^{1,2,3}, Reik V. Donner

^{3}, Mahesh Wickramasinghe

^{4}, István Z. Kiss

^{4}, Michael Small

^{2,5}and Jürgen Kurths

^{3,6,7}

### Abstract

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 electrochemicaloscillations 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.

Chaotic oscillations are a wide-spread phenomenon that can be observed in many natural and technological systems.

^{1,2}The complex dynamics of such systems is characterized by an exponential divergence of initially close trajectories as time proceeds.

^{3–5}As a consequence, long-term predictions of the amplitudes and phases are not possible. The degree of dynamical complexity of chaotic oscillators can be measured in terms of various quantities from nonlinear time series analysis and complex systems theory,

^{6,7}such as Lyapunov exponents, fractal dimensions, or entropies and related concepts from information theory. In addition to these widely applicable concepts, the phase coherence of chaotic oscillations has been recently recognized as an important complementary aspect. While this feature has been previously considered based on its manifestation in various nonlinear dynamical characteristics, in this work, we provide arguments that phase coherence should rather be viewed as a structural property of chaotic attractors in phase space. For a well-studied example of electrochemical chaos, we demonstrate that complex dynamics emerges without qualitative changes in attractor geometry related to a loss of phase coherence. This calls for the development of alternative geometric criteria for the presence of phase coherence replacing the present purely dynamics-based perspective.

This work has been partially funded by the Leibniz society (Project ECONS) and the Federal Ministry for Education and Research (BMBF) via the Potsdam Research Cluster for Georisk Analysis, Environmental Change and Sustainability (PROGRESS). Y.Z. has been supported by the National Natural Science Foundation of China (Grant No. 11135001) and a Hong Kong Polytechnic University Postdoctoral Fellowship. I.Z.K. and M.W. acknowledge support by the National Science Foundation under Grant No. CHE-0955555.

I. INTRODUCTION

II. DYNAMIC VS. GEOMETRIC VIEWS ON PHASE COHERENCE

A. Phase coherence and power spectrum

B. Definition of phase variables

C. Phase coherence as a structural attractorproperty

D. Examples

III. EXPERIMENTAL TIME SERIES

A. Experimental setup

B. Preprocessing of the data

C. Dynamical characteristics: Two examples

D. Phase definition and projective effects on phase coherenceanalysis

E. Phase coherence of experimental electrochemicaloscillations

IV. STRUCTURAL ATTRACTOR CHARACTERIZATION

A. Recurrence network analysis

B. Example cases

C. Changes in attractor geometry

D. Mathematical model

V. CONCLUSIONS

### Key Topics

- Attractors
- 84.0
- Coherence
- 63.0
- Thermodynamic properties
- 38.0
- Time series analysis
- 21.0
- Chaotic dynamics
- 15.0

## Figures

Effect of different projections of the Lorenz system (6) based on its *x*- (left panels) and *u*-variable (right panels): (a) (*x,z*)- and (b) (*u,z*)-projections, ((c) and (d)) periodograms as estimates of the PSD for the (c) *x*- and (d) *u*-coordinates, ((e) and (f)) Hilbert transforms (after correcting for the mean values), and ((g) and (h)) resulting time evolution of the detrended Hilbert phases.

Effect of different projections of the Lorenz system (6) based on its *x*- (left panels) and *u*-variable (right panels): (a) (*x,z*)- and (b) (*u,z*)-projections, ((c) and (d)) periodograms as estimates of the PSD for the (c) *x*- and (d) *u*-coordinates, ((e) and (f)) Hilbert transforms (after correcting for the mean values), and ((g) and (h)) resulting time evolution of the detrended Hilbert phases.

Projections of the reconstructed attractors for (a) case I and (b) caseII.

Projections of the reconstructed attractors for (a) case I and (b) caseII.

((a) and (b)) Periodogram as an estimator of the PSD, ((c) and (d)) recurrence time distributions (zooms for short times and low frequencies), and ((e) and (f)) estimated 2nd-order Rényi entropy for different values of *RR* for the chaotic electrochemical oscillations in ((a), (c), (e)) low-temperature case I and ((b), (d), (f)) high-temperature case II.

((a) and (b)) Periodogram as an estimator of the PSD, ((c) and (d)) recurrence time distributions (zooms for short times and low frequencies), and ((e) and (f)) estimated 2nd-order Rényi entropy for different values of *RR* for the chaotic electrochemical oscillations in ((a), (c), (e)) low-temperature case I and ((b), (d), (f)) high-temperature case II.

Phase coherence analysis in the high-temperature case II (cf. Fig. 2(b)): (a) Parts of the trajectory projected onto the plane of the reconstructed phase space. (b) Variation of the coherence index *CI* (Eq. (1)) in dependence on the rotation angle in the considered plane. ((c) and (d)) Parts of the trajectory of the original (*y*) and optimized () coordinate. The latter one has been obtained by rotating the plane about an angle of . ((e) and (f)) Reconstructed oscillations in the (*y,H*(*y*)) plane for the original and rotated coordinate. ((g) and (h)) Dynamics of the linearly detrended phase obtained from the original and reconstructed coordinate.

Phase coherence analysis in the high-temperature case II (cf. Fig. 2(b)): (a) Parts of the trajectory projected onto the plane of the reconstructed phase space. (b) Variation of the coherence index *CI* (Eq. (1)) in dependence on the rotation angle in the considered plane. ((c) and (d)) Parts of the trajectory of the original (*y*) and optimized () coordinate. The latter one has been obtained by rotating the plane about an angle of . ((e) and (f)) Reconstructed oscillations in the (*y,H*(*y*)) plane for the original and rotated coordinate. ((g) and (h)) Dynamics of the linearly detrended phase obtained from the original and reconstructed coordinate.

(a) Shannon entropy *H* of Poincaré intersection points through the surface (estimated using a histogram of 80 equi-sized bins), for experimental runs at different temperatures. (b) Coherence factor *CF* as a measure of phase coherence.

(a) Shannon entropy *H* of Poincaré intersection points through the surface (estimated using a histogram of 80 equi-sized bins), for experimental runs at different temperatures. (b) Coherence factor *CF* as a measure of phase coherence.

Color-coded representations of the local RN properties ((a) and (b)) and ((c) and (d)) ) for the experimental data of electrochemical oscillations in ((a) and (c)) low-temperature (case I) and ((b) and (d)) high-temperature regime (case II).

Color-coded representations of the local RN properties ((a) and (b)) and ((c) and (d)) ) for the experimental data of electrochemical oscillations in ((a) and (c)) low-temperature (case I) and ((b) and (d)) high-temperature regime (case II).

Probability distribution functions of the local RN properties ((a) and (b)) and ((c) and (d)) for the experimental data of electrochemical oscillations in the ((a) and (c)) low-temperature and ((b) and (d)) high-temperature regimes. Panels ((e) and (f)) display the associated scatter plots ( vs. ) as well as the values of the rank-order correlation coefficient (Spearman’s Rho) between both measures.

Probability distribution functions of the local RN properties ((a) and (b)) and ((c) and (d)) for the experimental data of electrochemical oscillations in the ((a) and (c)) low-temperature and ((b) and (d)) high-temperature regimes. Panels ((e) and (f)) display the associated scatter plots ( vs. ) as well as the values of the rank-order correlation coefficient (Spearman’s Rho) between both measures.

Behavior of RN-based characteristics for the electrochemical oscillations in dependence on the temperature as the unique control parameter varied in the experimental campaign (*RR* = 0.03): (a) global clustering coefficient , (b) network transitivity , (c) average path length , (d) assortativity coefficient , ((e) and (f)) standard deviation, and ((g) and (h)) skewness of the local clustering coefficient and logarithmic betweenness centrality (, and , respectively). Error bars indicate the mean values and standard deviations from 100 independent realizations of the RN obtained from *N* = 10 000 state vectors randomly selected from the whole embedded time series.

Behavior of RN-based characteristics for the electrochemical oscillations in dependence on the temperature as the unique control parameter varied in the experimental campaign (*RR* = 0.03): (a) global clustering coefficient , (b) network transitivity , (c) average path length , (d) assortativity coefficient , ((e) and (f)) standard deviation, and ((g) and (h)) skewness of the local clustering coefficient and logarithmic betweenness centrality (, and , respectively). Error bars indicate the mean values and standard deviations from 100 independent realizations of the RN obtained from *N* = 10 000 state vectors randomly selected from the whole embedded time series.

As in Fig. 8 for realizations of the numerical model for Ni dissolution.^{78}

As in Fig. 8 for realizations of the numerical model for Ni dissolution.^{78}

## Tables

Mean values and standard deviations (in brackets) of different RN characteristics for the experimental cases I and II, obtained from 100 independent samples of *N* = 10 000 state vectors randomly selected from the embedded time series. Note that all measures allow a discrimination between both cases with high confidence.

Mean values and standard deviations (in brackets) of different RN characteristics for the experimental cases I and II, obtained from 100 independent samples of *N* = 10 000 state vectors randomly selected from the embedded time series. Note that all measures allow a discrimination between both cases with high confidence.

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