Rui
Zhu
a and
Qian
Shu Li
*ab
aSchool of Chemical Engineering and Materials Science, Beijing Institute of Technology, Beijing 100081, People's Republic of China. E-mail: qsli@mh.bit.edu.cn
bNational Key Laboratory of Theoretical and Computational Chemistry, Jilin University, Changchun, Jilin 130023, People's Republic of China
First published on 13th December 2001
A Belousov–Zhabotinsky (BZ) reaction model subject to additive Gaussian white noise is investigated as the model system is located in the dynamical region of period-1 oscillation. The model is composed of three ordinary differential equations representing the time evolutions of HBrO2, Ce(IV), and BrCH(COOH)2
, respectively. Noise is separately added to the three equations. In all three cases the change in the Ce(IV) variable's output signal-to-noise ratio (SNR) as a function of the noise intensity shows non-monotonic behavior, indicating the occurrence of explicit internal signal stochastic resonance. However, its strength is quite different in each case, implying that the three reacting species play different roles in determining the dynamical behavior in the BZ reaction. The change in the peak height of the Ce(IV) variable's output frequency
spectrum and the shift of the peak's central frequency with increasing noise intensity are also examined.
In 1996, Schneider et al. first investigated chemical SR experimentally with input signal and noise in three nonlinear reaction systems, which are all in a steady state close to a Hopf bifurcation.11–13 Numerical simulations demonstrate that SR with input signal and noise can also occur in bistable chemical systems. However, contrary to the bistable state for the overdamped oscillator that is often used to explain the mechanism of SR, the chemical bistable state corresponds to a stable node and a stable limit cycle, respectively.16 In the numerical work on the photosensitive Belousov–Zhabotinsky reaction, two-parameter SR was found,20,21i.e., when one parameter is modulated by signal and the other by noise, the signal-to-noise ratio (SNR) goes through a maximum with increasing noise intensity. In the earlier studies of SR, as in the cases shown above, it was realized that the occurrence of SR needs three basic ingredients: (i) a threshold nonlinear system, (ii) a weak coherent input, and (iii) a noise input. With the development in SR studies, however, one is convinced that this is not the case. One example is the noise-free SR,27 where the role of noise is played by the chaos generated through the inherent dynamics of a deterministic chaotic system. Another more interesting example is the internal signal stochastic resonance (ISSR), which was first discovered when investigating numerically a two-dimensional autonomous model system in a state near a saddle-node bifurcation.28 When the control parameter is randomly modulated near the saddle-node bifurcation point, noise-induced coherent oscillation (NICO) occurs, the strength of which passes through a maximum with increasing noise intensity. Thereafter, Xin's group studied ISSR numerically in several chemical reaction models, which, however, are in a steady state near a Hopf bifurcation.14–18 Their simulations also showed that ISSR could occur in the two-parameter photosensitive Belousov–Zhabotinsky reaction.18 Very recently, they first presented the experimental observation of ISSR in the Belousov–Zhabotinsky reaction.19
To our knowledge, the systems studied in the previous chemical ISSR are single-threshold systems associated with Hopf bifurcation; and the occurrences of ISSR are closely related to NICO generated by crossing bifurcation. Recently, however, we found a new type of ISSR by simulating the Belousov–Zhabotinsky reaction system in the period-1 oscillatory state with only random modulation of the control parameter.29 It can occur without involving crossing bifurcation. Since in this new type of ISSR the signal comes from the intrinsic period-1 oscillation of a nonlinear system, we called this SR explicit ISSR. The internal signal investigated in the previous ISSR comes from NICO, which is not shown without external noise, so we called the previous type of ISSR implicit ISSR. Contrary to implicit ISSR, no sudden change of state is involved in explicit ISSR. In our previous work, we have compared their characteristics, and concluded that they have different underlying mechanisms.
Our previous studies on explicit ISSR were conducted with parametric perturbation, representing the effect of external noise on the system. In the present work, however, we will use additive perturbation, representing the effect of internal fluctuations on the system. We will use the same system as has been used in our previous work, i.e., the three-variable Belousov–Zhabotinsky reaction model developed by Gyorgyi and Field.30 Noise is added separately in the three equations, thus, there are three cases. Based on the signal to noise ratio variation with noise intensity obtained from the frequency spectrum, numerical simulations demonstrate that explicit ISSR can occur in all three cases, but its strength is significantly different in each case. The variation in frequency spectrum peak height and the shift of the peak's central frequency with noise intensity are examined for the three cases.
In all the three cases, we will examine the output behavior of the z variable, i.e., the Ce(IV) concentration, with increase in the intensity (D) of the additive noise. The choice of the Ce(IV) concentration variable here is based on the consideration of its role in determining the electric potential in the BZ reaction system. This is easily measured by a redox electrode in the real reaction system. Since our studies are conducted with additive perturbation of variables, they in fact investigate the effect of internal fluctuations of the variables on the BZ system.
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Fig. 1 Bifurcation diagram obtained from numerical simulations for the BZ reaction model of Gyorgyi and Field. Notations used: p1, period-1 oscillations; p2, period-2 oscillations; ch, a chaotic regime. The arrow indicates the period-1 oscillatory state at kf![]() ![]() ![]() ![]() |
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Fig. 2 log(z)
versus reduced time for the period-1 oscillation at kf![]() ![]() ![]() ![]() |
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Fig. 3 SNR as evaluated from the frequency spectra of time series of log(z) versus noise intensity. Gaussian white noise added to (a) dx/dτ (b) dz/dτ and (c) dv/dτ. |
The presence of such resonance peaks shows that the addition of noise with proper intensity can make the distribution of signals more centralized at the fundamental signals, indicating the constructive role of noise. However, it ought to be noted that the resonance peaks on the SNR curve tend to weaken with increasing noise intensity, indicating that the destructive role of noise still exists. We will see more clearly the two-fold role played by noise in the next section.
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Fig. 4 Spectrum peak height plotted against noise intensity. Gaussian white noise added to (a) dx/dτ (b) dz/dτ and (c) dv/dτ. |
A characteristic of implicit ISSR called the noise-induced frequency shift, in which the spectrum peak's central frequency is significantly shifted with the noise intensity, was reported in ref. 28. Thus, we will next examine the shift of the spectrum peak's central frequency in our three cases. As shown in Fig. 5, the all shift very little, and their values increase with increasing noise intensity except for several plateaus. Comparing Fig. 5 and Fig. 4, we see that the number of plateaus in Fig. 5 for any case is almost equal to the number of main peaks in Fig. 4 for the same case.
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Fig. 5 Central frequency of the spectrum peak versus noise intensity. Gaussian white noise added to (a) dx/dτ (b) dz/dτ and (c) dv/dτ. |
dx/dτ, dz/dτ, and dν/dτ represent in the BZ reaction the time evolutions of HBrO2, Ce(IV), and BrCH(COOH)2
, respectively; and the BZ system is investigated with added perturbations representing the effect of internal fluctuations on the system. Thus, the different values of the maximal SNR for the three cases indicate that, among HBrO2
, Ce(IV), and BrCH(COOH)2 in the BZ reaction, the internal fluctuation of Ce(IV) demonstrates the strongest effect on the system's output electric potential, while that of BrCH(COOH)2 demonstrates the weakest effect. In addition, comparing the three cases in Fig. 3–5, we can easily conclude that the larger the number of the main peaks on the spectrum peak height
curve (or the number of plateaus on the frequency curve) the stronger the obtained explicit ISSR.
Besides HBrO2, Ce(IV), and BrCH(COOH)2
, Br− is another important control intermediate to determine the dynamic behavior in the BZ reaction. However, the evolution equation of Br− is deleted for simplicity in the model of the present work. It can be argued that the additive perturbation of Br− should show explicit ISSR yet different in strength from the other three cases. This could be confirmed by doing the same work on the four-dimensional model32 of the BZ reaction (including the evolution equation of Br−).
In a real BZ reaction experiment all variables and even the parameters always fluctuate. According to our previous work on parametric perturbations29 and the present work on additive perturbations, we find that explicit ISSR could occur in both cases, i.e., external noises and internal fluctuations can cause the same effect. For the latter, the strengths of explicit ISSR for different variables could not only give the order of their sensitivities towards internal fluctuations but also examine their roles in determining the dynamic behavior in the BZ reaction from the viewpoint of perturbations.
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