Rapid convergence of time-averaged frequency in phase synchronized systems

Davidsen, Jörn; Kiss, István Z.; Hudson, John L.; Kapral, Raymond

doi:10.1103/physreve.68.026217

Cited by 7 publications

(5 citation statements)

References 17 publications

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“…1͑d͒ shows that the periods fall into a relatively small range of Ϯ16% and the distribution has bimodal character often seen in chaos obtained through period doubling scenario. 23 Figure 2 describes phase dynamics of the lowdimensional chaotic behavior at 10°C. The phase space using the derivative Hilbert transform is shown in Fig.…”

Section: Phase Coherent Chaotic Behavior At 10°cmentioning

confidence: 99%

Effect of temperature on precision of chaotic oscillations in nickel electrodissolution

Wickramasinghe

Kiss

2010

Chaos: An Interdisciplinary Journal of Nonlinear Science

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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.

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Section: Phase Coherent Chaotic Behavior At 10°cmentioning

confidence: 99%

Effect of temperature on precision of chaotic oscillations in nickel electrodissolution

Wickramasinghe

Kiss

2010

Chaos: An Interdisciplinary Journal of Nonlinear Science

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“…In this spirit, the observed bimodality for the considered experiments is closely related to the emergence of the chaotic attractor due to period-doubling bifurcations. 77 We note that this specific route to chaos actually gives rise to phasecoherent oscillations in the classical viewpoint based on the PSD. [19][20][21] The distinct behavior of C v and b v additionally confirms our previous statement that both measures provide complementary aspects of the local fragmentations of the systems' attractor.…”

Section: -8mentioning

confidence: 99%

Phase coherence and attractor geometry of chaotic electrochemical oscillators

Zou

Donner

Wickramasinghe

et al. 2012

Chaos: An Interdisciplinary Journal of Nonlinear Science

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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.

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“…The period distribution of the low-dimensional chaotic attractor is relatively broad and exhibits a multipeak structure, which is characteristic of chaotic behavior obtained from a period-doubling bifurcation route to chaos. [14] We use order parameters R 1 and R 2 to describe the extent of relative organization of one-and two-cluster states [Eq. (1)], [4,9]…”

mentioning

confidence: 99%

“…The observed period distribution of these elements is illustrated in Figure 1 c while the structure of the dynamical attractor can be seen in Figure 1 d. The period distribution of the low-dimensional chaotic attractor is relatively broad and exhibits a multipeak structure, which is characteristic of chaotic behavior obtained from a period-doubling bifurcation route to chaos. [14] We use order parameters R 1 and R 2 to describe the extent of relative organization of one-and two-cluster states [Eq. (1)], [4,9] where k = 1,2, " i is the imaginary unit, and f j is the phase of the j-th oscillator obtained with linear interpolation between the current maxima at which the phase is taken to be multiples of 2p.…”

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Engineering of Synchronization and Clustering of a Population of Chaotic Chemical Oscillators

et al. 2011

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Description, control, and design of weakly interacting dynamical units are challenging tasks that play a central role in many physical, chemical, and biological systems. [1][2][3][4][5] Control of temporal and spatial variations of reaction rates is especially daunting when the dynamical units exhibit deterministic chaotic oscillations that are sensitive to initial conditions and are long-term unpredictable. Control of spatiotemporal chaos by means of suppression of spiralwave turbulence to standing waves, cluster patterns, and uniform periodic oscillations, in the catalytic CO oxidation on a Pt (110) single-crystal surface has been successfully achieved with linear delayed feedback of the carbon monoxide partial pressure.[6] The same delayed global feedback methodology has also been applied to induce various dynamical clusters in electrochemical corrosion processes [7] and in the light-sensitive Belousov-Zhabotinsky (BZ) reaction.[8] A fundamental problem of tuning the complex structure of chaotic systems is choosing a feedback scheme and appropriate control parameters to steer the system to a desired structure of spatiotemporal reactivity. This objective requires the use of simple yet accurate models of nonlinear processes, their interactions, and responses to external perturbations. When the individual units exhibit periodic oscillatory behavior, the effect of weak feedback and coupling can be described by phase models [4,9] in which the state of each unit is represented by a single variable, the phase of oscillations.Typical phase-synchronized behavior of a population of periodic oscillators is dynamical differentiation (clustering) where the elements form groups (clusters) in which the phases are identical at all times but different from the phases of the elements in the other groups.[9]Phase models successfully described synchronization and dynamical differentiation (clustering) of oscillatory electrochemical [10] and BZ bead experiments.[11] Experiment-based phase models can also be used for synchronization engineering of periodic oscillators [12] where a carefully designed external feedback is applied to dial-up complex dynamical structures such as stable and sequentially visited dynamical cluster patterns and desynchronization.Herein, we consider populations of chaotic oscillators that exhibit strong cycle-to-cycle period variations. We show that the feedback scheme can be designed to obtain dynamical synchronization states of phase coherent chaotic oscillators in a quantitative manner. The method relies on the flexibility and versatility of synchronization engineering (originally developed for periodic oscillators [12] ) and on the observation that the long-term phase dynamics of chaotic oscillators is often similar to that of noisy periodic oscillators. [13] An experimental system of uncoupled, phase-coherent, chaotic oscillators was constructed by using an electrochemical cell consisting of 64 Ni working electrodes (99.98 % pure), a Pt mesh counter electrode, and a Hg/Hg 2 SO 4 /K 2 SO 4 (saturated) r...

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Rapid convergence of time-averaged frequency in phase synchronized systems

Cited by 7 publications

References 17 publications

Effect of temperature on precision of chaotic oscillations in nickel electrodissolution

Effect of temperature on precision of chaotic oscillations in nickel electrodissolution

Phase coherence and attractor geometry of chaotic electrochemical oscillators

Engineering of Synchronization and Clustering of a Population of Chaotic Chemical Oscillators

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