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Controlling complexity using forcing: simulations and experiments
Applied Mathematics and Computation 164 (2005) 467–491 www.elsevier.com/locate/amc
Controlling complexity using forcing: simulations and experiments P. Parmananda a
a,*
, M. Rivera a, B.J. Green b, J.L. Hudson
b
Facultad de Ciencias, UAEM, Av. Universidad. 1001, Col. Chamilpa, 62210 Cuernavaca, Morelos, Me´xico b Department of Chemical Engineering, Thornton Hall, University of Virginia, Charlottesville, Virginia 22903-2442, USA
Abstract We report the successful manipulation of non-linear dynamics using external forcing. In the case of temporal systems, a model system involving ordinary differential equations (odeÕs) was used for simulations. Experiments were carried out in a single anode electrochemical cell. Numerical and experimental results indicate that under the influence of external forcing, control of complexity and a change in periodicity of the autonomous dynamics can be achieved. A natural extension of this work involves analyzing the effects of global and local forcing on complex spatio-temporal behavior. A numerical model involving partial differential equations (pdeÕs) was used for simulating the dynamics of this extended system. An electrochemical cell involving multiple anodes was used for the corresponding experiments. In simulations and in experiments, suppression of spatio-temporal complexity is observed for the two forcing (global and local) methods. 2004 Elsevier Inc. All rights reserved. Keywords: Ordinary differential equations; Partial differential equations; Controlling complexity; External forcing; Electrochemical corrosion
*
Corresponding author. E-mail address: [email protected] (P. Parmananda).
0096-3003/$ - see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2004.06.035
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1. Introduction Manipulation of non-linear dynamics has been a topic of active research in the engineering community for a long-time [1]. However the interest of the physicists in general, and chaoticians in particular was invoked by the pioneering work of Ott, Grebogi and Yorke (OGY) [2]. In their landmark paper, they provided a general prescription for controlling chaos using a feedback technique. In OGYÕs control strategy, small time dependent perturbations are made to one of the control parameters, resulting in the stabilization of the system on one of the infinite unstable periodic orbits embedded in the chaotic attractor. Since the advent of OGY technique, dynamical control of chaotic behavior has been demonstrated [3–6] in various different real systems. These experiments use flexible feedback control strategies [2,7] to convert the observed chaotic behavior to periodic responses. However, there does exists an alternate control technique involving periodic modulation of a system parameter at an appropriate frequency [8–10]. This method, known as the non-feedback control, is generally considered to be less elegant than their feedback counterparts. This is mainly because the control signal, for the non-feedback control, does not go to zero subsequent to stabilization of the target state. Nevertheless, they possess an overwhelming advantage in that they do not require any prior knowledge of the system behavior. This makes them particularly appealing for a certain class of biological and chemical systems whose states are extremely difficult to measure in real time. In this article, we propose using forcing methods to control non-linear dynamics of electrochemical systems. Moreover, extending this technique to spatio-temporal systems enables the suppression of spatio-temporal complexity. The article is organized as follows: In Section 2, we present both numerical and experimental results involving control of temporal dynamics. Section 3, includes numerical and experimental results involving the control of spatio-temporal complexity when subjected to global forcing. Numerical and experimental results involving suppression of spatio-temporal complexity under the influence of local forcing are reported in Section 4. Finally, a brief discussion of the obtained results is presented in Section 5.
2. Control of temporal dynamics subjected to forcing In this section, we consider an autonomous dynamical system which is described by a general set of differential equations: x_ ¼ fðxðtÞ; pÞ;
ð1Þ
where x = (x1, x2, x3, . . ., xn) and p = (p1, p2, p3, . . . , pm) are the system variables and the control (bifurcation) parameters, respectively. Depending on the values
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of control parameters, a given system may exhibit a wealth of dynamical responses such as steady state(s), periodic or chaotic oscillations, etc. A discontinuous transition between these states is called a bifurcation, and the map showing the location of different dynamical states in the parameter space is called a bifurcation diagram. Considering that in our electrochemical experiments there is an easily measurable system variable x1 (e.g., the anodic current), the accessible control parameter p1 (e.g., the anodic potential) can be continuously perturbed such that p1 ðtÞ ¼ p1 ð0Þ þ c sinðxtÞ;
ð2Þ
where p1(0) is the initial parameter value and c sin(xt) is the superimposed sinusoidal modulation that transforms the unperturbed dynamics. The characteristics of the altered dynamics under the influence of this non-vanishing control are determined by c and x, the two parameters of the forcing function. Moreover, the dynamics remain altered as long as the control is being implemented. Subsequent to switching ‘‘off’’ the control, the system reverts back to exhibiting the initial unperturbed behavior. 2.1. Numerical results The forcing technique is first tested in a model for electrochemical corrosion [11] described by three dimensionless differential equations: Y_ ¼ pð1 hOH hO Þ qY ;
ð3Þ
h_ OH ¼ Y ð1 hOH hO Þ ½expðbhOH Þ þ rhOH þ 2shO ð1 hOH hO Þ;
ð4Þ
h_ O ¼ rhOH shO ð1 hOH hO Þ:
ð5Þ
Variables hO and hOH represent the fractions of the electrode surface covered by two different chemical species, while Y represents the concentration of metal ions in the electrolyte. Parameters p, q, r, s, and b are determined by chemical reaction rates in the model. Previous numerical studies have shown that depending on the parameter values this model can exhibit simple periodic or chaotic oscillations [11]. We numerically integrate these equations using a fourth order Runge–Kutta algorithm with a fixed stepsize (h = 2.0). For the purposes of control we change the value of p. For parameters values of Fig. 1, the model system exhibits chaotic oscillations. However, when subjected to the forcing term of Eq. (2), the dynamics are converted to period-1 oscillations. The parameters of the forcing term, determined by trial and error, are provided in the corresponding figure caption. It needs to be pointed out that this controlled period-1 oscillation of Fig. 1 is not the unstable period-1 orbit embedded with in the chaotic attractor. Such
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Fig. 1. Stabilization of period-1 dynamics in the numerical model (Eqs. (3)–(5)) exhibiting chaotic oscillations. The model parameters are [p, q, r, s, b] = [2.0 · 104, 1.0 · 103, 2.0 · 105, 9.7 · 105, 5.0], respectively and the forcing parameters are c = 5 · 105, x = 0.005 rad/s.
orbits are usually targeted using feedback control techniques and subsequent to their stabilization the control term goes to zero. The conversion of chaotic to periodic dynamics of Fig. 1 is representive of suppression of complexity via parametric entrainment. The perturbation is non-vanishing and subsequent to switching ‘‘off’’ the control, the system reverts back to exhibiting chaotic dynamics. Fig. 2 shows another such example involving the conversion of chaotic dynamics to a period-3 behavior. The system parameters remain the same, however, the control parameters c and x are different. These results indicate that the complexity of the temporal dynamics can indeed be suppressed using the forcing technique. Fig. 3, shows the conversion of period-1 oscillations to period-2 dynamics. The system parameters and the appropriate values of the forcing function (determined by trial and error), are provided in the corresponding figure caption. Similar to previous results, the dynamics remain altered as long as the forcing is being implemented. The timeseries of Fig. 3 indicate that the complexity of the autonomous dynamics can be increased using the forcing protocol. Another example of enhancing complexity using the forcing method is shown in Fig. 4. It depicts the conversion of the period-1 dynamics to a period-3 response. 2.2. Experimental results The experimental system was an EG&G Princeton Applied Research Model K60066 three-electrode electrochemical cell set up to study the potentiostatic
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Fig. 2. Stabilization of period-3 dynamics in the numerical model (Eqs. (3)–(5)) exhibiting chaotic oscillations. The model parameters are [p, q, r, s, b] = [2.0 · 104, 1.0 · 103, 2.0 · 105, 9.7 · 105, 5.0], respectively and the forcing parameters are c = 5 · 105, x = 0.0016 rad/s.
Fig. 3. Stabilization of period-2 dynamics in the numerical model (Eqs. (3)–(5)) exhibiting period-1 oscillations. The model parameters are [p, q, r, s, b] = [2.0 · 104, 1.0 · 103, 2.0 · 105, 9.61 · 105, 5.0], respectively and the forcing parameters are c = 5 · 105, x = 0.0025 rad/s.
electrodissolution of copper in an acetate buffer [12]. The anode is a rotating copper disk (5 mm diameter) shrouded by Teflon. The electrolyte is an acetate buffer, a mixture of 70 cm3 glacial acetic acid and 30 cm3 of 2 mol dm3 sodium acetate. The anodic potential is measured relative to a Saturated Calomel reference Electrode (SCE), while the cathode is a platinum foil disk (2.5 cm2 area).
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Fig. 4. Stabilization of period-3 dynamics in the numerical model (Eqs. (3)–(5)) exhibiting period-1 oscillations. The model parameters are [p, q, r, s, b] = [2.0 · 104, 1.0 · 103, 2.0 · 105, 9.61 · 105, 5.0], respectively and the forcing parameters are c = 5 · 105, x = 0.0017 rad/s.
Under potentiostatic conditions, the circuit potential is continuously adjusted by a potentiostat (EG&G Princeton Applied Research Model 362) to maintain a desired set value of the anodic potential V. The anodic current I is measured between the anode and cathode. Time series current data are collected and stored in a computer by sampling the anodic current using a data acquisition card with the sampling frequency fixed at 25 Hz. At the anodic potential and rotation of Fig. 5, the electrochemical system exhibits chaotic oscillations in anodic current (I). The anodic voltage (V) is modulated sinusoidally according to the following control formula, V ðtÞ ¼ V ð0Þ þ cðsinð2pmtÞÞ:
ð6Þ
The control parameters for the forcing function (c, m) are calculated using trial and error. The control is turned on at t = 50 s and consequently the chaotic dynamics is converted to period-1 (P1) oscillations. When the control is turned off at t = 70 s, the system moves away from the stabilized period-1 dynamics. After a transient of about 25 s during which the system recovers from the effect of control, it reverts back to exhibit original chaotic oscillations. This transient recovery period is observed both in experiments as well as in the numerical simulations. The same control formula (Eq. (6)) with a different set of forcing parameters can also stabilize a period-3 response as shown in Fig. 6. Consistent with numerical observations, experimental results also indicate that the forcing protocol (for appropriate choice of the control parameters) can increase the complexity of the observed dynamics. Figs. 7 and 8 show the transformation of the autonomous period-1 dynamics to period-2 and per-
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Fig. 5. Stabilization of period-1 dynamics in the experimental electrochemical cell exhibiting chaotic oscillations. The rotation rate is 2500 rpm, while the anodic potential V(0) is 0.770 V. The forcing parameters are c = 39.1 mV and m = 0.7 Hz. Anodic current is plotted over a period during which the external forcing is switched off, on and off again.
Fig. 6. Stabilization of period-3 dynamics in the experimental electrochemical cell exhibiting chaotic oscillations. The rotation rate is 2500 rpm, while the anodic potential V(0) is 0.770 V. The forcing parameters are c = 41.2 mV and m = 0.6 Hz. Anodic current is plotted over a period during which the external forcing is switched off, on and off again.
iod-3 dynamics, respectively. In both cases the dynamics remain altered for as long as the forcing protocol is being implemented. After the control is shut
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Fig. 7. Stabilization of period-2 dynamics in the experimental electrochemical cell exhibiting period-1 oscillations. The rotation rate is 1700 rpm, while the anodic potential V(0) is 0.680 V. The forcing parameters are c = 75.1 mV and m = 2.0 Hz. Anodic current is plotted over a period during which the external forcing is switched off, on and off again.
Fig. 8. Stabilization of period-3 dynamics in the experimental electrochemical cell exhibiting period-1 oscillations. The rotation rate is 1700 rpm, while the anodic potential V(0) is 0.680 V. The forcing parameters are c = 70.2 mV and m = 3.0 Hz. Anodic current is plotted over a period during which the external forcing is switched off, on and off again.
‘‘off’’, the system reverts back to exhibiting period-1 dynamics subsequent to a transient recovery period.
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For temporal systems, numerical and experimental results exhibit remarkable similarities. Both indicate that the forcing technique can be used to convert autonomous dynamics to controlled dynamics of lower (chaos ! P1) or higher complexity (P1 ! P2) in dissipative electrochemical systems. The parameters of the control function were always calculated using trial and error. Therefore, no prior knowledge of the system dynamics was required for successful implementation of the control method.
3. Control of spatio-temporal complexity using global forcing Coupling of non-linear oscillatory units is a simple way to create spatio-temporal chaos. The response of such spatio-temporally systems depends on the nature of coupling and dynamical evolution of the local oscillator. Appropriate tuning of system parameters can yield dynamics that are strongly disordered (maximally chaotic) with a rapid decay of correlations both in space and time. Taming of complex dynamical behavior typical of distributed dynamical systems can be obtained using feedback [13–15] or external forcing techniques [16,17]. Numerical results involving parametric resonance [17] indicate the suppression of natural turbulent dynamics in the presence of strong global forcing via stabilization of coherent structures (spirals, Turing patterns etc). Experimental confirmation of these simulations was reported for the light sensitive Belousov–Zhabotinsky (BZ) reaction [18]. In this section, we test the global forcing method for induction, propagation and maintenance of long-range ordered patterns. 3.1. Numerical results We choose the following model used for the description of CO-oxidation on a Pt(1 1 0) single crystal surface under UHV conditions [19,20]. The CO oxidation on Pt(1 1 0) proceeds via the Langmuir-Hinshelwood mechanism and associated surface structural changes. The underlying dynamics are determined by the surface coverages of CO and O and their interactions with the surface structure v. Spatial coupling of adjacent regions on the catalyst is provided by the surface diffusion of CO [19]. Since the CO and O coverages are always strictly anticorrelated for realistic parameters, they can be combined to a single coverage variable u which exhibits a S-shaped nullcline. For efficient numerical integration this nullcline can be simplified to a Z-shaped nullcline resulting in the following system of equations after rescaling of space and time uðu 1Þ vþb @tu ¼ u ð7Þ þ Dr2 u; a
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@ t v ¼ f ðuÞ v;
ð8Þ
where u and v are activator and inhibitor variables, respectively. The continuous and differentiable function f(u) exhibits the following threshold values (‘‘delayed inhibitor production’’): u < 1=3 ! f ðuÞ ¼ 0;
1=3 6 u 6 1 ! f ðuÞ ¼ 1 6:75uðu 1Þ2 ;
u > 1 ! f ðuÞ ¼ 1:
The function f(u) was fitted to experimental LEED and STM data such that the simulations are consistent with experimental observations [19]. Under appropriate parameter values in one spatial dimension, the model system exhibits transition to turbulence via backfiring of pulses. A bifurcation analysis reveals that the model system may exhibit travelling pulse behavior, amplitude turbulence and phase turbulence [19,20]. To integrate associated pdeÕs (Eqs. (7) and (8)) the system size was chosen to be 100 dimensionless units and then divided into 200 grid elements for simulation using explicit integration algorithm with constant time and space (100/200) steps subjected to periodic boundary conditions. To implement control, we perturb the entire array of oscillators with periodic sinusoidal forcing (c sinxt). Fig. 9(a) shows the space–time plot of the systems dynamics in gray-scale. Prior to the control, the dynamics exhibit amplitude turbulence. Subsequent to the implementation of the control, the dynamics are stabilized on a spatially homogeneous and a temporally periodic state. Fig. 9(b) shows the local timeseries of one of the oscillators. It shows the conversion of the local dynamics from irregular to regular subsequent to the initiation of the control. The control parameters of the forcing function, chosen by trial and error, are included in the figure caption along with the system parameters. 3.2. Experimental results We attempted an experimental confirmation of the numerical results suggesting that inception of order via global non-vanishing perturbations is possible. The system chosen for implementation of this global forcing strategy is a three-electrode electrochemical cell. This electrochemical system has been used previously to study the potentiostatic electrodissolution of iron in a sulfuric acid buffer [23] under ambient temperature (295–300K) conditions. The anode of the cell is an array of iron electrodes shrouded by epoxy. Electrolyte solution is a mixture of 1 M H2SO4 and 1 M Na2SO4. The anodic potential is measured relative to a Hg/Hg2SO4/K2SO4 reference electrode, while the cathode is a cylindrical platinum mesh encircling the array. Experiments were performed in an impinging jet cell as shown in Fig. 10. The diameter of the jet is 5 mm and the distance from the jet to the electrode surface is 6 mm. Each electrode
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Fig. 9. Control of spatio-temporal chemical chaos via stabilization of spatially homogeneous and temporally periodic states for the spatially extended system using global forcing. The system parameters are a = 0.84, = 0.12, b = 0.045 and D = 1/5.2 and the control parameters are c = 0.8 and x = 5. (a) Space time portrait prior and subsequent to (indicated by ‘‘on’’) implementation of the control signal. Every 20th step is plotted along the time axis. It illustrates induction and subsequent propagation of global order by virtue of global forcing. (b) The local time series of the 100th cell prior and subsequent to the implementation of the control. The controlled timeseries is a period-1 oscillation indicating inception of order.
is made from a pure iron wire of diameter 0.5 mm. The distance between the electrodes is 0.05 mm and they are embedded in epoxy in a ring configuration. The reference electrode is located next to the jet, that is, 6 mm from the plane of the working electrode and 4 mm from the axis; the separation is in a direction normal to the orientation of the array of working electrodes. Corrosion (electrodissolution) takes place only on the ends. Electrodes and the epoxy are sanded to a flat plane before each experiment with 180 grit silicon carbide sandpaper. The surface is then washed with deionized water and dried with compressed air. Electrochemical reaction is controlled by a bi-potentiostat (Pine Model AFRDE 4) and a waveform generator (HP 33120). Electrodes in the array are linked to the anode jacks of the bi-potentiostat through the ZRA (zero-resistance ammeters). The ZRA circuit consists of an operational amplifier and
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Fig. 10. Experimental set-up used to observe and subsequently control the spatio-temporal complexity observed during potentiostatic dissolution of iron in sulfuric acid.
associated feedback circuitry such that the individual currents of the electrodes in the array can be measured without altering their polarization potentials. To mimic the periodic forcing implemented in the numerical simulations, [21,22] all of the electrodes are forced sinusoidally in the potentiostatic mode. Data were recorded using a 32-channel data acquisition card (Keithley DAS180HC2) installed in a pentium PC. The outputs from the ZRA box are measured and recorded at the sampling rate of 2000 Hz. Fig. 11(a–c) show the 21 timeseries of the uncontrolled dynamics. Note that there are variations among the currents of individual electrodes. Time series of unforced chaotic oscillators in other geometries, a 61-hexagonal array and a ring of 29 electrodes, are discussed in [23,24], respectively. In all cases, including the ring configuration being considered here, the spatially extended system is heterogeneous, i.e., there are slight variations among the electrodes and in the spacings. Coupling among the electrodes is due to several factors including diffusion, convection, and migration. Thus both local and long-range [25,26] components of coupling dictate the system dynamics. Since, the numerical results involved identical oscillators with purely diffusive coupling, the experiments lack a one-to-one correspondence with the simulations. However, they retain the qualitative features of the simulations. Fig. 12(a–c) show the corresponding 21 timeseries of the controlled dynamics. Upon inspection, it is observed, that the complex dynamical behavior of each electrode is converted to regular dynamics under the influence of the superimposed non-vanishing
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Fig. 11. Experimental set-up used to observe and subsequently control the spatio-temporal complexity observed during potentiostatic dissolution of iron in sulfuric acid. (a–c) Uncontrolled timeseries of the 21 electrodes embedded in epoxy in a ring geometry.
(global) forcing. The characteristics of the forcing function, determined by trial and error, are provided in the figure caption. It is reasonable to expect that the global forcing would be able to suppress spatio-temporal complexity. However, it is a little surprising that the forcing
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control is successful despite the heterogeneity of the experimental system. This is an indication of the robustness of the forcing protocol. Finally, to reiterate, control using this non-feedback method can be achieved without any previous insight into the natural dynamics. This offers a tremendous advantage over their feedback counterparts.
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4. Control of spatio-temporal complexity using local forcing Results of the previous section indicate that it is possible to suppress spatiotemporal complexity under the influence of global forcing. However, it was recently reported [21] that high-dimensional chaotic dynamics typical of spatio-temporal systems could be converted to a well-defined ordered state
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without having to perturb the entire system. These simulations were carried out on a spatially extended system comprising of diffusively coupled periodic oscillators in the limit where the dynamics could be computed via integration of corresponding partial differential equations. A sinusoidal forcing function of
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the type A sin(xt) was superimposed on the evolution equation of one of the oscillators (via modulation of a control parameter or as an additive term). If the amplitude (A) of the perturbation exceeds a certain threshold and the forcing frequency is appropriate (the intrinsic frequency of an isolated oscillator is a good starting point), then the prevalent spatio-temporal complexity is replaced by an ordered state. The stabilized coherent state comprises of periodic
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pulse trains propagating from the point of stimulation to the end of the chain [21,22]. Following the induction of global order the local dynamics of each of the oscillators is rendered periodic. We attempt to implement this local control technique in the numerical model and in the experimental system of the previous section.
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Fig. 13. Control of spatio-temporal chemical chaos via stabilization of traveling pulse train solutions for the spatially extended system using local forcing. The system parameters are a = 0.84, = 0.12, b = 0.045 and D = 1/5.2 and the control parameters are c = 0.8 and x = 2.8 (a) Space time portrait prior and subsequent to (indicated by ‘‘on’’) implementation of the control signal. Every 20th step is plotted along the time axis. It illustrates induction and subsequent propagation of global order by virtue of local forcing. (b) The local time series of the 100th cell prior and subsequent to the implementation of the control. The controlled timeseries is a period-1 oscillation indicating inception of order.
4.1. Numerical results In this case one of the sites of the spatially extended system is perturbed with a local forcing. This involves forcing the evolution equation of one of the 200 oscillators (site) by a sinusoidal modulation c sin(xt). Under the influence of this local perturbation the altered dynamics of one of the oscillators (for example i = 1) (the evolution equations for the remaining oscillators are unchanged) is represented by 1u1 ðu1 1Þ vþb @ t u1 ¼ u1 ð9Þ þ Dr2 u1 þ c sinðxtÞ; a @ t v ¼ f ðu1 Þ v:
ð10Þ
The characteristics of the forcing function can be chosen by trial and error. Fig. 13(a) shows the space time plot for the extended system under the effect of this
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(a)
(b)
(c)
(d)
(e)
(f)
(g) Fig. 14. The timeseries of the eight electrode array configuration in the absence of control. (a–g) correspond to electrodes 2 through 8, respectively.
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local forcing. For the appropriate choice of the control parameters, the model dynamics exhibit the induction of order and its subsequent propagation up until complete suppression of spatio-temporal complexity is achieved. Fig. 13(b) shows the local timeseries of the oscillator furthest away from the forcing. It indicates that a complete suppression of the spatio-temporal complexity can be achieved. The controlled state comprises of stable wave trains propagating in one spatial dimension. These coherent waves are initiated at the point of stimulation and traverse the entire extended system. The control perturbations to the first site remain non-zero. This forcing technique is extremely powerful as it attains global order via local non-vanishing perturbations. The ordered state continues to persists as long as the control signal is ‘‘ON’’. 4.2. Experimental Results In this section, we report the experimental confirmation of the numerical results suggesting that the inception of global order via local non-vanishing perturbations is possible. The system chosen for implementation of this local forcing strategy is the three-electrode electrochemical cell described in the previous subsection. The anode of the cell for these experiments is an array of eight iron electrodes shrouded by epoxy. The other electrodes and the electrolyte solution remain the same. Each electrode is made from a pure iron wire of diameter 0.5 mm. The distance between the electrodes is 0.05 mm and they are embedded in epoxy in a straight line configuration as shown in Fig. 10. Thus, the length of the eight-electrode configuration is 4.95 mm. Fig. 14(a–g) shows a section of the time series for the seven response electrodes in the uncontrolled case where no forcing is imposed. The other electrode (#1) which subsequently is forced (sinusoidally) in the galvanostatic mode is held at zero current. Note that there are variations among the currents of the seven electrodes. The appropriate perturbation frequency x to implement the control was obtained from examining the Fourier spectra of the natural dynamics and was calculated to be x = 25 Hz. Some results obtained with a forcing frequency of 25 Hz and an amplitude of 7.5 mA applied to electrode #1 are shown in Fig. 15. Fig. 15(a) is the forced electrode. Fig. 15(b–h) show the dynamics for the remaining electrodes after transients have died out. The chaotic dynamics have been converted to period-1 oscillations for all the electrodes. The forcing signal is able to maintain control and suppress the spatio-temporal complexity. Similar to numerical observations the control emanates from the forcing electrode and spreads down the array. However, instead of a smooth propagation of order observed in simulations, experimental results are as shown in Fig. 16. The forcing signal was initiated at time (t = 6 s). Almost instantly the second and third electrode start exhibiting periodic dynamics. The
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(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
Fig. 15. The time series of the eight electrode array configuration subjected to local periodic stimulus. (a–h) correspond to electrodes 1 through 8, respectively.
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40
Time(s)
30
PERIODIC
20
CHAOTIC
10
0
2
3
4
5
6
7
8
Electrode Number Fig. 16. Propagation of the control signal down the array. Control was initiated at time = 6 s and complete suppression of spatio-temporal chaos is attained at time = 30 s. This time-lag between the third and the fourth electrode was observed consistently.
fourth electrode joins in after a little delay. After an elapsed time of about 20 s the fifth electrode joins in and almost immediately electrodes six, seven and eight start to exhibit periodic dynamics as shown in Fig. 15(b–h). It should be pointed out that although the forced dynamics are in different amplitude ranges, the frequency of the period-1 oscillations is identical for all the response electrodes and to that of the forcing signal (x = 25 Hz). This is similar to phase-synchronization of coupled oscillators as defined by Fujigaki et al. [27]. When the forcing was turned off the electrodes went back to exhibiting chaotic dynamics. The experiments were repeated several times under these conditions. In most cases the observed response was as described above, i.e., all electrodes were brought to a periodic state. Occasionally, however, only the electrodes (one to four) nearest the forcing electrode became periodic. It appears that the coupling is sometimes not strong enough to overcome fluctuations induced by system inhomogeneities and drift.
5. Discussion The results of the previous three sections indicate that forcing is indeed a viable method to manipulate the oscillatory dynamics of non-linear systems. For temporal systems this method not only can lead to the suppression of complexity but also change the periodicity of the regular dynamics. This involved
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decreasing and/or increasing the complexity of the autonomous regular behavior. This method is then extended to spatially extended systems. Control of spatio-temporal chaos is complicated due to the existence of numerous unstable spatial modes, but, is more important too, because of its possible applications in plasma, laser devices and chemical systems where both spatial and temporal dependences need to be considered. The results involving suppression of spatio-temporal chaos seem to be a natural extension of the suppression of the temporal dynamics. However, the fact that local forcing can also achieve the suppression of spatio-temporal complexity is indeed impressive. These results could also be considered as an example of phase-synchronization [27] where the forced coupled oscillators are synchronized with each other in phase but are different in amplitudes and location in state space. Finally, as pointed out by Baier et al. [21] this work is of interest in an information theoretic context in biology. Local temporal information of biological relevance could be encoded in a meaningful ordered spatial pattern that creates corresponding temporal signals at distinct sites. Furthermore the pattern formation inside a cell could act as a biologically relevant encoding mechanism to transfer extracellular signals to targeted sites of biochemical action. Experimental evidence of such encoding mechanism in biological cells was found by Camacho and Lechlieter [28]. They reported experimental formation of a regular spatial calcium wave pattern following a local receptor activation by applying an external concentration level of bombesin to Xenopus oocytes. Similar to our experimental observations, their results indicate that transduction of information is possible for only finite ranges of receptor activation. This implies that the extended system acts as a non-linear frequency filter such that only a range of temporal signals get translated to an ordered spatial pattern. Finally in their experiments brief receptor activations do not spread over the whole cell but are destroyed in the vicinity of the point of activation. This is consistent with our observations.
Acknowledgement This work has been supported by CONACyT(Mexico).
References [1] K. Ogata, Modern Control Engineering, second ed., Prentice-Hall, Englewood Cliffs, NJ, 1990. [2] E. Ott, C. Grebogi, J.A. Yorke, Controlling chaos, Phys. Rev. Lett. 64 (1990) 1196. [3] W.L. Ditto, S.N. Rauseo, M.L. Spano, Experimental control of chaos, Phys. Rev. Lett. 65 (1990) 3211.
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[4] E.R. Hunt, Stabilizing high-period orbits in a chaotic system: the diode resonator, Phys. Rev. Lett. 67 (1991) 1953. [5] R. Roy, T. Murphy Jr., T.D. Maier, Z. Gills, E.R. Hunt, Dynamical control of a chaotic laser: experimental stabilization of a globally coupled system, Phys. Rev. Lett. 68 (1992) 1259. [6] P. Parmananda, P. Sherard, R.W. Rollins, H.D. Dewald, Control of chaos in an electrochemical cell, Phys. Rev. E 47 (1993) R3003. [7] R.W. Rollins, P. Parmananda, P. Sherard, Controlling chaos in highly dissipative systems––a simple recursive control algorithm, Phys. Rev. E 47 (1993) R780. [8] R. Mettin, T. Kurz, Optimized periodic control of chaotic systems, Phys. Lett. A 206 (1995) 331. [9] R. Mettin, A. Hubler, A. Scheeline, W. Lauterborn, Parametric entrainment control of chaotic systems, Phys. Rev. E 51 (1995) 4065–4075. [10] P. Parmananda, M. Rivera, R. Madrigal, Altering oscillatory dynamics of an electrochemical system using external forcing, Electrochim. Acta 44 (1999) 4677. [11] J. Kevin McCoy, P. Parmananda, R.W. Rollins, A..J. Markworth, Chaotic dynamics in a model of metal passivation, J. Mater. Res. 8 (1993) 1858. [12] H.D. Dewald, P. Parmananda, R.W. Rollins, Periodic and chaotic current oscillations at a copper electrode in an acetate electrolyte, J. Electrochem. Soc. (1993) 1969. [13] G. Hu, Z. Qu, Controlling spatiotemporal chaos in coupled map lattice systems, Phys. Rev. Lett. 72 (1994) 68. [14] D. Auerbach, Controlling extended systems of chaotic elements, Phys. Rev. Lett. 72 (1994) 1184. [15] P. Parmananda, M. Hildebrand, M. Eiswirth, Suppressing turbulence using feedback control techniques, Phys. Rev. E 56 (1997) 239. [16] Y. Braiman, J.F. Lindner, W.L. Ditto, Taming spatiotemporal chaos with disorder, Nature (London) 378 (1997) 465. [17] P. Coullet, K. Emillson, Strong resonances of spatially distributed oscillators: a laboratory to study patterns and defects, Physica D 61 (1992) 119. [18] V. Petrov, Q. Quyang, H.L. Swinney, Resonant pattern formation in a chemical system, Nature (London) 388 (1997) 655. [19] M. Ba¨r, C. Zu¨licke, M. Eiswirth, G. Ertl, Spiral waves in a surface reaction: Model calculations, J. Chem. Phys. 100 (1994) 1202. [20] M. Ba¨r, M. Eiswirth, Turbulence due to spiral breakup in a continuous excitable media, Phys. Rev. E 48 (1993) R1635. [21] G. Baier, S. Sven, J-P Chen, A.A. Hoff, Local stimulation induces long-range order in spatiotemporal disorder, J. Chem. Phys. 110 (6) (1999) 3251. [22] P. Parmananda, M. Eiswirth, Suppression of chemical turbulence using feedbacks and forcing, J. Phys. Chem. 103 (28) (1999) 5510. [23] Z. Fei, J.L. Hudson, Chaotic oscillations on arrays of iron electrodes, Ind. Eng. Chem. Res. 37 (1998) 2172. [24] Z. Fei, B.J. Green, J.L. Hudson, Spatiotemporal patterns on a ring array of electrode, J. Phys. Chem. B 103 (1999) 2178. [25] G. Fla¨tgen, K. Krischer, Accelerating fronts in an electrochemical system due to global coupling, Phys. Rev. E 51 (1995) 3997. [26] K. Krischer, in: B.E. Conway Jr. (Ed.), Modern Aspects in Electrochemistry, Number 32, Kluwer Academic/Plenum Press, New York, 1999. [27] H. Fujigaki, M. Nishi, T. Shimada, Synchronization of nonlinear systems with distinct parameters: phase synchronization and metamorphosis, Phys. Rev. E 53 (1996) 3192. [28] P. Camacho, J.D. Lechleiter, in: G.R. Brock, K. Ackrill (Eds.), Calcium Waves, Gradients and Oscillations, Wiley, Chichester, 1995.
Controlling complexity using forcing: simulations and experiments P. Parmananda a
a,*
, M. Rivera a, B.J. Green b, J.L. Hudson
b
Facultad de Ciencias, UAEM, Av. Universidad. 1001, Col. Chamilpa, 62210 Cuernavaca, Morelos, Me´xico b Department of Chemical Engineering, Thornton Hall, University of Virginia, Charlottesville, Virginia 22903-2442, USA
Abstract We report the successful manipulation of non-linear dynamics using external forcing. In the case of temporal systems, a model system involving ordinary differential equations (odeÕs) was used for simulations. Experiments were carried out in a single anode electrochemical cell. Numerical and experimental results indicate that under the influence of external forcing, control of complexity and a change in periodicity of the autonomous dynamics can be achieved. A natural extension of this work involves analyzing the effects of global and local forcing on complex spatio-temporal behavior. A numerical model involving partial differential equations (pdeÕs) was used for simulating the dynamics of this extended system. An electrochemical cell involving multiple anodes was used for the corresponding experiments. In simulations and in experiments, suppression of spatio-temporal complexity is observed for the two forcing (global and local) methods. 2004 Elsevier Inc. All rights reserved. Keywords: Ordinary differential equations; Partial differential equations; Controlling complexity; External forcing; Electrochemical corrosion
*
Corresponding author. E-mail address: [email protected] (P. Parmananda).
0096-3003/$ - see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2004.06.035
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1. Introduction Manipulation of non-linear dynamics has been a topic of active research in the engineering community for a long-time [1]. However the interest of the physicists in general, and chaoticians in particular was invoked by the pioneering work of Ott, Grebogi and Yorke (OGY) [2]. In their landmark paper, they provided a general prescription for controlling chaos using a feedback technique. In OGYÕs control strategy, small time dependent perturbations are made to one of the control parameters, resulting in the stabilization of the system on one of the infinite unstable periodic orbits embedded in the chaotic attractor. Since the advent of OGY technique, dynamical control of chaotic behavior has been demonstrated [3–6] in various different real systems. These experiments use flexible feedback control strategies [2,7] to convert the observed chaotic behavior to periodic responses. However, there does exists an alternate control technique involving periodic modulation of a system parameter at an appropriate frequency [8–10]. This method, known as the non-feedback control, is generally considered to be less elegant than their feedback counterparts. This is mainly because the control signal, for the non-feedback control, does not go to zero subsequent to stabilization of the target state. Nevertheless, they possess an overwhelming advantage in that they do not require any prior knowledge of the system behavior. This makes them particularly appealing for a certain class of biological and chemical systems whose states are extremely difficult to measure in real time. In this article, we propose using forcing methods to control non-linear dynamics of electrochemical systems. Moreover, extending this technique to spatio-temporal systems enables the suppression of spatio-temporal complexity. The article is organized as follows: In Section 2, we present both numerical and experimental results involving control of temporal dynamics. Section 3, includes numerical and experimental results involving the control of spatio-temporal complexity when subjected to global forcing. Numerical and experimental results involving suppression of spatio-temporal complexity under the influence of local forcing are reported in Section 4. Finally, a brief discussion of the obtained results is presented in Section 5.
2. Control of temporal dynamics subjected to forcing In this section, we consider an autonomous dynamical system which is described by a general set of differential equations: x_ ¼ fðxðtÞ; pÞ;
ð1Þ
where x = (x1, x2, x3, . . ., xn) and p = (p1, p2, p3, . . . , pm) are the system variables and the control (bifurcation) parameters, respectively. Depending on the values
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of control parameters, a given system may exhibit a wealth of dynamical responses such as steady state(s), periodic or chaotic oscillations, etc. A discontinuous transition between these states is called a bifurcation, and the map showing the location of different dynamical states in the parameter space is called a bifurcation diagram. Considering that in our electrochemical experiments there is an easily measurable system variable x1 (e.g., the anodic current), the accessible control parameter p1 (e.g., the anodic potential) can be continuously perturbed such that p1 ðtÞ ¼ p1 ð0Þ þ c sinðxtÞ;
ð2Þ
where p1(0) is the initial parameter value and c sin(xt) is the superimposed sinusoidal modulation that transforms the unperturbed dynamics. The characteristics of the altered dynamics under the influence of this non-vanishing control are determined by c and x, the two parameters of the forcing function. Moreover, the dynamics remain altered as long as the control is being implemented. Subsequent to switching ‘‘off’’ the control, the system reverts back to exhibiting the initial unperturbed behavior. 2.1. Numerical results The forcing technique is first tested in a model for electrochemical corrosion [11] described by three dimensionless differential equations: Y_ ¼ pð1 hOH hO Þ qY ;
ð3Þ
h_ OH ¼ Y ð1 hOH hO Þ ½expðbhOH Þ þ rhOH þ 2shO ð1 hOH hO Þ;
ð4Þ
h_ O ¼ rhOH shO ð1 hOH hO Þ:
ð5Þ
Variables hO and hOH represent the fractions of the electrode surface covered by two different chemical species, while Y represents the concentration of metal ions in the electrolyte. Parameters p, q, r, s, and b are determined by chemical reaction rates in the model. Previous numerical studies have shown that depending on the parameter values this model can exhibit simple periodic or chaotic oscillations [11]. We numerically integrate these equations using a fourth order Runge–Kutta algorithm with a fixed stepsize (h = 2.0). For the purposes of control we change the value of p. For parameters values of Fig. 1, the model system exhibits chaotic oscillations. However, when subjected to the forcing term of Eq. (2), the dynamics are converted to period-1 oscillations. The parameters of the forcing term, determined by trial and error, are provided in the corresponding figure caption. It needs to be pointed out that this controlled period-1 oscillation of Fig. 1 is not the unstable period-1 orbit embedded with in the chaotic attractor. Such
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Fig. 1. Stabilization of period-1 dynamics in the numerical model (Eqs. (3)–(5)) exhibiting chaotic oscillations. The model parameters are [p, q, r, s, b] = [2.0 · 104, 1.0 · 103, 2.0 · 105, 9.7 · 105, 5.0], respectively and the forcing parameters are c = 5 · 105, x = 0.005 rad/s.
orbits are usually targeted using feedback control techniques and subsequent to their stabilization the control term goes to zero. The conversion of chaotic to periodic dynamics of Fig. 1 is representive of suppression of complexity via parametric entrainment. The perturbation is non-vanishing and subsequent to switching ‘‘off’’ the control, the system reverts back to exhibiting chaotic dynamics. Fig. 2 shows another such example involving the conversion of chaotic dynamics to a period-3 behavior. The system parameters remain the same, however, the control parameters c and x are different. These results indicate that the complexity of the temporal dynamics can indeed be suppressed using the forcing technique. Fig. 3, shows the conversion of period-1 oscillations to period-2 dynamics. The system parameters and the appropriate values of the forcing function (determined by trial and error), are provided in the corresponding figure caption. Similar to previous results, the dynamics remain altered as long as the forcing is being implemented. The timeseries of Fig. 3 indicate that the complexity of the autonomous dynamics can be increased using the forcing protocol. Another example of enhancing complexity using the forcing method is shown in Fig. 4. It depicts the conversion of the period-1 dynamics to a period-3 response. 2.2. Experimental results The experimental system was an EG&G Princeton Applied Research Model K60066 three-electrode electrochemical cell set up to study the potentiostatic
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Fig. 2. Stabilization of period-3 dynamics in the numerical model (Eqs. (3)–(5)) exhibiting chaotic oscillations. The model parameters are [p, q, r, s, b] = [2.0 · 104, 1.0 · 103, 2.0 · 105, 9.7 · 105, 5.0], respectively and the forcing parameters are c = 5 · 105, x = 0.0016 rad/s.
Fig. 3. Stabilization of period-2 dynamics in the numerical model (Eqs. (3)–(5)) exhibiting period-1 oscillations. The model parameters are [p, q, r, s, b] = [2.0 · 104, 1.0 · 103, 2.0 · 105, 9.61 · 105, 5.0], respectively and the forcing parameters are c = 5 · 105, x = 0.0025 rad/s.
electrodissolution of copper in an acetate buffer [12]. The anode is a rotating copper disk (5 mm diameter) shrouded by Teflon. The electrolyte is an acetate buffer, a mixture of 70 cm3 glacial acetic acid and 30 cm3 of 2 mol dm3 sodium acetate. The anodic potential is measured relative to a Saturated Calomel reference Electrode (SCE), while the cathode is a platinum foil disk (2.5 cm2 area).
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Fig. 4. Stabilization of period-3 dynamics in the numerical model (Eqs. (3)–(5)) exhibiting period-1 oscillations. The model parameters are [p, q, r, s, b] = [2.0 · 104, 1.0 · 103, 2.0 · 105, 9.61 · 105, 5.0], respectively and the forcing parameters are c = 5 · 105, x = 0.0017 rad/s.
Under potentiostatic conditions, the circuit potential is continuously adjusted by a potentiostat (EG&G Princeton Applied Research Model 362) to maintain a desired set value of the anodic potential V. The anodic current I is measured between the anode and cathode. Time series current data are collected and stored in a computer by sampling the anodic current using a data acquisition card with the sampling frequency fixed at 25 Hz. At the anodic potential and rotation of Fig. 5, the electrochemical system exhibits chaotic oscillations in anodic current (I). The anodic voltage (V) is modulated sinusoidally according to the following control formula, V ðtÞ ¼ V ð0Þ þ cðsinð2pmtÞÞ:
ð6Þ
The control parameters for the forcing function (c, m) are calculated using trial and error. The control is turned on at t = 50 s and consequently the chaotic dynamics is converted to period-1 (P1) oscillations. When the control is turned off at t = 70 s, the system moves away from the stabilized period-1 dynamics. After a transient of about 25 s during which the system recovers from the effect of control, it reverts back to exhibit original chaotic oscillations. This transient recovery period is observed both in experiments as well as in the numerical simulations. The same control formula (Eq. (6)) with a different set of forcing parameters can also stabilize a period-3 response as shown in Fig. 6. Consistent with numerical observations, experimental results also indicate that the forcing protocol (for appropriate choice of the control parameters) can increase the complexity of the observed dynamics. Figs. 7 and 8 show the transformation of the autonomous period-1 dynamics to period-2 and per-
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Fig. 5. Stabilization of period-1 dynamics in the experimental electrochemical cell exhibiting chaotic oscillations. The rotation rate is 2500 rpm, while the anodic potential V(0) is 0.770 V. The forcing parameters are c = 39.1 mV and m = 0.7 Hz. Anodic current is plotted over a period during which the external forcing is switched off, on and off again.
Fig. 6. Stabilization of period-3 dynamics in the experimental electrochemical cell exhibiting chaotic oscillations. The rotation rate is 2500 rpm, while the anodic potential V(0) is 0.770 V. The forcing parameters are c = 41.2 mV and m = 0.6 Hz. Anodic current is plotted over a period during which the external forcing is switched off, on and off again.
iod-3 dynamics, respectively. In both cases the dynamics remain altered for as long as the forcing protocol is being implemented. After the control is shut
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Fig. 7. Stabilization of period-2 dynamics in the experimental electrochemical cell exhibiting period-1 oscillations. The rotation rate is 1700 rpm, while the anodic potential V(0) is 0.680 V. The forcing parameters are c = 75.1 mV and m = 2.0 Hz. Anodic current is plotted over a period during which the external forcing is switched off, on and off again.
Fig. 8. Stabilization of period-3 dynamics in the experimental electrochemical cell exhibiting period-1 oscillations. The rotation rate is 1700 rpm, while the anodic potential V(0) is 0.680 V. The forcing parameters are c = 70.2 mV and m = 3.0 Hz. Anodic current is plotted over a period during which the external forcing is switched off, on and off again.
‘‘off’’, the system reverts back to exhibiting period-1 dynamics subsequent to a transient recovery period.
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For temporal systems, numerical and experimental results exhibit remarkable similarities. Both indicate that the forcing technique can be used to convert autonomous dynamics to controlled dynamics of lower (chaos ! P1) or higher complexity (P1 ! P2) in dissipative electrochemical systems. The parameters of the control function were always calculated using trial and error. Therefore, no prior knowledge of the system dynamics was required for successful implementation of the control method.
3. Control of spatio-temporal complexity using global forcing Coupling of non-linear oscillatory units is a simple way to create spatio-temporal chaos. The response of such spatio-temporally systems depends on the nature of coupling and dynamical evolution of the local oscillator. Appropriate tuning of system parameters can yield dynamics that are strongly disordered (maximally chaotic) with a rapid decay of correlations both in space and time. Taming of complex dynamical behavior typical of distributed dynamical systems can be obtained using feedback [13–15] or external forcing techniques [16,17]. Numerical results involving parametric resonance [17] indicate the suppression of natural turbulent dynamics in the presence of strong global forcing via stabilization of coherent structures (spirals, Turing patterns etc). Experimental confirmation of these simulations was reported for the light sensitive Belousov–Zhabotinsky (BZ) reaction [18]. In this section, we test the global forcing method for induction, propagation and maintenance of long-range ordered patterns. 3.1. Numerical results We choose the following model used for the description of CO-oxidation on a Pt(1 1 0) single crystal surface under UHV conditions [19,20]. The CO oxidation on Pt(1 1 0) proceeds via the Langmuir-Hinshelwood mechanism and associated surface structural changes. The underlying dynamics are determined by the surface coverages of CO and O and their interactions with the surface structure v. Spatial coupling of adjacent regions on the catalyst is provided by the surface diffusion of CO [19]. Since the CO and O coverages are always strictly anticorrelated for realistic parameters, they can be combined to a single coverage variable u which exhibits a S-shaped nullcline. For efficient numerical integration this nullcline can be simplified to a Z-shaped nullcline resulting in the following system of equations after rescaling of space and time uðu 1Þ vþb @tu ¼ u ð7Þ þ Dr2 u; a
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@ t v ¼ f ðuÞ v;
ð8Þ
where u and v are activator and inhibitor variables, respectively. The continuous and differentiable function f(u) exhibits the following threshold values (‘‘delayed inhibitor production’’): u < 1=3 ! f ðuÞ ¼ 0;
1=3 6 u 6 1 ! f ðuÞ ¼ 1 6:75uðu 1Þ2 ;
u > 1 ! f ðuÞ ¼ 1:
The function f(u) was fitted to experimental LEED and STM data such that the simulations are consistent with experimental observations [19]. Under appropriate parameter values in one spatial dimension, the model system exhibits transition to turbulence via backfiring of pulses. A bifurcation analysis reveals that the model system may exhibit travelling pulse behavior, amplitude turbulence and phase turbulence [19,20]. To integrate associated pdeÕs (Eqs. (7) and (8)) the system size was chosen to be 100 dimensionless units and then divided into 200 grid elements for simulation using explicit integration algorithm with constant time and space (100/200) steps subjected to periodic boundary conditions. To implement control, we perturb the entire array of oscillators with periodic sinusoidal forcing (c sinxt). Fig. 9(a) shows the space–time plot of the systems dynamics in gray-scale. Prior to the control, the dynamics exhibit amplitude turbulence. Subsequent to the implementation of the control, the dynamics are stabilized on a spatially homogeneous and a temporally periodic state. Fig. 9(b) shows the local timeseries of one of the oscillators. It shows the conversion of the local dynamics from irregular to regular subsequent to the initiation of the control. The control parameters of the forcing function, chosen by trial and error, are included in the figure caption along with the system parameters. 3.2. Experimental results We attempted an experimental confirmation of the numerical results suggesting that inception of order via global non-vanishing perturbations is possible. The system chosen for implementation of this global forcing strategy is a three-electrode electrochemical cell. This electrochemical system has been used previously to study the potentiostatic electrodissolution of iron in a sulfuric acid buffer [23] under ambient temperature (295–300K) conditions. The anode of the cell is an array of iron electrodes shrouded by epoxy. Electrolyte solution is a mixture of 1 M H2SO4 and 1 M Na2SO4. The anodic potential is measured relative to a Hg/Hg2SO4/K2SO4 reference electrode, while the cathode is a cylindrical platinum mesh encircling the array. Experiments were performed in an impinging jet cell as shown in Fig. 10. The diameter of the jet is 5 mm and the distance from the jet to the electrode surface is 6 mm. Each electrode
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Fig. 9. Control of spatio-temporal chemical chaos via stabilization of spatially homogeneous and temporally periodic states for the spatially extended system using global forcing. The system parameters are a = 0.84, = 0.12, b = 0.045 and D = 1/5.2 and the control parameters are c = 0.8 and x = 5. (a) Space time portrait prior and subsequent to (indicated by ‘‘on’’) implementation of the control signal. Every 20th step is plotted along the time axis. It illustrates induction and subsequent propagation of global order by virtue of global forcing. (b) The local time series of the 100th cell prior and subsequent to the implementation of the control. The controlled timeseries is a period-1 oscillation indicating inception of order.
is made from a pure iron wire of diameter 0.5 mm. The distance between the electrodes is 0.05 mm and they are embedded in epoxy in a ring configuration. The reference electrode is located next to the jet, that is, 6 mm from the plane of the working electrode and 4 mm from the axis; the separation is in a direction normal to the orientation of the array of working electrodes. Corrosion (electrodissolution) takes place only on the ends. Electrodes and the epoxy are sanded to a flat plane before each experiment with 180 grit silicon carbide sandpaper. The surface is then washed with deionized water and dried with compressed air. Electrochemical reaction is controlled by a bi-potentiostat (Pine Model AFRDE 4) and a waveform generator (HP 33120). Electrodes in the array are linked to the anode jacks of the bi-potentiostat through the ZRA (zero-resistance ammeters). The ZRA circuit consists of an operational amplifier and
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Fig. 10. Experimental set-up used to observe and subsequently control the spatio-temporal complexity observed during potentiostatic dissolution of iron in sulfuric acid.
associated feedback circuitry such that the individual currents of the electrodes in the array can be measured without altering their polarization potentials. To mimic the periodic forcing implemented in the numerical simulations, [21,22] all of the electrodes are forced sinusoidally in the potentiostatic mode. Data were recorded using a 32-channel data acquisition card (Keithley DAS180HC2) installed in a pentium PC. The outputs from the ZRA box are measured and recorded at the sampling rate of 2000 Hz. Fig. 11(a–c) show the 21 timeseries of the uncontrolled dynamics. Note that there are variations among the currents of individual electrodes. Time series of unforced chaotic oscillators in other geometries, a 61-hexagonal array and a ring of 29 electrodes, are discussed in [23,24], respectively. In all cases, including the ring configuration being considered here, the spatially extended system is heterogeneous, i.e., there are slight variations among the electrodes and in the spacings. Coupling among the electrodes is due to several factors including diffusion, convection, and migration. Thus both local and long-range [25,26] components of coupling dictate the system dynamics. Since, the numerical results involved identical oscillators with purely diffusive coupling, the experiments lack a one-to-one correspondence with the simulations. However, they retain the qualitative features of the simulations. Fig. 12(a–c) show the corresponding 21 timeseries of the controlled dynamics. Upon inspection, it is observed, that the complex dynamical behavior of each electrode is converted to regular dynamics under the influence of the superimposed non-vanishing
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(global) forcing. The characteristics of the forcing function, determined by trial and error, are provided in the figure caption. It is reasonable to expect that the global forcing would be able to suppress spatio-temporal complexity. However, it is a little surprising that the forcing
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control is successful despite the heterogeneity of the experimental system. This is an indication of the robustness of the forcing protocol. Finally, to reiterate, control using this non-feedback method can be achieved without any previous insight into the natural dynamics. This offers a tremendous advantage over their feedback counterparts.
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4. Control of spatio-temporal complexity using local forcing Results of the previous section indicate that it is possible to suppress spatiotemporal complexity under the influence of global forcing. However, it was recently reported [21] that high-dimensional chaotic dynamics typical of spatio-temporal systems could be converted to a well-defined ordered state
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without having to perturb the entire system. These simulations were carried out on a spatially extended system comprising of diffusively coupled periodic oscillators in the limit where the dynamics could be computed via integration of corresponding partial differential equations. A sinusoidal forcing function of
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the type A sin(xt) was superimposed on the evolution equation of one of the oscillators (via modulation of a control parameter or as an additive term). If the amplitude (A) of the perturbation exceeds a certain threshold and the forcing frequency is appropriate (the intrinsic frequency of an isolated oscillator is a good starting point), then the prevalent spatio-temporal complexity is replaced by an ordered state. The stabilized coherent state comprises of periodic
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pulse trains propagating from the point of stimulation to the end of the chain [21,22]. Following the induction of global order the local dynamics of each of the oscillators is rendered periodic. We attempt to implement this local control technique in the numerical model and in the experimental system of the previous section.
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Fig. 13. Control of spatio-temporal chemical chaos via stabilization of traveling pulse train solutions for the spatially extended system using local forcing. The system parameters are a = 0.84, = 0.12, b = 0.045 and D = 1/5.2 and the control parameters are c = 0.8 and x = 2.8 (a) Space time portrait prior and subsequent to (indicated by ‘‘on’’) implementation of the control signal. Every 20th step is plotted along the time axis. It illustrates induction and subsequent propagation of global order by virtue of local forcing. (b) The local time series of the 100th cell prior and subsequent to the implementation of the control. The controlled timeseries is a period-1 oscillation indicating inception of order.
4.1. Numerical results In this case one of the sites of the spatially extended system is perturbed with a local forcing. This involves forcing the evolution equation of one of the 200 oscillators (site) by a sinusoidal modulation c sin(xt). Under the influence of this local perturbation the altered dynamics of one of the oscillators (for example i = 1) (the evolution equations for the remaining oscillators are unchanged) is represented by 1u1 ðu1 1Þ vþb @ t u1 ¼ u1 ð9Þ þ Dr2 u1 þ c sinðxtÞ; a @ t v ¼ f ðu1 Þ v:
ð10Þ
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local forcing. For the appropriate choice of the control parameters, the model dynamics exhibit the induction of order and its subsequent propagation up until complete suppression of spatio-temporal complexity is achieved. Fig. 13(b) shows the local timeseries of the oscillator furthest away from the forcing. It indicates that a complete suppression of the spatio-temporal complexity can be achieved. The controlled state comprises of stable wave trains propagating in one spatial dimension. These coherent waves are initiated at the point of stimulation and traverse the entire extended system. The control perturbations to the first site remain non-zero. This forcing technique is extremely powerful as it attains global order via local non-vanishing perturbations. The ordered state continues to persists as long as the control signal is ‘‘ON’’. 4.2. Experimental Results In this section, we report the experimental confirmation of the numerical results suggesting that the inception of global order via local non-vanishing perturbations is possible. The system chosen for implementation of this local forcing strategy is the three-electrode electrochemical cell described in the previous subsection. The anode of the cell for these experiments is an array of eight iron electrodes shrouded by epoxy. The other electrodes and the electrolyte solution remain the same. Each electrode is made from a pure iron wire of diameter 0.5 mm. The distance between the electrodes is 0.05 mm and they are embedded in epoxy in a straight line configuration as shown in Fig. 10. Thus, the length of the eight-electrode configuration is 4.95 mm. Fig. 14(a–g) shows a section of the time series for the seven response electrodes in the uncontrolled case where no forcing is imposed. The other electrode (#1) which subsequently is forced (sinusoidally) in the galvanostatic mode is held at zero current. Note that there are variations among the currents of the seven electrodes. The appropriate perturbation frequency x to implement the control was obtained from examining the Fourier spectra of the natural dynamics and was calculated to be x = 25 Hz. Some results obtained with a forcing frequency of 25 Hz and an amplitude of 7.5 mA applied to electrode #1 are shown in Fig. 15. Fig. 15(a) is the forced electrode. Fig. 15(b–h) show the dynamics for the remaining electrodes after transients have died out. The chaotic dynamics have been converted to period-1 oscillations for all the electrodes. The forcing signal is able to maintain control and suppress the spatio-temporal complexity. Similar to numerical observations the control emanates from the forcing electrode and spreads down the array. However, instead of a smooth propagation of order observed in simulations, experimental results are as shown in Fig. 16. The forcing signal was initiated at time (t = 6 s). Almost instantly the second and third electrode start exhibiting periodic dynamics. The
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fourth electrode joins in after a little delay. After an elapsed time of about 20 s the fifth electrode joins in and almost immediately electrodes six, seven and eight start to exhibit periodic dynamics as shown in Fig. 15(b–h). It should be pointed out that although the forced dynamics are in different amplitude ranges, the frequency of the period-1 oscillations is identical for all the response electrodes and to that of the forcing signal (x = 25 Hz). This is similar to phase-synchronization of coupled oscillators as defined by Fujigaki et al. [27]. When the forcing was turned off the electrodes went back to exhibiting chaotic dynamics. The experiments were repeated several times under these conditions. In most cases the observed response was as described above, i.e., all electrodes were brought to a periodic state. Occasionally, however, only the electrodes (one to four) nearest the forcing electrode became periodic. It appears that the coupling is sometimes not strong enough to overcome fluctuations induced by system inhomogeneities and drift.
5. Discussion The results of the previous three sections indicate that forcing is indeed a viable method to manipulate the oscillatory dynamics of non-linear systems. For temporal systems this method not only can lead to the suppression of complexity but also change the periodicity of the regular dynamics. This involved
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decreasing and/or increasing the complexity of the autonomous regular behavior. This method is then extended to spatially extended systems. Control of spatio-temporal chaos is complicated due to the existence of numerous unstable spatial modes, but, is more important too, because of its possible applications in plasma, laser devices and chemical systems where both spatial and temporal dependences need to be considered. The results involving suppression of spatio-temporal chaos seem to be a natural extension of the suppression of the temporal dynamics. However, the fact that local forcing can also achieve the suppression of spatio-temporal complexity is indeed impressive. These results could also be considered as an example of phase-synchronization [27] where the forced coupled oscillators are synchronized with each other in phase but are different in amplitudes and location in state space. Finally, as pointed out by Baier et al. [21] this work is of interest in an information theoretic context in biology. Local temporal information of biological relevance could be encoded in a meaningful ordered spatial pattern that creates corresponding temporal signals at distinct sites. Furthermore the pattern formation inside a cell could act as a biologically relevant encoding mechanism to transfer extracellular signals to targeted sites of biochemical action. Experimental evidence of such encoding mechanism in biological cells was found by Camacho and Lechlieter [28]. They reported experimental formation of a regular spatial calcium wave pattern following a local receptor activation by applying an external concentration level of bombesin to Xenopus oocytes. Similar to our experimental observations, their results indicate that transduction of information is possible for only finite ranges of receptor activation. This implies that the extended system acts as a non-linear frequency filter such that only a range of temporal signals get translated to an ordered spatial pattern. Finally in their experiments brief receptor activations do not spread over the whole cell but are destroyed in the vicinity of the point of activation. This is consistent with our observations.
Acknowledgement This work has been supported by CONACyT(Mexico).
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