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Edited by: Ralph G. Andrzejak, Universitat Pompeu Fabra, Spain

Reviewed by: Jan Frederik Totz, Massachusetts Institute of Technology, United States; Tanmoy Banerjee, University of Burdwan, India

This article was submitted to Dynamical Systems, a section of the journal Frontiers in Applied Mathematics and Statistics

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

We investigate the formation of weak chimera states in modular networks of electrochemical oscillations during the electrodissulution of nickel in sulfuric acid. In experiment and simulation, we consider two globally coupled populations of highly non-linear oscillators which are weakly coupled through a collective resistance. Without cross coupling, the system exhibits bistability between a one- and a two-cluster state, whose frequencies are distinct. For weak cross coupling and initial conditions for the one- and two-cluster states for populations 1 and 2, respectively, weak chimera dynamics are generated. The weak chimera state exhibits localized frequency synchrony: The oscillators in each population are frequency-synchronized while the two populations are not. The chimera state is very robust: The behavior is maintained for hundreds of cycles for the rather heterogeneous natural frequencies of the oscillators. The experimental results are confirmed with numerical simulations of a kinetic model for the chemical process. The features of the chimera states are compared to other previously observed chimeras with oscillators close to Hopf bifurcation, coupled with parallel resistances and capacitances or with a non-linear delayed feedback. The experimentally observed synchronization patterns could provide a mechanism for generation of chimeras in biological systems, where robust response is essential.

Synchronization of oscillatory chemical reactions is an important dynamical phenomenon with relevance to many physical and biological processes [

Understanding what coupling properties—topology, delay, symmetry, and non-linearity —influence synchronization in dynamical models [

Chimeras, synchrony patterns with coexisting domains of coherence and incoherence in networks of identical oscillators, have attracted a tremendous amount of attention in the last decades [

The theoretically predicted chimera states challenged the fundamental understanding of the non-linear dynamics of chemical reactions and the experimental techniques that enabled the construction of networks of coupled chemical reactions. Can we design networks and choose experimental conditions favorable for the chimera state?

As the theory of chimera states is quickly growing [

In this paper, we report the occurrence of weak chimera states in a modular network of electrochemical oscillators with the electrodissolution of nickel in sulfuric acid. First, for comparison with previous results, an overview is given on the characteristics of chimera states in the nickel electrodissolution system [

Schematics of the experimental setups (top) for the different types of network topologies (bottom). _{ind}: individual resistance. _{coll}: collective resistance. _{g}: group resistance.

A standard three-electrode electrochemical cell for the ring network with non-local coupling [_{2}SO_{4} saturated K_{2}SO_{4} is the reference, and a platinum rod is the counter electrode. The electrodes were immersed in a 3 M H_{2}SO_{4} solution. The cell temperature was maintained at 10°C by a circulating bath. The working electrode array has 1 mm diameter wires, embedded in epoxy, with a spacing of 2 mm. With this large spacing the potential drop in the electrolyte is sufficiently small (about 0.1 mV), so that without the presence of additional coupling, the oscillations do not show synchronization [_{ind}) for each wire. The potentiostat sets the constant circuit potential, and the currents, measured from the potential drops across the individual resistances, were digitized using a National Instrument PCI 6255 data acquisition board at a rate of 200 Hz (Note that each wire has the same circuit potential in this configuration).

The properties of individual oscillations for a wire of a given diameter can be changed by the circuit potential (

The experimental setup with non-linear feedback shown in _{σ,k}(_{2}SO_{4} sat K_{2}SO_{4} reference electrode, was set with a multichannel potentiostat interfaced with a real-time Labview controller. The electrode potentials _{σ,k}(_{σ,k}(_{σ,k}(_{σ,k}(_{σ,k}(_{ind}, with _{ind} = 1 kOhm. The electrode potentials were adjusted for offset with,

where _{κσ} determines the network topology,

is the feedback. For each population, _{11} = _{22} = 1. Coupling between the population is set to _{12} = _{21} = ε, where ε is the cross-coupling factor. The linear and quadratic feedback gains, _{1} and _{2}, respectively, are applied to induce the required dynamics. The delay τ_{Ex} was set to be equal to half of the period of the uncoupled oscillators. See [_{1}, _{2}, and τ_{Ex}. _{0} = 1, 160 mV and the natural frequency (i.e., the frequency of the oscillation without coupling) was about 0.45 Hz. In a typical experiment of about 500 oscillations, the natural frequency change is about 2–3 mHz.

_{ind}) was added to each electrode. Additionally, two group resistances (_{g}) and a collective resistance (_{coll}) were used to generate the intra- and inter-population coupling, respectively.

To put our results in context, we start out with reviewing the chimera states observed earlier with non-local ring network (section Chimera State with Non-local Ring Network Close to Hopf Bifurcation) and weak chimeras with non-linear feedback (section Weak Chimera with Non-linear Feedback). In section Weak Chimera in Modular Networks with Strongly Non-linear Oscillators, new results are presented in a modular network of highly non-linear oscillators coupled through differences in the electrode potentials.

Here we considered oscillations in the experimental system that occur close to Hopf bifurcation [

Experimental traditional chimera state in a non-locally coupled regular network. _{ind} = 1 kOhm,

Weak chimeras can arise in modular oscillator networks consisting of multiple populations with stronger coupling within populations and weaker coupling between different populations [

We used a synchronization engineering [

Experimental weak chimera state with non-linear feedback. _{1, 2} − φ_{1, 1}), population 2 (red thick line, φ_{2, 2} − φ_{2, 1}), and for two elements between the populations (blue dashed line, φ_{2, 2} − φ_{1, 1}). Feedback parameters: _{1} = 0.22, _{2} = 2.0 1/V.

As shown in

We now consider networks of two coupled populations with a larger number of oscillators per population and inherent non-linearities through the phase response curve and the oscillators' waveforms. For a phase description, these properties lead to non-sinusoidal phase interaction, which can give multistability between in-phase synchrony and other cluster states with global coupling [

We used the kinetic scheme proposed by Haim et al. [_{ind}) and global (collective) (_{coll}) resistance, the charge and mass balance equations are the following [

where _{F}(

and Γ_{i} is the surface capacity. Γ_{i} were randomly chosen between 9.999 ×10^{−3} and 10.001 ×10^{−3} for simulating the heterogeneities of the different natural frequencies of the oscillators [_{h} = 1600, a = 0.3, b = 6 × 10^{−5} and c = 0.001 are kinetic parameters. The global coupling occurs through the electrode potential equation (last term in Equation 3). _{c}/(_{ind}_{ind} + 40_{c} [

^{−4} for the one- and two-cluster states for 24.3 ≤

Numerical simulation: Cluster formation with global coupling (^{−4}).

The behavior at

The space time plot for the one- and two-cluster state (the left and right panel, respectively) are shown in ^{−3}. Because there is bistability between the one-cluster and the two-cluster states with differing frequency, the conditions may favor the formation of weak chimera states in networks. We note that oscillator heterogeneity was added to the model to better represent the experimental scenario. The same bistability also occurs for uniform populations (i.e., with Γ_{i} = 0.01 for all oscillators).

As a simple modular network obtained from the globally coupled oscillator populations, we introduce some cross coupling between two populations. For the electrode potential, the equations are:

(The equations for surface coverages are the same, i.e., Equation 4). There is a strong global coupling within the populations,

With ε = 0.05 with initial conditions corresponding to the one (or two)-cluster states for all the oscillators, the expected one (or two) cluster state was obtained. ^{−3}. The grayscale plot (

Numerical simulation: Weak chimera state in the modular network with weak cross coupling. ^{−4} and ε = 0.05. _{1} − φ_{2}), populations 2 red thin line, (φ_{41} − φ_{45}), and between populations (blue dashed, φ_{1} − φ_{41}).

Without coupling, the oscillators exhibit slight heterogeneity, and the natural frequencies have a standard deviation of about 14 mHz. To confirm the weak chimera state, we performed a set of experiments following the guidelines developed in the simulations. First, we used only population 1 (40 electrodes), and coupled them globally (i.e., _{coll} = 0 Ohm, _{g} = 10 Ohm in

Experiments: Cluster formation with global coupling. _{ind} = 600 Ohm, _{g} = 10 Ohm, _{coll} = 0 Ohm,

As an example of the behavior observed in this region, the dynamics is shown at

Now we consider the modular network with two populations of 40 electrodes. The oscillators in each population are coupled with _{g}. As shown in _{coll}. The resistance _{coll} induces global coupling between every electrode pair, with coupling strength _{coll} = _{coll}/[(_{ind} + 40_{g})_{eq}], where _{eq} = _{ind} + 40_{g} + 80_{coll}. The group resistance induces coupling only within a population, i.e., _{g} = _{g}/[_{ind}(_{ind}+40_{g})]. The total coupling thus _{g} + _{coll}, and ε = _{g}/

Experiments: Weak chimera state in modular network with weak cross coupling. _{1} − φ_{2}), populations 2 red thin line (φ_{41} − φ_{45}), and between populations (blue dashed, φ_{1} − φ_{41}). _{ind} = 580 Ohm, _{g} = 9 Ohm, _{coll} = 0.5 Ohm,

We also performed a long-term experiment to check for the robustness of the chimera state. The chimera state was stable for about 1,000 cycles, after which a one-cluster state was observed. In this parameter region, the system parameters exhibit a slow drift toward the Hopf bifurcation point. One explanation for the loss of the chimera state is that during this slow drift the oscillations become less non-linear for the chimera state to occur as the parameters leave the region where bistability is present.

We also confirmed that by increasing the coupling strength, the chimera state breaks down. While for ε = 0.1 stable chimera state occurs, with ε = 0.2 (with similar coupling strengths and initial conditions) only in-phase behavior can be observed in the experiments.

Robust weak chimera states were observed in a modular network of two populations of globally coupled electrochemical oscillators with simple resistive cross coupling between populations that is sufficiently weak (ε < 0.2). There are important differences in the observed chimera states compared to those in our previous studies [

In identifying the experimental conditions for the chimera state, we relied on our previous study [

Similar weak chimera states could be observed in many other chemical systems. For example, other electrochemical systems and the BZ reaction can generate rich variety of clusters, in particular, when the sign of the coupling strength can also be varied (e.g., excitatory and inhibitory coupling) [

IK and CB conceived the presented idea. CB encouraged IK and JO-E to investigate the weak chimeras in modular networks. IK and JO-E planned the experiments and the numerical simulations. JO-E performed the experiments and numerical simulations, and processed the data. JO-E took the lead in writing the manuscript. IK and CB revised the manuscript. IK supervised the project. All authors provided critical feedback and helped shape the research, analysis, and manuscript.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.