators exhibit weak positive nonisochronicity; this is in contrast with the strong negative nonisochronicity obtained with cross coupling. The proposed coupling configurations reveal the importance of the nonisochronicity level of oscillations for the experimentally observed synchronization patterns and also provide efficient ways of tuning the nonisochronicity level of the oscillations. This latter feature can be exploited to design synchronization features with a combination of resistive (difference) and capacitive (differential) coupling.

Phase coherence and attractor geometry of chaotic electrochemical oscillators.

Chaotic attractors are known to often exhibit not only complex dynamics but also a complex geometry in phase space. In this work, we provide a detailed characterization of chaotic electrochemical oscillations obtained experimentally as well as numerically from a corresponding mathematical model. Power spectral density and recurrence time distributions reveal a considerable increase of dynamic complexity with increasing temperature of the system, resulting in a larger relative spread of the attractor in phase space. By allowing for feasible coordinate transformations, we demonstrate that the system, however, remains phase-coherent over the whole considered parameter range. This finding motivates a critical review of existing definitions of phase coherence that are exclusively based on dynamical characteristics and are thus potentially sensitive to projection effects in phase space. In contrast, referring to the attractor geometry, the gradual changes in some fundamental properties of the system commonly related to its phase coherence can be alternatively studied from a purely structural point of view. As a prospective example for a corresponding framework, recurrence network analysis widely avoids undesired projection effects that otherwise can lead to ambiguous results of some existing approaches to studying phase coherence. Our corresponding results demonstrate that since temperature increase induces more complex chaotic chemical reactions, the recurrence network properties describing attractor geometry also change gradually: the bimodality of the distribution of local clustering coefficients due to the attractor's band structure disappears, and the corresponding asymmetry of the distribution as well as the average path length increase.

Dynamics of electrochemical oscillators with electrode size disparity: asymmetrical coupling and anomalous phase synchronization.

Experiments are carried out in dual electrode oscillatory Ni electrodissolution in which the two electrodes have different surface areas. The transition to phase synchronization is analyzed as asymmetrical coupling strength, induced by placing a cross resistance between the electrodes, is varied. It is shown that because of nonisochronicity (phase shear, i.e., strong dependence of period on amplitude) of the oscillators, anomalous phase synchronization effects can be observed: advanced/delayed synchronization and, to a lesser extent, frequency difference enhancement. The type of synchronization is strongly affected by the underlying heterogeneities of the oscillators; in the experiments with a slow driver (large surface area) electrode the synchronization is advanced, with a fast driver electrode the synchronization is delayed with respect to symmetrical coupling. The findings thus reveal that the interplay of asymmetrical coupling with the types of inherent heterogeneities plays an important role for the interpretation of size effects in the dynamical behavior of a nonlinear chemical reaction.

Phase synchronization of three locally coupled chaotic electrochemical oscillators: enhanced phase diffusion and identification of indirect coupling.

Experiments are carried out with three locally coupled phase coherent chaotic electrochemical oscillators (A-B-C) in nickel dissolution in sulfuric acid. As the interaction strength is increased among the electrodes, an onset of synchronization is observed where the frequencies become identical and the phase differences are bounded. The precision of the period of the oscillators is characterized by phase diffusion coefficients from phases and phase differences. The transition to synchronization with increase of coupling strength was found to be accompanied by enhanced phase fluctuations that cause the precision of the oscillations to deteriorate. A parallel synchrony analysis showed that the direct (between A and B and B and C) and the indirect (between A and C) couplings can be correctly identified with the use of a partial phase synchrony index; therefore, the network topology can be deduced from dynamical measurements. Numerical simulations with a locally coupled model for electrochemical chaos confirm the presence of enhanced phase fluctuations close to the transition to synchronization and the usefulness of the partial phase synchrony index for differentiation of direct from indirect interactions in a small network of oscillators.

Effect of temperature on precision of chaotic oscillations in nickel electrodissolution.

We investigate the effects of temperature on complexity features of chaotic electrochemical oscillations using the anodic electrodissolution of nickel in sulfuric acid. The precision of the "period" of chaotic oscillation is characterized by phase diffusion coefficient (D). It is shown that reduced phase diffusion coefficient (D/frequency) exhibits Arrhenius-type dependency on temperature with apparent activation energy of 108 kJ/mol. The reduced Lyapunov exponent of the attractor exhibits no considerable dependency on temperature. These results suggest that the precision of electrochemical oscillations deteriorates with increase in temperature and the variation of phase diffusion coefficient does not necessarily correlate with that of Lyapunov exponent. Modeling studies qualitatively simulate the behavior observed in the experiments: the precision of oscillations in the chaotic Ni dissolution model can be tuned by changes of a time scale parameter of an essential variable, which is responsible for the development of chaotic behavior.

Distinguishing direct from indirect interactions in oscillatory networks with multiple time scales.

We propose a method to infer the coupling structure in networks of nonlinear oscillatory systems with multiple time scales. The method of partial phase synchronization allows us to infer the coupling structure for coupled nonlinear oscillators with one well-defined time scale. The case of oscillators with multiple time scales has remained a challenge until now. Here, we introduce partial recurrence based synchronization analysis to tackle this challenge. We successfully apply the proposed method to model systems and experimental data from coupled electrochemical oscillators. The statistical significance of the results is evaluated based on a surrogate hypothesis test.

Frequency of negative differential resistance electrochemical oscillators: theory and experiments.

An approximate formula for the frequency of oscillations is theoretically derived for skeleton models for electrochemical systems exhibiting negative differential resistance (NDR) under conditions close to supercritical Hopf bifurcation points. The theoretically predicted omega infinity (k/R)1/2 relationship (where R is the series resistance of the cell and k is the rate constant of the charge transfer process) was confirmed in experiments with copper and nickel electrodissolution. The experimentally observed Arrhenius-type dependence of frequency on temperature can also be explained with the frequency equation. The experimental validity of the frequency equation indicates that 'apparent' rate constants can be extracted from frequency measurements of electrochemical oscillations; such method can aid future modeling of complex responses of electrochemical cells.