Effect of temperature on precision of chaotic ... - Semantic Scholar

CHAOS 20, 023125 共2010兲

Effect of temperature on precision of chaotic oscillations in nickel electrodissolution Mahesh Wickramasinghe and István Z. Kissa兲 Department of Chemistry, Saint Louis University, 3501 Laclede Ave., St. Louis, Missouri 63103, USA

共Received 22 March 2010; accepted 7 May 2010; published online 21 June 2010兲 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 共D兲. It is shown that reduced phase diffusion coefficient 共D / frequency兲 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. © 2010 American Institute of Physics. 关doi:10.1063/1.3439209兴 Rhythmic processes often occur in important physical, chemical, and biological systems. In biology, many cyclic processes are referred to as “clocks” because they provide timing information. Changes of temperature can have an effect on both the period and the “stability” of the clock. Period has been known to exhibit Arrheniustype (exponential) dependency in most (but not all) chemical systems; this dependency is often counterbalanced in many biological systems that produce robust, “temperature compensated” clocks with constant period in a large temperature window. The other important clock property is the precision of the clock: how well the peak-to-peak period repeats itself. This property is especially important with complex rhythms such as chaos where deterministic processes regulate large changes in peak-to-peak periods. We report experimental results on temperature dependence on the precision of a chaotic chemical clock, the electrodissolution of Ni in sulfuric acid. We show that the precision of the chaotic current oscillations deteriorates with increase in temperature. It is also found that the predictability and precision of the chemical chaotic clock are quite different dynamical properties whose variation with temperature does not necessarily follow each other. I. INTRODUCTION

Rhythmic variations of physical quantities 共e.g., potential, current兲 and concentrations of substances underlie many complex biological1,2 and chemical3–5 phenomena. These rhythms often provide time clue for starting or ending important processes. Therefore, the precision of the period of a rhythm is an important quality marker for how well the Author to whom correspondence should be addressed. Telephone: ⫹1-314977-2139. Fax: ⫹1-314-977-2521. Electronic mail: [email protected].

a兲

1054-1500/2010/20共2兲/023125/7/$30.00

cyclic process functions as a clock. Strictly periodic processes can deteriorate because of either stochastic 共“external”兲 influences that are often considered as “noise,” or because of some inherent deterministic dynamical processes that result in a complex rhythm with deterministic fluctuations in amplitudes and period. In neuroscience precision has been considered as the “jitter” in the average spike time characterized by the inverse of the coefficient of variation of the periods of noisy periodic oscillations.6 Motivated by this biological example, we shall consider precision of oscillations as a quantity related to the extent of changes of periods of the subsequent cycles. Deterministic chaos is an example of complex rhythm in which long-term predictions of amplitudes and periods are not possible. Measures of complexity of chaotic dynamics are characterized by several quantities.4,7,8 Lyapunov exponents describe the degree of exponential divergence of nearby state space trajectories and thus are related to the predictability of the behavior. Dimensions and entropies characterize the topological/informational complexity of the attractor underlying the chaotic variations. These measures capture some important properties of chaotic behavior; however, the precision of the chaotic clock can be better characterized by quantities related to time scale variations.9 In the past decades, the concept of “phase” of chaotic oscillations proved to be a useful concept through which entrainment and synchronization properties were successfully characterized.9–12 Short-term fluctuations of phase are deterministic; however, long-term variations are similar to that of a periodic oscillator in the presence of noise.13,14 The dynamics of phase of a chaotic oscillator is generally diffusive and the phase performs a random walk for which a phase diffusion coefficient 共D兲 can be defined.9 This quantity effectively characterizes the precision of the “period” of a chaotic oscillator.

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© 2010 American Institute of Physics

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Chaos 20, 023125 共2010兲

M. Wickramasinghe and I. Z. Kiss

In this paper, we use phase diffusion coefficient to characterize the precision of a chaotic oscillatory chemical process: the electrodissolution of Ni in sulfuric acid. We investigate the effect of an important system parameter, temperature, on the precision of the chaotic current oscillations. The variation of phase diffusion coefficient with temperature is compared to that of traditional complexity measures including Lyapunov exponent and information dimension. Numerical simulations are carried out with an ordinary differential equation model15 of the experimental system to support the findings that temperature can strongly affect the precision of chaotic oscillations.

C. Phase diffusion coefficient

In order to evaluate the extent of phase fluctuations, the phase variance9,17 is defined as

␴2共␶兲 = 具关⌽共t + ␶兲 − ⌽共t兲兴2典, where ⌽共t兲 = ␾共t兲 − 2␲␻t is the “detrended phase.” To evaluate the variance the data file was divided in segments with ␶ time lengths and initial time of t0. ␴2 is obtained as the variance of the ⌽共t0 + ␶兲 − ⌽共t0兲 values for the segments. The phase diffusion coefficient D is obtained from the slope of a linear least squares fit17,18 to the ␴2共␶兲 versus ␶ plot,

␴ 2共 ␶ 兲 = B 1␶ + B 2 II. EXPERIMENTAL METHODS

and D = B1/2.

A. Experimental setup

A standard electrochemical cell consisting of a nickel working electrode 共Goodfellow Cambridge Ltd., 99.98%, 1.0 mm diameter兲, a Hg/ Hg2SO4/saturated K2SO4 reference electrode, and a platinum counterelectrode is used in the experiment. The experiments are carried out in 4.5 mol/L sulfuric acid solution at temperatures in the range of 10– 30 ° C maintained by a Neslab RTE-7 circulating bath. The working electrode is embedded in epoxy so that the reaction takes place only at the end. The nickel electrode, connected to a potentiostat 共ACM Instruments, Gill AC兲 through an external resistance 共350 ⍀ ⱕ R ⱕ 1300 ⍀兲, is polarized at a constant electrode potential V. The current is digitized with a National Instruments 共PCI 6255兲 data acquisition board with a maximum data acquisition rate of 1000 Hz. A typical data file consisted of about 1250 oscillations with 185 data points per cycle. The solution is stirred slowly with a magnetic stirrer in order to remove O2 formed by water electrolysis. Prior to the experiment, the electrode is wet polished with series sandpapers 共P180-P4000兲 with a Buehler Metaserv 3000 polisher.

The Hilbert transform of the current i共t兲 1 PV ␲

冕

⬁

−⬁

i共␶兲 − 具i典 d␶ t−␶

共1兲

was used in defining phase10 ␾共t兲,

␾共t兲 = arctan

dH共t兲/dt . di共t兲/dt

共2兲

PV in Eq. 共1兲 implies that the integral should be evaluated in the sense of Cauchy principle value. 具i典 is the temporal average value of the current, i共t兲. The definition of phase requires a unique center of rotation in the phase space using the Hilbert transform.10 The derivative approach applied here was previously shown to be more effective than the standard definition of phase that uses Hilbert transform12,16 in Eq. 共2兲. The frequency of an oscillator was obtained from a linear fit of ␾共t兲 versus t,

␻=

冓 冔

1 d␾共t兲 . 2␲ dt

D. Model and numerical methods

Numerical simulations were carried out using an electrochemical model proposed by Haim et al.15 that describes the dynamics of anodic Ni dissolution. The model includes three dimensionless variables: ␪, total surface coverage of NiOH + NiO, ␩, surface coverage of NiO, and e, the potential of the electrode. The model with potentiostatic mode of cell operation19 is as follows: de v − e = − iF共␪, ␩兲, dt r ⌫1

共4a兲

exp共0.5e兲 d␪ = 共1 − ␪兲 − bCh␩ exp共e兲, dt 1 + Ch exp共e兲

␣⌫2

d␩ = exp共2e兲共␪ − ␩兲 − cCh␩ exp共e兲, dt

共4b兲

共4c兲

where v is the dimensionless circuit potential and r is the dimensionless resistance. The Faradic current 共iF,兲 is given as

B. Frequency and phase of oscillation

H共t兲 =

共3兲

iF =

冉

冊

Ch exp共0.5e兲 + a exp共e兲 共1 − ␪兲. 1 + Ch exp共e兲

共5兲

Parameter values Ch = 1600, a = 0.3, b = 6 ⫻ 10−5, c = 0.001, ⌫1 = 0.01, ⌫2 = 0.8 were used in the present study. In the original formulation the time scale parameter ␣ = 1; the physical meaning of this parameter is discussed later. The ordinary differential equations were solved numerically with 20 XPPAUT program package applying a fourth-order Runge– Kutta method with variable step size. Maximal Lyapunov exponents 共␭兲 were determined using lyap_r method in TISEAN 共Ref. 8兲 package. This method requires an embedding dimension, delay time, and number of iteration time as parameters for a certain time series. We determined maximal Lyapunov exponents for embedding dimensions 兵7, 8, 11其 and 兵5, 7, 9, 11其 for experimental and stimulated time series, respectively. The reported values are the mean values obtained at the given embedding dimensions. The delay time was determined using the autocorrelation function8 of the normalized current 共i − 具i典兲. The time at

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FIG. 1. Experiments: Phase coherent chaotic dynamics of a single electrode at 10 ° C. 共a兲 Time series of the current. 共b兲 Attractor using time-delay coordinates. 共c兲 Peak-to-peak period of the nth oscillation in current time series. 共d兲 Histogram of periods. The average period 具T典 = 0.90 s. R = 1300 ⍀, V = 1293 mV.

which the autocorrelation function reached a value of exp共−1兲 was taken as delay time. 共The iteration time was the same as delay time.兲 Information dimension 共di兲 was determined with the fixed mass method of TISEAN,8 which requires delay time, minimal embedding dimension 共5兲, maximal embedding dimension 共11兲, minimal time separation 共same as delay time兲, and number of reference points 共1000兲 as parameters. III. RESULTS A. Experiments 1. Phase coherent chaotic behavior at 10 ° C

The potentiostatic dissolution of Ni exhibits chaotic dynamics upon addition of an appropriate external resistance to the circuit.21,22 The time series of current oscillations at 10 ° C is shown in Fig. 1共a兲. At this low temperature, chaotic oscillations are relatively smooth without large, abrupt variations in amplitude. The reconstructed attractor with time delay coordinates is shown in Fig. 1共b兲. In accordance with previous findings21 the attractor has low-dimensional character with information dimension of 2.3 and reduced Lyapunov exponent 共␭r = ␭ / ␻兲 of 1.5 1/cycle. 共The term “reduced” will be used throughout this paper to denote frequency normalized quantities; in this example the reduced Lyapunov exponent is expressed per cycle instead of the more common 1/s unit.兲 The peak-to-peak periods of oscillations are shown in Fig. 1共c兲; similar to previous results23 the peak-to-peak cycles often follow a pattern where long and short period cycles alternate. The peak-to-peak period histogram in Fig. 1共d兲 shows that the periods fall into a relatively small range of ⫾16% 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. 2共a兲. The oscillations exhibit a clear center of rotation around the

Chaos 20, 023125 共2010兲

FIG. 2. 共Color online兲 Experiments: Phase dynamics of phase coherent chaotic oscillations. 共a兲 Phase space using the derivative Hilbert transform. 共b兲 Phase vs time. 共c兲 Detrended phase vs time. 共d兲 Phase variance vs time.

origin; the phase thus can be obtained as the unwrapped angle to the phase point at a given time. The phase 关Fig. 2共b兲兴 increased approximately linearly versus time; from the slope we obtained a frequency of ␻ = 1.108 Hz. The detrended phase, obtained by removing linear trend in phase angle, is shown in Fig. 2共c兲. The phase fluctuations are very small: for experimental time of 1500 s corresponding to 1350 oscillations the largest deviation 共5 rad兲 was less than 1 cycle. The phase diffusion coefficient was obtained from a linear fit of the phase variance versus time plot, as shown in Fig. 2共d兲. As expected from the qualitative dynamical features, the reduced phase diffusion coefficient 共Dr = D / ␻兲 at 10 ° C was found to be a small value of 0.0047 rad2. The phase analysis indicates that at the low temperature the chaotic behavior is phase coherent and thus the chaotic oscillations are very precise. 2. Non-phase-coherent chaotic behavior at 30 ° C

The dynamics of metal dissolution are strongly affected when temperature is increased to 30 ° C. In order to obtain chaotic oscillations, the external resistance and circuit potential have to be changed from the values applied at 10 ° C. During the experiments we started with a low resistance where only period-1 behavior can be observed in the oscillatory region of circuit potential. The resistance was then increased so that a period-2 behavior is identified in a region of circuit potential. Further increase of external resistance typically resulted in a period-doubling route to chaos as the circuit potential was changed. The data are obtained from approximately the middle of the chaotic circuit potential region. The presented results thus describe characteristics of emerging chaotic behavior; the applied resistance and circuit potential values are different at various temperatures. The time series of current oscillations is shown in Fig. 3共a兲. The mean current levels increased from a value of 0.225 mA at 10 ° C to a value of 1.005 mA at 30 ° C. The topological structure of reconstructed attractor 关Fig. 3共b兲兴 seems to be more complicated than that observed at 10 ° C

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Chaos 20, 023125 共2010兲

M. Wickramasinghe and I. Z. Kiss

FIG. 3. Experiments: Non-phase-coherent chaotic dynamics of a single electrode at 30 ° C. 共a兲 Time series of the current. 共b兲 Attractor using time-delay coordinates. 共c兲 Instantaneous period of the nth oscillation in current time series. 共d兲 Histogram of periods. 具T典 = 0.17 s. R = 400 ⍀, V = 1410 mV.

关Fig. 1共b兲兴; this qualitative observation is supported by the increased information dimension that was found to be 2.7. However, the reduced Lyapunov exponent decreased from 1.5 to 1.3 1/cycle. The relatively simple alternation of long and short cycles seen at 10 ° C 关Fig. 1共c兲兴 is broken at 30 ° C 关Fig. 3共c兲兴: a series of long and a series of short cycles tend to alternate. The histogram of peak-to-peak periods 关Fig. 3共d兲兴 exhibits a larger spread of ⫾34% and more distinct bimodal character at 30 ° C than those seen at 10 ° C in Fig. 1共d兲. Phase dynamics at 30 ° C is portrayed in Fig. 4. The oscillations exhibit a clear center of rotation around origin in the derivative Hilbert space 关Fig. 4共a兲兴; however, unique center of rotation was not found in the traditional Hilbert space 共not shown兲. The frequency of oscillations was found to be ␻ = 6.017 Hz obtained from the phase versus time plot in

FIG. 5. 共Color online兲 Experiments: The dependency of mean current and frequency of chaotic oscillations on temperature. 共a兲 The dependency of mean current on temperature. 共b兲 Arrhenius plot of mean current vs temperature. 共c兲 The dependency of frequency of current oscillations vs temperature. 共d兲 Arrhenius plot of frequency of current oscillations vs temperature. Experiments are carried out at the following temperatures: 10.0, 15.0, 20.0, 22.5, 25.0, 27.5, and 30.0 ° C.

Fig. 4共b兲. The detrended phase 关Fig. 4共c兲兴 exhibits large fluctuations that are typically seen with random walk; during an experiment time of 1800 oscillations a phase deviation 共11 rad兲 close to two full cycles was seen. The reduced phase diffusion coefficient was found to be 0.089 rad2 关see Fig. 4共d兲兴: an 18-fold increase from the coefficient at 10 ° C. The chaotic oscillations exhibited higher complexity when temperature was increased from 10 to 30 ° C. At the elevated temperature the time series develops more complicated oscillation pattern and the chaotic attractor has a somewhat increased value of information dimension. Most importantly, the “traditional” Hilbert transform fails to define phase and the phase diffusion coefficient is large; these results imply that at elevated temperature the chaotic behavior becomes non-phase-coherent. 3. Temperature effects

FIG. 4. 共Color online兲 Experiments: Phase dynamics of non-phase-coherent chaotic oscillations. 共a兲 Phase space using the derivative Hilbert transform. 共b兲 Phase vs time. 共c兲 Detrended phase vs time. 共d兲 Phase variance vs time.

To quantitatively characterize the effect of temperature on the chaotic dynamics we have carried out experiments at seven different temperatures between 10 and 30 ° C. The dependency of mean current and the frequency of chaotic oscillations on temperature are shown in Figs. 5共a兲 and 5共c兲, respectively. The mean current is proportional to the overall rate of dissolution; a previous study24 showed that with simple periodic current oscillations in nickel electrodissolution, the variations of currents with temperature follow that of the frequency. Similar to the periodic oscillations, both mean current and frequency exhibited strong increase with temperature for the chaotic oscillations; the ln 具i典 versus 1 / T 关Fig. 5共b兲兴 and the ln ␻ versus 1 / T 关Fig. 5共d兲兴 plots reveal Arrhenius-type temperature dependencies. The apparent activation energies obtained from the slopes of Arrhenius plots were 54 kJ/mol for mean current and 61 kJ/mol for frequency. The apparent activation energies of current and fre-

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Effect of temperature on precision

FIG. 6. 共Color online兲 Experiments: Complexity measures of chaotic oscillations. 共a兲 Reduced phase diffusion coefficient, Dr = D / ␻, vs temperature. 共b兲 Arrhenius plot of reduced phase diffusion coefficient vs temperature. 共c兲 Lyapunov exponent 共circle兲 and reduced Lyapunov exponent 共triangle兲, ␭r = ␭ / ␻, vs temperature. 共d兲 Information dimension vs temperature.

quency of chaotic oscillation thus have slightly increased values compared to the Ea = 46 kJ/ mol 共current兲 and 49 kJ/mol 共frequency兲 obtained for periodic oscillations.24 The dependency of reduced phase diffusion coefficient on temperature is shown in Fig. 6共a兲. The reduced phase diffusion coefficient exhibits strong, Arrhenius-type temperature dependence 关Fig. 6共b兲兴; the apparent activation energy was found to be 108 kJ/mol, almost twice than those obtained for current or frequency. In contrast, a traditional measure of complexity of chaos, the reduced Lyapunov exponent exhibited no considerable dependency on temperature, as shown in Fig. 6共c兲. Although the Lyapunov exponent 共with unit of 1/s兲 does increase with temperature, the increase is due to the increase of frequency. Therefore, when the reduced Lyapunov exponent is considered 共with unit of 1/cycle兲 only minor variations are seen with temperature. The information dimension of chaotic attractor did increase from a value of 2.34 to 2.6 from 10 to 20 ° C. Beyond 20 ° C, the dimension had a small increase to the value of about 2.7 observed at 30 ° C. B. Numerical simulations

A greatly simplified ordinary differential equation model 关Eqs. 共4兲 and 共5兲兴 has already been developed for simulation of chaotic Ni electrodissolution.15,19 In this model the kinetic parameters 共a , b , c , ⌫1 , ⌫2兲 were not determined in independent experiments but instead were optimized in a parameter search to simulate experimental bifurcation structure. In order to model temperature dependence of chaotic behavior, the temperature dependence 共pre-exponential factor and activation energy兲 of all five parameters needs to be known. Because of the relatively large number of unknown parameters, we do not directly simulate the effect of temperature on Eqs. 共4兲 and 共5兲, but rather explain the experimentally observed dependence of phase coherence of chaotic oscillations as follows. 共A similar approach was used for interpretation of temperature induced chaos.25兲

Chaos 20, 023125 共2010兲

FIG. 7. 共Color online兲 Numerical simulations: The dependency of chaotic dynamics on time scale parameter ␣ in electrochemical model 关Eq. 共4兲兴. 共a兲 Frequency of chaotic oscillations vs ␣. 共b兲 Reduced phase diffusion coefficient vs ␣. 共c兲 Reduced Lyapunov exponent vs ␣. 共d兲 Information dimension vs ␣.

In a previous study24 it was shown that the periods of electrochemical oscillations are determined by time scale parameters of the two essential variables. In Ni dissolution, these variables are ␪ and e; the time scales of these variables are strongly affected by parameters ⌫1 and r, respectively. When chaotic behavior develops, dynamical changes occur in the third variable ␩. Our hypothesis is that the variation of this third variable will be responsible for producing complex rhythms; therefore, the precision of chaotic behavior could be strongly affected by ␣, a time scale of variable ␩. This time scale parameter incorporates temperature dependence of parameters that affects the time scale of ␩, i.e., ⌫2, 具e典, and c in a simplified manner. We carried out numerical simulations in which the effect of ␣ parameter on chaotic dynamics is studied while all other parameters are kept constant. In this way, the relative effect of time scale parameter ␣ is investigated on characteristics of chaotic behavior. Chaotic behavior is studied at parameter values r = 50 and v = 60.8: this parameter set with ␣ = 1 generates lowdimensional chaotic behavior similar to that observed in the experiments at 10 ° C. Equations 共4兲 and 共5兲 were integrated and the currents j共t兲 = 共v − e共t兲兲 / r were processed the same way as the experimental data. Below ␣ = 1 chaotic oscillations did not occur; above ␣ = 1.344 bursting oscillations occur; therefore, the numerical study was limited to 1.000ⱕ ␣ ⱕ 1.344. Figure 7共a兲 shows the dependence of frequency of the chaotic oscillator on ␣. The frequency is practically independent of parameter ␣; the variation is less than ⫾2%. This lack of variation is quite expected from the theory of frequency of electrochemical oscillators:24 ␣ affects the time scale of a variable 共␩兲 that is not essential in generation of periodic oscillations; therefore, it will have only minor effects on the frequency. The reduced phase diffusion coefficient increased with increasing ␣ 关see Fig. 7共b兲兴. Dr exhibited a tenfold increase while ␣ was increased from 1.000 to 1.344. There is large

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M. Wickramasinghe and I. Z. Kiss

scatter of data for parameter region close to the appearance of bursting oscillations, 1.250ⱕ ␣ ⬍ 1.344; the behavior in this region exhibits intermittent dynamics with long periodic windows. Note that the variation of Dr with ␣ is approximately linear. Temperature is expected to affect the kinetic parameters in Arrhenius manner; therefore, the Arrhenius type of experimental variation of Dr can be explained by linear dependency on ␣ which in turn has Arrhenius type of dependency on temperature. The reduced Lyapunov exponent 关Fig. 7共c兲兴 increases with increasing ␣. Note, however, that the variation is not very strong and mainly occurs at the endpoints of the parameter region of the chaotic oscillations; there is small increase from ␭r = 0.34 to 0.39 while ␣ is increased from 1.050 to 1.219. The information dimension 关Fig. 7共d兲兴 has a slightly increasing trend as the dimension increases from 1.69 to 1.82 as ␣ is varied. Numerical simulations thus indicate that the time scale parameter 共␣兲 corresponding to “third” variable responsible for chaotic oscillations 共␩兲 has strong effect on phase diffusion and relatively weak effect on Lyapunov exponent and information dimension. These effects are similar to those observed in the experiments. IV. DISCUSSION

We have presented experimental results showing that with increase of temperature the precision of chaotic current oscillations deteriorates in nickel electrodissolution. The effect of temperature on dynamics of oscillatory systems has been studied extensively.1,24,26–33 In particular, the studies of dependence of period on temperature revealed Arrheniustype dependence in many chemical, and temperature compensated dependence in many biological1,32 and some chemical systems.26,27 With nickel electrodissolution the frequency increases with temperature and follows Arrhenius-type dependence for simple periodic24 and, as it is shown in this study, for chaotic oscillations as well. In addition to the oscillations speeding up, the precision of chaotic oscillations was greatly affected by temperature. The precision of biological clocks is an important dynamical property; for example, it was shown that the circadian gene expression of cyanobacteria, in contrast to that of mammalian suprachiasmatic nuclei cells, is impressively precise and the high precision is built in cellular property.34 The temperature dependence of precision of biological clocks would reveal additional dynamical characteristics and thus provide valuable information for improving modeling studies. Manifestations of breakdown of clock precision with changes in temperature have been reported in homogeneous chemical systems in other contexts. Temperature induced route to chaos was found in the H2O2 – HSO3− – S2O32− flow reaction system.25 In the oscillatory electrochemical urea system two-frequency quasiperiodic oscillation is transformed to three-frequency oscillations with increase in temperature.35 Our experiments show that the phase coherent chaotic dissolution at low temperature becomes non-phasecoherent at elevated temperatures. The synchronization properties of phase coherent chaotic oscillations are quite

different from those of non-phase-coherent oscillators;12,16 therefore, nickel dissolution could be used as an experimental test bed for studying different, phase diffusion dependent routes to synchrony. In the experiments Dr exhibited more pronounced temperature dependence on temperature than those of the traditional complexity measures. We observed very small dependence of Lyapunov exponents while phase diffusion greatly changed: the precision of the chaotic behavior deteriorated with temperature increase but the predictability remained nearly the same. The different behaviors of these measures can be interpreted by considering the useful concept of characterizing complexity of chaos based on the properties of embedded unstable periodic orbits 共UPOs兲.7 Since the Lyapunov exponent and phase diffusion are determined primarily by the stability and period of UPOs, respectively, it is quite possible that predictability and precision of a chaotic attractor are not always strongly correlated quantities. We consider phase diffusion coefficient as a complexity measure that complements rather than replaces other traditional complexity measures such as the Lyapunov exponent. We have carried out numerical simulations and showed that phase diffusion coefficient can be effectively tuned by changing the time scale parameter of the “third” variable responsible for chaotic behavior. We have carried out numerical simulations similar to those of the chemical model but with chaotic Rössler oscillator.12 We found that the precision of chaotic oscillations in the Rössler system could be effectively tuned by introducing a time scale parameter for variable z. Of course, changes in phase diffusion could be expected by changes of parameters; for example, a parameter in equation for the y variable was found to affect phase diffusion coefficient.12 We note, however, that by critically slowing down the z variable bursting behavior can be produced and thus the overall complexity of the system can be increased in a simple manner. Our experiments indicate that the precision of chaotic oscillators measured by the phase diffusion coefficient is an important complexity measure that gives information about dynamical changes affected by external parameters. In addition, phase fluctuations in coupled oscillators can be applied to characterize the transition to synchronization.17,18 The application of phase diffusion coefficient is thus encouraged in analysis of chaotic physical, chemical, and biological experimental systems, especially when phase of the signals can be relatively easily defined and long time series data are available. ACKNOWLEDGMENTS

Acknowledgments are made to the Donors of the American Chemical Society Petroleum Research Fund and the President’s Research Fund of Saint Louis University for support of this research. A. T. Winfree, The Geometry of Biological Time 共Springer-Verlag, Berlin, 1980兲. 2 A. Goldbeter, Biochemical Oscillations and Cellular Rhythms: The Molecular Bases of Periodic and Chaotic Behavior 共Cambridge University Press, Cambridge, England, 1996兲. 1

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