⇦Chinese Physics Letters, 2017, Vol. 34, No. 7, Article code 070201 Resonance Analyses for a Noisy Coupled Brusselator Model * Pei-Rong Guo(郭培荣)1, Hai-Yan Wang(王海燕)2**, Jin-Zhong Ma(马晋忠)1 Affiliations 1School of Natural and Applied Sciences, Northwestern Polytechnical University, Xi'an 710072 2School of Marine Science and Technology, Northwestern Polytechnical University, Xi'an 710072 Received 5 January 2017 *Supported by the National Natural Science Foundation of China under Grant No 61571365.
**Corresponding author. Email: hywang@nwpu.edu.cn Citation Text: Guo P R, Wang H Y and Ma J Z 2017 Chin. Phys. Lett. 34 070201 Abstract We discuss the dynamical behavior of a chemical network arising from the coupling of two Brusselators established by the relationship between products and substrates. Our interest is to investigate the coherence resonance (CR) phenomena caused by noise for a coupled Brusselator model in the vicinity of the Hopf bifurcation, which can be determined by the signal-to-noise ratio (SNR). The CR in two coupled Brusselators will be considered in the presence of the Gaussian colored noise and two uncorrelated Gaussian white noises. Simulation results show that, for the case of single noise, the SNR characterizing the degree of temporal regularity of coupled model reaches a maximum value at some optimal noise levels, and the noise intensity can enhance the CR phenomena of both subsystems with a similar trend but in different resonance degrees. Meanwhile, effects of noise intensities on CR of the second subsystem are opposite for the systems under two uncorrelated Gaussian white noises. Moreover, we find that CR might be a general phenomenon in coupled systems. DOI:10.1088/0256-307X/34/7/070201 PACS:02.50.-r, 43.60.Cg © 2017 Chinese Physics Society Article Text Coherence resonance, a kind of stochastic resonance[1-4] without input signals, was first found in a limit cycle model,[5] and was shown to occur in excitable media, e.g., the FitzHugh–Nagumo model,[6] and the Brusselator model.[7] The coherence resonance (CR) phenomena of a single model have been widely studied in excitable systems with random perturbations,[8,9] and this research has provided the basis for further investigation of CR in coupled nonlinear stochastic systems that have attracted considerable attention due to their great importance in a number of fields such as physics, chemical reactions and biology. For example, Liu et al.[10] demonstrated that when identical or slightly nonidentical chaotic oscillators are coupled together, the temporal regularity of some measured signal characterizing the degree of the synchronization among the oscillators can be modulated by external noise in the sense of CR. Gong et al.[11] also studied the effect of the deviation of a particular kind of non-Gaussian colored noise on the collective firing behavior in bidirectionally coupled deterministic HH neurons, and the phenomenon of noise-induced CR shows that there is an optimal deviation of the non-Gaussian noise by which the coupled neurons can perform the most regular firing behavior. Werner et al.[12] stated that the CR can be observed in directionally noise-driven coupled rings with an odd or even number of elements, etc. For a Brusselator model, Hou et al.[13] interpreted how noise influences CR in the vicinity of the Hopf bifurcation. Shi[14] described the cooperation effect of noise and an external signal on implicit and explicit coherence resonances in a Brusselator system, in which the system stays in a steady state or an oscillatory state. Based on the existing methods and theories about a single Brusselator model, Zhou and Zhang[15] considered the echo wave and coexisting phenomena in linearly coupled Brusselators and put forward a new method to prove the existence of the echo wave. Li et al.[16] studied the cooperative effects of parameter heterogeneity and coupling on the CR of two unidirectional coupled Brusselator subsystems, and also investigated the CR of a two-way coupled Brusselator system under a multiplicative noise at only one end. The above-researched coupled Brusselator model is based on the subsystem coupled to their neighbor and the coupling strength can be adjusted by the coefficient of the subsystem. It is different from a coupled model established by the relationship between products and substrates of two Brusselator subsystems proposed by Tyson.[17] This coupled model studied by Tyson has attracted the attention of scholars, and the deterministic behaviors, such as subharmonic resonance, multiply-periodic and almost-periodic oscillations, associated Hopf bifurcation and double-Hopf bifurcations, have been studied for a long time.[17,18] The aim of this work is to study how the noise excitations influence the CR behaviors of this coupled system, in which the system is subjected to an additive Gaussian colored noise and two uncorrelated additive Gaussian white noises, respectively. We expect to give a detailed analysis of the connection of the resonance behavior between two subsystems of this coupled Brusselator system. To investigate the nonlinear phenomena in a chemical network, Tyson[17] proposed a model consisting of two Brusselators coupled in series, in which the two outputs $U$ and $E$ of the subsystem (Eqs. (1a)–(1d)) serve as the inputs of the subsystem (Eqs. (1e)–(1g)), expressed by $$\begin{align} &A\stackrel{k_1}{\longrightarrow}X,~~ \tag {1a}\\ &B+X\stackrel{k_2}{\longrightarrow}Y+U,~~ \tag {1b}\\ &2X+Y\stackrel{k_3}{\longrightarrow}3X,~~ \tag {1c}\\ &X\stackrel{k_4}{\longrightarrow}E,~~ \tag {1d}\\ &E+U\stackrel{k_5}{\longrightarrow}V+F,~~ \tag {1e}\\ &2U+V\stackrel{k_6}{\longrightarrow}3U,~~ \tag {1f}\\ &U\stackrel{k_7}{\longrightarrow}G,~~ \tag {1g} \end{align} $$ where $k_1$–$k_7$ are rate constants. The subsystems (Eqs. (1a)–(1d)) are called the 'first' Brusselator and the subsystems (Eqs. (1e)–(1g)) are called the 'second' Brusselator. The first Brusselator describes the transformation of the initial substrates $A$ and $B$ into products $U$ and $E$, and the second Brusselator produces $F$ and $G$ from two outputs of the first Brusselator. The rate equations for the five intermediates $X$, $Y$, $E$, $U$ and $V$ are $$\begin{align} \dot {X}=\,&k_1 A-k_2 BX+k_3 X^2Y-k_4 X, \\ \dot {Y}=\,&k_2 BX-k_3 X^2Y, \\ \dot {E}=\,&k_4 X-k_5 EU, \\ \dot {U}=\,&k_2 BX-k_5 EU+k_6 U^2V-k_7 U, \\ \dot {V}=\,&k_5 EU-k_6 U^2V.~~ \tag {2} \end{align} $$ By scaling and changes of variables to system (2), a free coupled Brusselator model is given as $$\begin{align} \dot {x}=\,&a-bx+x^2y-x, \\ \dot {y}=\,&bx-x^2y, \\ \dot {e}=\,&x-eu, \\ \dot {u}=\,&bx-eu+u^2v-u, \\ \dot {v}=\,&eu-u^2v,~~ \tag {3} \end{align} $$ where the variables $x$, $y$, $e$, $u$, $v$, $a$ and $b$ represent the dimensionless concentrations of the reactants $X$, $Y$, $E$, $U$, $V$, $A$ and $B$, respectively, and $a,b>0$ are the fixed concentrations. We have set all the rate constants as unity for convenience. Letting the right side of Eq. (3) be zero, we obtain the only equilibrium point, $$ (x_{\rm s},y_{\rm s},e_{\rm s},u_{\rm s},v_{\rm s})=(a,\frac{b}{a},\frac{1}{b},ab,\frac{1}{ab^2}). $$ Applying the translation $({\bar {x},\bar {y},\bar {e},\bar {u},\bar {v}})^{\rm T}=({x-x_{\rm s},y-y_{\rm s},e-e_{\rm s},u-u_{\rm s},v-v_{\rm s} })^{\rm T}$ to Eq. (3) yields $$\begin{alignat}{1} \dot {\bar {x}}=\,&({b-1})\bar {x}+a^2\bar {y}+\frac{b}{a}\bar {x}^2+2a\bar {x}\bar {y}+\bar {x}^2\bar {y}, \\ \dot {\bar {y}}=\,&-b\bar {x}-a^2\bar {y}-\frac{b}{a}\bar {x}^2-2a\bar {x}\bar {y}-\bar {x}^2\bar {y}, \\ \dot {\bar {e}}=\,&\bar {x}-ab\bar {e}-\frac{1}{b}\bar {u}-\bar {e}\bar {u}, \\ \dot {\bar {u}}=\,&b\bar {x}-ab\bar {e}+\Big({\frac{1}{b}-1}\Big)\bar {u}+a^2b^2\bar {v}\\ &+\frac{1}{ab^2}\bar {u}^2-\bar {e}\bar {u}+2ab\bar {u}\bar {v}+\bar {u}^2\bar {v}, \\ \dot {v}=\,&ab\bar {e}-\frac{1}{b}\bar {u}-a^2b^2\bar {v}-\frac{1}{ab^2}\bar {u}^2\\ &+\bar {e}\bar {u}-2ab\bar {u}\bar {v}-\bar {u}^2\bar {v}.~~ \tag {4} \end{alignat} $$ Then the unique equilibrium of Eq. (4) is $({\bar {x}_{\rm s},\bar {y}_{\rm s},\bar {e}_{\rm s},\bar {u}_{\rm s},\bar {v}_{\rm s} })=({0,0,0,0,0})$, and the coefficient matrix of the linear part of Eq. (4) is $$ C=\left(\begin{matrix} {b-1}& {a^2}& 0& 0& 0\\ {-b}& {-a^2}& 0& 0& 0\\ 1& 0& {-ab}& {-\frac{1}{b}}& 0\\ b& 0& {-ab}& {\frac{1}{b}-1}& {a^2b^2}\\ 0& 0& {ab}& {-\frac{1}{b}}& {-a^2b^2}\\ \end{matrix}\right). $$ Thus the characteristic polynomial of the linearized system evaluated at the equilibrium can be found as $$\begin{align} 0=\,&|\lambda E-C| \\ =\,&[\lambda ^2+({a^2-b+1})\lambda +a^2] \\ &\times [\lambda ^3+({a^2b^2+ab+1-\frac{1}{b}})\lambda ^2\\ &+({a^3b^3+a^2b^2+ab-2a})\lambda +a^3b^3], \end{align} $$ where $E$ is an identity matrix, and $\lambda$ stands for eigenvectors. It is obvious that the fifth-degree polynomial can be decomposed into a quadratic part and a cubic part, corresponding to a normal isolated Brusselator and the second Brusselator, respectively. This is not surprising because there is only unidirectional coupled in system (1), e.g., the presence of species $E$, $U$ and $V$ has no effect on the kinetics of species $X$ and $Y$. The concrete forms of three conditions are as follows: $$\begin{alignat}{1} F_1 (a,b)=\,&a^2-b+1 < 0,~~ \tag {5a}\\ F_2 (a,b)=\,&a^2b^3+ab^2+b-1 < 0,~~ \tag {5b}\\ F_3 (a,b)=\,&a^4b^6+2a^3b^5+2a^2b^4-3a^2b^3\\ &+2ab^3-3ab^2+b^2-3b+2 < 0.~~ \tag {5c} \end{alignat} $$ Since the first Brusselator is completely oblivious to the second, using the Routh–Hurwitz criteria[17] we know that the first Brusselator will lose stability when $a$ and $b$ satisfy inequality (5a), and the second will be unstable if $a$ and $b$ satisfy one of the conditions (5b) and (5c) without affecting the stability of the first. A stability analysis with respect to the parameters $a$ and $b$ are shown in Fig. 1, which gives the region of instability (shaded) of the two subsystems simultaneously for a coupled Brusselator model. Here $Q$ is a critical point of the double-Hopf bifurcation that occurred in the entire coupled model.
cpl-34-7-070201-fig1.png
Fig. 1. Stability analysis in parameter space.
We take the approximate value of $Q$ as $Q({a=0.479,b=1.228})$. In what follows, we discuss the CR phenomenon near a double-Hopf bifurcation of the coupled Brusselator model in the presence of noises by calculating SNRs for different variables. The model specifically includes two driving ways of noise: (1) The first Brusselator is subjected to an additive Gaussian colored noise. (2) The two Brusselators are subjected to two uncorrelated additive Gaussian white noises, respectively. We give a computing method of the signal-to-noise ratio (SNR) based on Refs. [19,20] with the expression $$\begin{align} SNR=\frac{P-P_{\rm noise} }{P_{\rm noise}},~~ \tag {6} \end{align} $$ where $P_{\rm noise}$ represents the power spectrum of noise, and $P$ is the total power spectrum of the system without the power spectrum of external signals. A real chemical reaction will be subjected to external or internal random fluctuations, and sometimes these fluctuations play an important role in promoting the occurrence of the coherence motion. To consider the effects of random factors on a chemical reaction, a Gaussian colored noise is added to the first Brusselator, expressed by $$\begin{align} \dot {x}=\,&a-bx+x^2y-x+\zeta (t), \\ \dot {y}=\,&bx-x^2y, \\ \dot {e}=\,&x-eu, \\ \dot {u}=\,&bx-eu+u^2v-u, \\ \dot {v}=\,&eu-u^2v,~~ \tag {7} \end{align} $$ where $\zeta (t)$ denotes an exponential Gaussian colored noise satisfying $$ \langle {\zeta (t)} \rangle =0,~ \langle \zeta (t)\zeta (s)\rangle =\frac{D}{\tau }\exp \Big(-\frac{|{t-s}|}{\tau }\Big), $$ which can be generated by the following linear process $$ \dot {\zeta }(t)=-\frac{\zeta }{\tau }+\frac{\eta (t)}{\tau}, $$ with $\tau$ being the correlation time of the noise, and $\eta (t)$ being a Gaussian white noise with zero mean and intensity $D$. Next, a numerical simulation is carried out to investigate the connection of CR between two subsystems when the first Brusselator of the coupled model is subjected to a Gaussian colored noise excitation. Based on Eq. (6), SNRs can be computed by the second-order stochastic Runge–Kutta algorithm under the initial conditions $({x_0,y_0,e_0,u_0,v_0 })=({0,0,0,0,0})$, $t_0=0$, $a=0.479$, $b=1.228$, and the time step $h=0.01$. If the SNR nonlinearly changes with increasing stochastic parameters and passes through a maximum at an intermediate optimal value where the oscillating reaction becomes most regular, which represents the presence of CR near the bifurcation point $Q$ in model (7), the influences of different parameters on CR can be analyzed by changing features of SNR. In the simulation of SNRs we find that the change of SNRs as a function of the noise intensity $D$ is not obvious with different $\tau$. Thus we only consider the effects of different noise intensities $D$ on SNRs as a function of the correlation time $\tau$ in this section.
cpl-34-7-070201-fig2.png
Fig. 2. The simulation results of SNRs as a function of the correlation time $\tau$ for different noise intensities $D=0.001$, 0.005 and 0.01 about variables $x$ and $y$ in the first Brusselator (a) and variables $u$ and $v$ in the second Brusselator (b).
Figure 2(a) shows the SNRs of variables $x$ and $y$ of the first Brusselator model under different noise intensities $D$. It is obvious that the maximum values of SNRs become larger with the increase of $D$, indicating that the first Brusselator has CR phenomena under the Gaussian colored noise excitation, and the greater the noise intensity is, the more obvious the CR phenomena are. Our simulation results also show the slightly right-skewing of the maximum values of SNRs with the increase of $D$, meaning that the CR phenomena are more intense with the increase of the noise intensity and the correlation time synchronously. Furthermore, we find that the SNRs of variables $x$ and $y$ change with the same trend but in different degrees, and the directing effects of noise on variable $x$ make CR more intense. In Fig. 2(b), the SNRs of variables $u$ and $v$ are also presented for different intensities $D$. Simulation results show that the change trend of the SNRs about variables $u$ and $v$ is almost similar to Fig. 2(a). However, the maximum value of SNR of variable $v$ is hardly changed with a larger $D$. Moreover, according to Fig. 2(b), it is observed that the values of SNRs are greater compared with Fig. 2(a). It is noteworthy that when the first Brusselator is excited by a Gaussian colored noise, two subsystems of the coupled Brusselator model show CR phenomena with nearly the same trend but in different degrees, and indirect effects of noise on the first Brusselator make the CR of the second more intense. The influences of environmental factors on a chemical reaction have the characteristics of random and uncertainties, thus these influences can be described by different noises. For the purpose of studying and learning the effects of noises on a system with a greater depth, next two uncorrelated additive Gaussian white noises are added to two subsystems of Eq. (3), respectively. The resulting stochastic model is $$\begin{align} \dot {x}=\,&a-bx+x^2y-x+g_1 (t), \\ \dot {y}=\,&bx-x^2y, \\ \dot {e}=\,&x-eu, \\ \dot {u}=\,&bx-eu+u^2v-u+g_2 (t), \\ \dot {v}=\,&eu-u^2v,~~ \tag {8} \end{align} $$ where $g_1 (t)$ and $g_2 (t)$ are uncorrelated Gaussian white noises with $$\begin{align} \langle {g_1 (t)} \rangle =\,&\langle {g_2 (t)} \rangle =0, \\ \langle {g_1 (t)g_1 ({t'})} \rangle =\,&2D_1 \delta (t-t'), \\ \langle {g_2 (t)g_2 ({t'})} \rangle =\,&2D_2 \delta (t-t'). \end{align} $$ Since the first Brusselator is completely oblivious to the second, in the following we only investigate CR of the second Brusselator in the vicinity of the Hopf bifurcation when two Brusselators are subjected to two independent Gaussian white noises, respectively. Similarly, SNRs of the second Brusselator can also be computed by the second-order stochastic Runge–Kutta algorithm based on Eq. (6). Observing the changing features of SNRs, we can identify CR phenomena and can analyze the combined effect of two independent Gaussian white noises on the CR of the second Brusselator. In the simulation of the SNRs of the second Brusselator we find that the change of SNRs as a function of the noise intensity $D_2$ is not obvious with different $D_1$. This can be interpreted through the above results that indirect effects of additive Gaussian colored noise on the first Brusselator make CR of the second more intense. Thus we only consider the effects of different noise intensities $D_2$ on SNRs as a function of the noise intensity $D_1$. Figure 3 clearly gives the SNRs as a function of $D_1$ under different noise intensities of $g_2(t)$. Simulation results show that when the value of $D_2$ is determined, the SNRs of $u$ and $v$ increase at first and then decrease, which explains that $g_1 (t)$ in the first Brusselator is advantageous to the occurrence of CR of the second. When the value of $D_2$ is relatively small, such as $D_2 =0.0001$ and 0.001, SNRs of $u$ and $v$ are nearly unchanged, indicating that the noise in the first Brusselator controls the CR of the coupled model in this case. If $D_2 =0.01$, SNRs of $u$ and $v$ increase slowly with the slight reduction of maximum value, which means that $g_2(t)$ plays a role in the second Brusselator. With the increase of the noise intensity, $D_2 =0.1$, the SNR of variable $u$ is largely decreased with an unobvious maximum value, which means that the CR of variable $u$ is inhibited by the excitation in itself. Meanwhile, the SNR of variable $v$ is largely increased also with an unobvious maximum value, which means that $g_2 (t)$ plays a leading role to restrain the CR of variable $v$.
cpl-34-7-070201-fig3.png
Fig. 3. Simulation results of SNRs as a function of the noise intensity $D_1$ for different noise intensities $D_2 =0.0001$, 0.001, 0.01 and 0.1 about variables $u$ and $v$ in the second Brusselator.
By virtue of the above analysis, we obtain that the two uncorrelated noises in two subsystems of the coupled Brusselator model have the opposite effects on CR of the second Brusselator. In summary, numerical simulations have been carried out to investigate the connection of CR of two subsystems when the first Brusselator is excited by an additive Gaussian colored noise. The cooperative effects of two independent Gaussian white noises on the CR of the second Brusselator in the vicinity of Hopf bifurcation are also researched. It is found that two subsystems of the coupled model exhibit CR phenomena with the same trend in different degrees, and CR is more obvious with the increase of the Gaussian colored noise intensity. Additionally, we observe that the effects of two uncorrelated additive Gaussian white noises on the CR of the second Brusselator are opposite, i.e., the perturbation in the first Brusselator is advantageous to the occurrence of CR of the second, but the Gaussian white noise in the second is disadvantageous to the occurrence of CR of itself. Our results provide a good foundation to study the CR phenomena of coupled systems with different types of excitations, such as multiplicative Gaussian noise,[21,22] non-Gaussian Lévy noise.[23-25] We expect that our results might be general phenomena in one-way coupled systems, and they would have extensive applicability and practical significance in the studies of many other coupled systems. References Active flutter suppression control law design method based on balanced proper orthogonal decomposition reduced order modelStochastic resonance in a genetic toggle model with harmonic excitation and L?vy noiseStochastic resonance in periodic potentials driven by colored noiseStochastic resonance without external periodic forceDemonstration of the Casimir Force in the 0.6 to 6 ? m RangeIntrinsic noise induced resonance in presence of sub-threshold signal in BrusselatorCoherence Resonance in a Single-Walled Carbon Nanotube Ion ChannelCoherence Resonance in Coupled Chaotic OscillatorsCoherence resonance induced by the deviation of non-Gaussian noise in coupled Hodgkin?Huxley neuronsCoherence resonance and stochastic resonance in directionally coupled ringsCoherence resonance induced by colored noise near Hopf bifurcationThe cooperation effect of noise and an external signal on implicit and explicit coherence resonances in the brusselator systemEcho waves and coexisting phenomena in coupled brusselatorsCooperative effects of parameter heterogeneity and coupling on coherence resonance in unidirectional coupled brusselator systemBIFURCATION AND STABILITY ANALYSES FOR A COUPLED BRUSSELATOR MODELSpectrum of the vortex bound states of the Dirac and Schr?dinger Hamiltonian in the presence of superconducting gapsCell biology and immunology lessons taught by Legionella pneumophilaNoise-induced dynamics in a delayed bistable system with correlated noisesThe Switch in a Genetic Toggle System with L?vy NoiseL?vy-noise-induced transport in a rough triple-well potentialEffects of L?vy noise and immune delay on the extinction behavior in a tumor growth model
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