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Phase coherence and attractor geometry of chaotic electrochemical oscillators
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Image of FIG. 1.
FIG. 1.

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.

Image of FIG. 2.
FIG. 2.

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

Image of FIG. 3.
FIG. 3.

((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.

Image of FIG. 4.
FIG. 4.

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.

Image of FIG. 5.
FIG. 5.

(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.

Image of FIG. 6.
FIG. 6.

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).

Image of FIG. 7.
FIG. 7.

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.

Image of FIG. 8.
FIG. 8.

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.

Image of FIG. 9.
FIG. 9.

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


Generic image for table
Table I.

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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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Phase coherence and attractor geometry of chaotic electrochemical oscillators