Birhythmicity, intrinsic entrainment, and minimal chimeras in an 1 electrochemical experiment

The coexistence of limit cycles in phase space, so called birhythmicity, is a phenomenon known to exist in many systems 6 in various disciplines. Yet, detailed experimental investigations are rare, as are studies on the interaction between 7 birhythmic components. In this article, we present experimental evidence for the existence of birhythmicity during 8 the anodic electrodissolution of Si in a ﬂuoride-containing electrolyte using weakly illuminated n-type Si electrodes. 9 Moreover, we demonstrate several types of interaction between the coexisting limit cycles, in part resulting in peculiar 10 dynamics. The two limit cycles exhibit vastly different sensitivities with respect to a small perturbation of the electrode 11 potential, rendering the coupling essentially unidirectional. A manifestation of this is an asymmetric 1:2 intrinsic 12 entrainment of the coexisting limit cycles on an individual uniformly oscillating electrode. In this state, the phase 13 space structure mediates the locking of one of the oscillators to the other one across the separatrix. Furthermore, the 14 transition scenarios from one limit cycle to the other one at the borders of the birhythmicity go along with different types 15 of spatial symmetry breaking. Finally, the master-slave type coupling promotes two (within the experimental limits) 16 identical electrodes initialized on the different limit cycles to adopt states of different complexity: one of the electrodes 17 exhibits irregular, most likely chaotic, motion, while the other one exhibits period-1 oscillations. The coexistence of 18 coherence and incoherence is the characteristic property of a chimera state, the two coupled electrodes constituting an 19 experimental example of a smallest chimera state in a minimal network conﬁguration. 20


I. INTRODUCTION 46
The discovery of birhythmicity in physical systems dates 47 back to at least 1976, when it was reported to exist in a model 48 of a continuous stirred tank reactor with consecutive exother-49 mic systems or coupled birhythmic oscillators have been studied with normal form type equations.These include wave phenomena in spatially extended reaction-diffusion models and ensembles of coupled birhythmic (phase) oscillators [20][21][22][23][24][25][26][27] .The latter were also found to promote the occurrence of chimera states, an interesting prediction which awaits experimental validation.
In this paper we investigate the nonlinear dynamics occurring during silicon electrodissolution in a fluoride-containing electrolyte.This system exhibits a plethora of dynamical phenomena, such as oscillations [28][29][30] , phase clusters and chemical turbulence 31 , or chimera states 32 .Moreover, it has been found that the system exhibits two types of limit cycles, which were coined low amplitude oscillations (LAOs) and high amplitude oscillations (HAOs), respectively 33 .Although the electrochemical mechanism leading to either of these oscillations is not yet known, experiments suggested that they arise due to two different main feedback loops in the system 33 .Later Tosolini et al. reported the coexistence of chaotic attractors and speculated that the bichaoticity is linked to an intrinsic birhythmicity, the interaction between the coexisting oscillators in phase space causing both of the limit cycles to become unstable and give rise to chaotic attractors 34 .Here, we will continue on this path and show that the electrodissolution of silicon does indeed exhibit the coexistence of two stable limit cycles, yet in a drastically different parameter range than the bichaotic one.
Instead of using p-doped silicon as our working electrode as in Refs.33 and 34, we use n-doped silicon.The electrooxidation reaction proceeds through the following net reaction: (1)     where 1 ≤ λ VB ≤ 4 denotes the number of charge carriers that come from the valence band of the silicon.Since at least the first oxidation requires a valence band hole, the electrooxidation of n-doped Si requires the illumination of the electrode with a wavelength that is larger than the band gap.The illumination intensity is thus an additional bifurcation parameter in our study.The oxide formed in reaction (1) is chemically etched in the overall reaction 35 SiO The interaction between oxidation and etching kinetics are believed to cause the oscillatory behavior 30 , yet the corresponding feedback loops could not yet be identified.
The rest of the article is structured as follows.In section II we introduce the experimental setup.In section III we first shows the results obtained with one electrode, where birhythmicity is illustrated in phase space, physical space and in the frequency domain.Then, coupling experiments with two electrodes are presented.Implications of the experimental data concerning intrinsic and extrinsic coupling of the birhythmic system are discussed in section IV.
FIG. 1. Sketch of the experimental setup (not to scale) with its three parts, the laser illumination setup, with a spatial light modulator (SLM) as centerpiece, the ellipsometric imaging setup, allowing spatially resolved in situ monitoring of the electrode surface, and the electrochemical setup, consisting of an electrochemical cell and a potentiostat.Note that we connected a resistor R ext between the working electrode (WE) and the potentiostat.

II. EXPERIMENTAL SYSTEM 130
The experimental setup is sketched in Fig. length with the ellipsomicroscopic surface imaging setup 36,37 . 146 The resulting ellipsometric intensity signal ξ will be presented In contrast to this stepwise transition, the transition from a LAO to a HAO at the low illumination border is abrupt and takes place on the entire electrode at the same time.In   pling, any change in current at one electrode causes the potential drops across both electrode/electrolyte interfaces U el,1/2 , to change according to: where U is the externally applied voltage, and j 1/2 and A 1/2 are the current densities and areas of the respective electrodes.
The use of the SLM allows us to employ different initialization protocols to the two electrodes so that we can initialize each electrode in either a HAO or a LAO state independently.In Clearly, the slightly different parameters of the two electrodes (such as a minor difference in their electrode areas) lead to a small difference of their natural frequencies, and the coupling via the external resistor does not suffice to synchronize them.
The picture is different in the case of the LAOs.When we initialize both electrodes using the LAO protocol, they typically exhibit phase synchronization, as can be seen in Fig. 7 b) (Multimedia view).

147 as a percentage
Fig. 3 a) are depicted.The LAOs remain spatially homoge-212

Fig. 4
Fig. 4 c)-d) (Multimedia view) an example of such a transition is shown.Once the illumination has been reduced below a critical value, the electrode attains a HAO as soon as the current reaches the new maximal current level imposed by the reduced illumination.If we expand our parameter space by also changing the applied voltage U, we obtain the 2D phase diagram shown in Fig. 5. Here, the HAOs are marked with crosses and the LAOs with circles.The red and blue areas indicate the regions where only HAOs respectively LAOs were found, and the striped area marks the birhythmic region.Note that we only include measurements at the edges of and not within the birhythmic region for clarity.Evidently, the birhythmic region extends over a large region in this parameter plane, demonstrating that birhythmicity is a robust feature of the system.

Fig. 6 a
Fig. 6 a) depicts an experiment that we initialized in a LAO at a low voltage, and b) one that we initialized in a HAO at high voltages.In each spectrogram the main frequency and the second frequency at each voltage are marked with a solid and a dashed line, respectively.In the spectrogram in Fig. 6 a) we see that the main frequency of the initial LAO at 3.9 V vs MSE decreases before the system transitions to the faster oscillating HAO state at 5.8 V vs MSE.This transition from LAOs to HAOs occurs again quasi-simultaneously on the entire electrode, just as in the case when we varied the illumination, cf.Fig. 4 c)-d) (Multimedia view).As the system undergoes a transition to a HAO, the main frequency abruptly jumps from 16 mHz to 26 mHz.The frequency of the HAO first stays approximately constant and then starts to increase at about 5 V vs MSE.The increase in frequency is accompanied by the emergence of a subharmonic mode.We observe a similar behavior when the system is initialized in a HAO state and the voltage is swept towards lower values (Fig.6 b)).First, the frequency hardly changes with decreasing voltage until at about 5 V vs MSE where it starts to increase and a first subharmonic peak emerges.In a small voltage interval around 4 V vs MSE this first subharmonic peak is accompanied by a sub-subharmonic frequency.The emergence of the subharmonic frequencies is accompanied by

337
FIG. 7. Exemplary time series of the spatially averaged ellipsometric intensity of two electrodes coupled electrically through an external resistor.a) Both electrode 1 ξ 1 and electrode 2 ξ 2 in a HAO state.b) Both electrodes in an LAO state.c) Electrode 1 in a HAO state and electrode 2 in a chaotic state.Here, electrode 2 was initialized using the LAO-initialization protocol.A 1 = 11.43 mm 2 , A 2 = 10.55 mm 2 .R ext (A 1 + A 2 ) = 1 kΩcm 2 , I Ill = 1.0 mW/cm 2 , U = 5.75 V vs MSE.(Multimedia view).

Fig. 7 (
Fig. 7 a) (Multimedia view) depicts time series of the two electrodes when they are both initialized with the HAO protocol.It can be seen that the oscillations on the two electrodes are slightly out of phase at t = 0 s, are in phase at t = 185 s, and have drifted to an antiphase configuration at t = 550 s.

Fig. 7 c
Fig. 7 c) (Multimedia view) depicts an example where electrode 1 was initialized with the HAO protocol and electrode 2 with the LAO protocol.In this case, electrode 1 assumes a periodic HAO which is very close to the one of case a).In contrast, electrode 2 does not oscillate in a simple periodic LAO state.Instead it exhibits a more complex temporal behavior.The frequency spectrum of the time series (not shown) exhibits a strongly enhanced background distribution around the main oscillation frequency, suggesting that the dynamics is chaotic.This counterintuitive coexistence of chaos and order is not FIG. 8. Spectrograms of two coupled electrodes, obtained from the spatially averaged ellipsometric intensity signal and a quasistationary cyclic voltage scan (dU/dt = 0.1 mV/s) at illumination intensity I Ill = 0.94 mW/cm 2 and R ext (A 1 + A 2 ) = 1 kΩcm 2 .The solid line indicates the main frequency and the dashed line indicates the second frequency.a) Electrode 1, initialized with the HAO protocol, A 1 = 5.53 mm 2 .b) Electrode 2, initialized with the LAO protocol, A 2 = 6.17 mm 2 .