- Email: [email protected]
Stochastic resonance in a mono-stable system subject to frequency mixing periodic force and noise
Chaos, Solitons and Fractals 40 (2009) 401–407 www.elsevier.com/locate/chaos
Stochastic resonance in a mono-stable system subject to frequency mixing periodic force and noise Bingchang Zhou *, Wei Xu Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, China Accepted 25 July 2007
Abstract The phenomenon of stochastic resonance (SR) in a biased mono-stable system driven by multiplicative and additive white noise and two periodic fields is investigated. Analytic expressions of the signal-to-noise ratio (SNR) for fundamental harmonics and higher harmonics are derived by using the two-state theory. It is shown that the SNR is a non-monotonic function of the intensities of the multiplicative and additive noises, as well as the bias of the mono-stable system and SR appears at both fundamental harmonics and higher harmonics. Moreover, the higher the order of mixed harmonics is, the smaller the SNR values are, that is, the suppression exists for higher harmonics. Ó 2007 Elsevier Ltd. All rights reserved.
1. Introduction Since Benzi and co-workers [1] discovered a phenomenon that they termed stochastic resonance (SR), this phenomenon has been extensively investigated from both the theoretical and experimental points of view. Now the SR paradigm has drawn considerable attention in such diverse fields as electronic and magnetic systems, climatology, chemistry, laser physiology, neuroscience, biophysics, solid-state physics and even sociology [2–10]. Recently, there have appeared some extensions of SR, such as ASR [11], DSR [12], non-Markovian SR [13], QSR [14] and CSR [15,16]. In order to describe SR, McNamara and co-workers [4,5] introduced a two-state model and obtained the signal-to-noise ratio (SNR) in adiabatic limit to characterize the SR. Gammaitoni and co-workers [17] have suggested the escape time distribution to describe SR. Dykman et al. [18] and Hu et al. [2] introduced the linear-response theory and perturbation theory to investigate the SR. The above-mentioned papers only involved the system driven by one periodic force. Later, the phenomenon of SR in a bistable system with several periodic forces attracted great attention. Gitterman [19] studied the theoretical results of the bistable oscillator driven by two periodic fields. Grigoreko et al. [20,21] reported the response of a bistable system with a frequency mixing force and investigated the SR with the experiment of an iron garnet thin film. Most of the studies on SR have been on the bistable potential with different noises. However, there are a lot of mono-stable systems [22–24] in many real physical systems. Dykman et al. [22] and Evstigneev et al. [23] studied the SR based on linear-response theory in a mono-stable over-damped system. Guo and co-workers [24] investigated the *
Corresponding author. E-mail address: [email protected] (B. Zhou).
0960-0779/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.07.094
402
B. Zhou, W. Xu / Chaos, Solitons and Fractals 40 (2009) 401–407
SR by using the adiabatic approximation theory. In this paper, we study the SR in mono-stable system with frequency mixing periodic force and find the SR for higher harmonics by applying the two-state theory.
2. SNR for the mono-stable system Consider a mono-stable over-damped system driven by multiplicative noise n(t), additive noise g(t) and mixed periodic signal F(t). The model is generally described by the following Langevin equation dx ¼ ax3 þ r þ xnðtÞ þ gðtÞ þ F ðtÞ; dt
ð1Þ
where a > 0, r, which denotes the bias of the mono-stable system, is a constant. The noise terms n(t) and g(t) are the uncorrelated noise which are characterized by their mean and variance hnðtÞi ¼ hgðtÞi ¼ 0; hnðtÞnðsÞi ¼ 2Ddðt sÞ; hgðtÞgðsÞi ¼ 2Qdðt sÞ;
ð2aÞ ð2bÞ ð2cÞ
where D and Q are the multiplicative and additive noise intensities, respectively. The mixed periodic signal is given by F ðtÞ ¼ f1 cosðX1 tÞ þ f2 cosðX2 tÞ:
ð3Þ
According to Eqs. (1)–(3), the corresponding FPK is oP ðx; tÞ o o2 ¼ Aðx; tÞP ðx; tÞ þ 2 BðxÞP ðx; tÞ; ox ot ox
ð4Þ
Aðx; tÞ ¼ Dx ax3 þ r þ F ðtÞ;
ð5Þ
where BðxÞ ¼ Dx2 þ Q:
Now according to Eqs. (4) and (5), the stationary distribution function can be written as V ðxÞ 1=2 exp ; P st ðxÞ ¼ N ½BðxÞ D where N is the normalization constant, V(x) is the rectified potential function and has the form Z x D ½U 0 ðxÞ þ r þ F ðtÞ dx V ðxÞ ¼ 1 BðxÞ
ð6Þ
ð7Þ
with U 0 ðxÞ ¼
dU ¼ ax3 Dx: dx
ð8Þ
From Eqs. (7) and (8), we can see that thepmono-stable system (1) can be regarded as an equivalent bistable system for ffiffiffiffiffiffiffiffiffi the case of D 5 0. Assuming that x ¼ D=a and x0 = 0 are the stable and unstable points in the equivalent bistable system. The transition rates out of x± can be obtained based on the adiabatic condition jU 00 ðx0 ÞU 00 ðx Þj1=2 V ðx Þ V ðx0 Þ exp W ¼ ð9Þ ¼ W 0 expðqF ðtÞÞ; 2p D where W±0 denotes the characteristic switching frequency of the equivalent bistable system when it is only driven by additive and multiplicative noise, which is given by D DU W 0 ¼ pffiffiffi exp rq ; ð10aÞ 2D 2p 1 D q ¼ pffiffiffiffiffiffiffi arctan pffiffiffiffiffiffi ; ð10bÞ DQ aQ 2 a D þ aQ DU ¼ D 1 þ ln 1 : ð10cÞ Q aQ
B. Zhou, W. Xu / Chaos, Solitons and Fractals 40 (2009) 401–407
403
From Ref. [5], the equivalent bistable case can be regarded as a two-state system, with n± denoting the occupation probabilities of the stable states x±. The occupation probabilities n± satisfies the following master equation dnþ =dt nþ W þ W : ð11Þ ¼ dn =dt W þ W n The distribution density P(x,t) of the two-state system is written as P ðx; tÞ ¼ nþ ðtÞdðx xþ Þ þ n ðtÞdðx x Þ:
ð12Þ
Under the adiabatic limit condition, we can expand Eq. (9) in series with the small parameter = qF(t) and obtain n ¼ n0 þ n1 þ n2 þ ;
ð13aÞ
W ¼ W 0 þ W 1 þ W 2 þ :
ð13bÞ
We can obtain from Eqs. (12) and (13) dnþ0 ¼ W 0 ðW þ0 þ W 0 Þnþ0 ; dt dnþ1 W þ0 W 1 W þ1 W 0 ðW þ0 þ W 0 Þnþ1 ; ¼ dt W þ0 þ W 0 dnþ2 ¼ ðW 1 þ W þ1 Þnþ1 ðW þ0 þ W 0 Þnþ2 ; dt
ð14aÞ ð14bÞ ð14cÞ
with n+0 denotes the occupation probability of stable state x+ when the signal F(t) = 0. By solving Eqs. (14a)–(14c), we can obtain the expression of n±0, n±1 and n±2. The averaged autocorrelation function is given by Z 2p=X X hhxðt þ sÞxðtÞiit ¼ limt0 !1 x2þ nþ ðt þ sjxþ ; tÞnþ ðtjx0 ; t0 Þ þ xþ x nþ ðt þ sjx ; tÞn ðtjx0 ; t0 Þ 2p 0 þ xþ x n ðt þ sjxþ ; tÞnþ ðtjx0 ; t0 Þ þ x2 n ðt þ sjx ; tÞn ðtjx0 ; t0 Þ dt: ð15Þ By applying the Fourier transform of the autocorrelation function, we can get the expression of the power spectrum S(X) in the case of f1 5 0 and f2 = 0 SðXÞ ¼ S 1 ðXÞdðx XÞ þ S 2 ðx; XÞ;
ð16Þ
Fig. 1. SNR1 as a function of multiplicative noise intensity D with different values of r (a = 1, f1 = 0.3, f2 = 0.1, X1 = 0.3, X2 = 0.1, Q = 0.2).
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B. Zhou, W. Xu / Chaos, Solitons and Fractals 40 (2009) 401–407
where S 1 ðXÞ ¼
4pDb2 ; aðc2 þ X2 Þ
S 2 ðx; XÞ ¼
4Dc þ x2 Þ
aðc2
ð17aÞ 1 2b2 2 2 cosh ðrqÞ c þ X2
! ð17bÞ
with
Fig. 2. SNR2 as a function of multiplicative noise intensity D with different values of r (a = 1, f1 = 0.3, f2 = 0.1, X1 = 0.3, X2 = 0.1, Q = 0.2).
Fig. 3. SNR1 as a function of additive noise intensity Q with different values of r (a = 1, f1 = 0.3, f2 = 0.1, X1 = 0.3, X2 = 0.1, D = 0.4).
B. Zhou, W. Xu / Chaos, Solitons and Fractals 40 (2009) 401–407
Df1 q DU b ¼ pffiffiffi exp ; 2D 2p coshðrqÞ pffiffiffi DU 2D coshðrqÞ: c¼ exp 2D p
405
ð18aÞ ð18bÞ
Here S1(X) is the power density associated with the output signal. S2(x,X) is the power spectrum connected with the noise background in the adiabatic limit case. Then the analytic expression of signal-to-noise ratio (SNR) for the fundamental harmonic, which is defined as the ratio of the power density of signal to the noise background, is pDf 2 q2 DU SNR ¼ pffiffiffi 1 exp ð19Þ ½1 k1 2D 2 2 coshðrqÞ with k¼
2D2 f12 q2 exp ðDU=DÞ : pðX2 þ 2D2 expðDU=DÞcosh2 ðrqÞ=p2 Þ
ð20Þ
Similar to above analysis, in the case of weak modulation force X1, X2 W+0 + W0 the expressions of SNR for higher harmonics are obtained pf 2 f 2 q6 D3 DU exp ð21Þ SNR1 ðX ¼ X1 þ X2 Þ ¼ pffiffiffi1 2 ½1 k1 tanh2 ðrqÞ; 2D 2 coshðrqÞ pffiffiffi 2 4 8 3 DU 2pf1 f2 q D ð22Þ exp SNR2 ðX ¼ X1 þ 2X2 Þ ¼ ½1 k1 tanh2 ½1 þ 3tanh2 ðrqÞ: coshðrqÞ 2D
3. Discussion and conclusion Up to now, the expressions of the SNR for fundamental harmonic and higher harmonics are derived by using the two-state theory. The SNR for the fundamental harmonic has been discussed [24] and the SR has been detected. Now we only discuss the influence of multiplicative noise intensity D, additive noise intensity Q, the bias of the mono-stable system r on the SNR for higher harmonics and draw some conclusions. In Figs. 1 and 2, we plot the curves of SNR as a function of multiplicative noise intensity D with different values of r for higher harmonics. There is a single peak in the curve of SNR with increasing D and SR phenomenon appears. The
Fig. 4. SNR2 as a function of additive noise intensity Q with different values of r (a = 1, f1 = 0.3, f2 = 0.1, X1 = 0.3, X2 = 0.1, D = 0.4).
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B. Zhou, W. Xu / Chaos, Solitons and Fractals 40 (2009) 401–407
values of SNR1 in Fig. 1 are larger than those of SNR2 in Fig. 2 when the other parameters choose the same values, which is consistent with the result obtained in Ref. [25]. Therefore, the higher the order of mixed harmonics is, the smaller the SNR value is. Moreover, the position of the resonance peak moves to the left with increasing r. Figs. 3 and 4 show the SNR as a function of additive noise intensity Q with different values of r for higher harmonics. There is a pronounced single peak in the curve and SR displays for this case, which is agree with the result of Refs. [24,25]. We can see there is a strong suppression of higher harmonics at certain values of the noise level. Meanwhile, the position of the peak moves to the right with increasing r, which is opposite to the results obtained in Figs. 1 and 2. The curves of SNR versus the bias of the mono-stable system r with varied D are shown in Figs. 5 and 6 for higher harmonics. It is an interesting phenomenon there is a peak in the curve and SR appears, which is opposite to the result obtained in an asymmetric bistable system. Moreover, the result in Fig. 5 explains the phenomenon in Fig. 1.
Fig. 5. SNR1 as a function of r with different values of D (a = 1, f1 = 0.3, f2 = 0.1, X1 = 0.3, X2 = 0.1, Q = 0.2).
Fig. 6. SNR2 as a function of r with different values of D (a = 1, f1 = 0.3, f2 = 0.1, X1 = 0.3, X2 = 0.1, Q = 0.2).
B. Zhou, W. Xu / Chaos, Solitons and Fractals 40 (2009) 401–407
407
Acknowledgement This work is supported by National Natural Science Foundation of China (Grant nos. 10472091 and 10332030).
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]
Benzi R, Sutera A, Vulpiani A. J Phys A: Math Gen 1981;14:L453. Hu G, Nicolis G, Nicolis C. Phys Rev A 1990;42:2030. Hu G, Haken H, Xie FG. Phys Rev Lett 1996;77:1925. McNamara B, Wiesenfeld K, Roy R. Phys Rev Lett 1988;60:2626. McNamara B, Wiesenfeld K. Phys Rev A 1989;39:4854. Gammaitoni L, Hanggi P, Jung P, Marchesoni F. Rev Mod Phys 1998;70:223. Oya T, Asai T, Amemiya Y. Chaos Solitons & Fractals 2007;32:855. Kang YM, Xie Y, Xu JX. Chaos Solitons & Fractals 2006;27:715. Jin YF, Xu W, Xu M. Chaos Solitons & Fractals 2005;26:1183. Babinec P. Chaos Solitons & Fractals 2002;13:1767. Collins JJ, Chow CC, Capela AC, Imhoff TT. Phys Rev E 1996;54:5575. Zaikin AA, Kurths J, Schimansky-Geier L. Phys Rev Lett 2000;85:227. Goychuk I, Hanggi P. Phys Rev Lett 2003;91:070601. Goychuk I, Hanggi P. Phys Rev E 1999;59:5137. Gammaitoni L, Locher M, Bulsara A, Hanggi P, Neff J, Wiesenfeld K, et al. Phys Rev Lett 1999;82:4574. Locher M, Inchiosa ME, Neff J, Bulsara A, Wiesenfeld K, Gammaitoni L, et al. Phys Rev E 2000;62:317. Gammaitoni L, Marchesoni F, Santucci S. Phys Rev Lett 1989;62:349. Dykman MI, Mannella R, McClintock PVE, Stocks NG. Phys Rev Lett 1990;65:2606. Gitterman M. J Phys A: Math Gen 2004;37:5729. Grigorenko AN, Nikitin PI, Roschepkin GV. J Appl Phys 1996;79:6113. Grigorenko AN, Nikitin PI, Roschepkin GV. J Exp Theor Phys 1997;85:343. Dykman MI, Luchinsky DG, Manella R, McClintock PVE, Stein ND, Stocks NG. Nuovo Cimento D 1995;17:661. Evstigneev M, Reimann P, Pankov V, Prince RH. Europhys Lett 2004;65:7. Guo F, Zhou YR, Jiang SQ, Gu TX. J Phys A: Math Gen 2006;39:13861. Jung P, Talkner P. Phys Rev E 1995;51:2640.
Stochastic resonance in a mono-stable system subject to frequency mixing periodic force and noise Bingchang Zhou *, Wei Xu Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, China Accepted 25 July 2007
Abstract The phenomenon of stochastic resonance (SR) in a biased mono-stable system driven by multiplicative and additive white noise and two periodic fields is investigated. Analytic expressions of the signal-to-noise ratio (SNR) for fundamental harmonics and higher harmonics are derived by using the two-state theory. It is shown that the SNR is a non-monotonic function of the intensities of the multiplicative and additive noises, as well as the bias of the mono-stable system and SR appears at both fundamental harmonics and higher harmonics. Moreover, the higher the order of mixed harmonics is, the smaller the SNR values are, that is, the suppression exists for higher harmonics. Ó 2007 Elsevier Ltd. All rights reserved.
1. Introduction Since Benzi and co-workers [1] discovered a phenomenon that they termed stochastic resonance (SR), this phenomenon has been extensively investigated from both the theoretical and experimental points of view. Now the SR paradigm has drawn considerable attention in such diverse fields as electronic and magnetic systems, climatology, chemistry, laser physiology, neuroscience, biophysics, solid-state physics and even sociology [2–10]. Recently, there have appeared some extensions of SR, such as ASR [11], DSR [12], non-Markovian SR [13], QSR [14] and CSR [15,16]. In order to describe SR, McNamara and co-workers [4,5] introduced a two-state model and obtained the signal-to-noise ratio (SNR) in adiabatic limit to characterize the SR. Gammaitoni and co-workers [17] have suggested the escape time distribution to describe SR. Dykman et al. [18] and Hu et al. [2] introduced the linear-response theory and perturbation theory to investigate the SR. The above-mentioned papers only involved the system driven by one periodic force. Later, the phenomenon of SR in a bistable system with several periodic forces attracted great attention. Gitterman [19] studied the theoretical results of the bistable oscillator driven by two periodic fields. Grigoreko et al. [20,21] reported the response of a bistable system with a frequency mixing force and investigated the SR with the experiment of an iron garnet thin film. Most of the studies on SR have been on the bistable potential with different noises. However, there are a lot of mono-stable systems [22–24] in many real physical systems. Dykman et al. [22] and Evstigneev et al. [23] studied the SR based on linear-response theory in a mono-stable over-damped system. Guo and co-workers [24] investigated the *
Corresponding author. E-mail address: [email protected] (B. Zhou).
0960-0779/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.07.094
402
B. Zhou, W. Xu / Chaos, Solitons and Fractals 40 (2009) 401–407
SR by using the adiabatic approximation theory. In this paper, we study the SR in mono-stable system with frequency mixing periodic force and find the SR for higher harmonics by applying the two-state theory.
2. SNR for the mono-stable system Consider a mono-stable over-damped system driven by multiplicative noise n(t), additive noise g(t) and mixed periodic signal F(t). The model is generally described by the following Langevin equation dx ¼ ax3 þ r þ xnðtÞ þ gðtÞ þ F ðtÞ; dt
ð1Þ
where a > 0, r, which denotes the bias of the mono-stable system, is a constant. The noise terms n(t) and g(t) are the uncorrelated noise which are characterized by their mean and variance hnðtÞi ¼ hgðtÞi ¼ 0; hnðtÞnðsÞi ¼ 2Ddðt sÞ; hgðtÞgðsÞi ¼ 2Qdðt sÞ;
ð2aÞ ð2bÞ ð2cÞ
where D and Q are the multiplicative and additive noise intensities, respectively. The mixed periodic signal is given by F ðtÞ ¼ f1 cosðX1 tÞ þ f2 cosðX2 tÞ:
ð3Þ
According to Eqs. (1)–(3), the corresponding FPK is oP ðx; tÞ o o2 ¼ Aðx; tÞP ðx; tÞ þ 2 BðxÞP ðx; tÞ; ox ot ox
ð4Þ
Aðx; tÞ ¼ Dx ax3 þ r þ F ðtÞ;
ð5Þ
where BðxÞ ¼ Dx2 þ Q:
Now according to Eqs. (4) and (5), the stationary distribution function can be written as V ðxÞ 1=2 exp ; P st ðxÞ ¼ N ½BðxÞ D where N is the normalization constant, V(x) is the rectified potential function and has the form Z x D ½U 0 ðxÞ þ r þ F ðtÞ dx V ðxÞ ¼ 1 BðxÞ
ð6Þ
ð7Þ
with U 0 ðxÞ ¼
dU ¼ ax3 Dx: dx
ð8Þ
From Eqs. (7) and (8), we can see that thepmono-stable system (1) can be regarded as an equivalent bistable system for ffiffiffiffiffiffiffiffiffi the case of D 5 0. Assuming that x ¼ D=a and x0 = 0 are the stable and unstable points in the equivalent bistable system. The transition rates out of x± can be obtained based on the adiabatic condition jU 00 ðx0 ÞU 00 ðx Þj1=2 V ðx Þ V ðx0 Þ exp W ¼ ð9Þ ¼ W 0 expðqF ðtÞÞ; 2p D where W±0 denotes the characteristic switching frequency of the equivalent bistable system when it is only driven by additive and multiplicative noise, which is given by D DU W 0 ¼ pffiffiffi exp rq ; ð10aÞ 2D 2p 1 D q ¼ pffiffiffiffiffiffiffi arctan pffiffiffiffiffiffi ; ð10bÞ DQ aQ 2 a D þ aQ DU ¼ D 1 þ ln 1 : ð10cÞ Q aQ
B. Zhou, W. Xu / Chaos, Solitons and Fractals 40 (2009) 401–407
403
From Ref. [5], the equivalent bistable case can be regarded as a two-state system, with n± denoting the occupation probabilities of the stable states x±. The occupation probabilities n± satisfies the following master equation dnþ =dt nþ W þ W : ð11Þ ¼ dn =dt W þ W n The distribution density P(x,t) of the two-state system is written as P ðx; tÞ ¼ nþ ðtÞdðx xþ Þ þ n ðtÞdðx x Þ:
ð12Þ
Under the adiabatic limit condition, we can expand Eq. (9) in series with the small parameter = qF(t) and obtain n ¼ n0 þ n1 þ n2 þ ;
ð13aÞ
W ¼ W 0 þ W 1 þ W 2 þ :
ð13bÞ
We can obtain from Eqs. (12) and (13) dnþ0 ¼ W 0 ðW þ0 þ W 0 Þnþ0 ; dt dnþ1 W þ0 W 1 W þ1 W 0 ðW þ0 þ W 0 Þnþ1 ; ¼ dt W þ0 þ W 0 dnþ2 ¼ ðW 1 þ W þ1 Þnþ1 ðW þ0 þ W 0 Þnþ2 ; dt
ð14aÞ ð14bÞ ð14cÞ
with n+0 denotes the occupation probability of stable state x+ when the signal F(t) = 0. By solving Eqs. (14a)–(14c), we can obtain the expression of n±0, n±1 and n±2. The averaged autocorrelation function is given by Z 2p=X X hhxðt þ sÞxðtÞiit ¼ limt0 !1 x2þ nþ ðt þ sjxþ ; tÞnþ ðtjx0 ; t0 Þ þ xþ x nþ ðt þ sjx ; tÞn ðtjx0 ; t0 Þ 2p 0 þ xþ x n ðt þ sjxþ ; tÞnþ ðtjx0 ; t0 Þ þ x2 n ðt þ sjx ; tÞn ðtjx0 ; t0 Þ dt: ð15Þ By applying the Fourier transform of the autocorrelation function, we can get the expression of the power spectrum S(X) in the case of f1 5 0 and f2 = 0 SðXÞ ¼ S 1 ðXÞdðx XÞ þ S 2 ðx; XÞ;
ð16Þ
Fig. 1. SNR1 as a function of multiplicative noise intensity D with different values of r (a = 1, f1 = 0.3, f2 = 0.1, X1 = 0.3, X2 = 0.1, Q = 0.2).
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B. Zhou, W. Xu / Chaos, Solitons and Fractals 40 (2009) 401–407
where S 1 ðXÞ ¼
4pDb2 ; aðc2 þ X2 Þ
S 2 ðx; XÞ ¼
4Dc þ x2 Þ
aðc2
ð17aÞ 1 2b2 2 2 cosh ðrqÞ c þ X2
! ð17bÞ
with
Fig. 2. SNR2 as a function of multiplicative noise intensity D with different values of r (a = 1, f1 = 0.3, f2 = 0.1, X1 = 0.3, X2 = 0.1, Q = 0.2).
Fig. 3. SNR1 as a function of additive noise intensity Q with different values of r (a = 1, f1 = 0.3, f2 = 0.1, X1 = 0.3, X2 = 0.1, D = 0.4).
B. Zhou, W. Xu / Chaos, Solitons and Fractals 40 (2009) 401–407
Df1 q DU b ¼ pffiffiffi exp ; 2D 2p coshðrqÞ pffiffiffi DU 2D coshðrqÞ: c¼ exp 2D p
405
ð18aÞ ð18bÞ
Here S1(X) is the power density associated with the output signal. S2(x,X) is the power spectrum connected with the noise background in the adiabatic limit case. Then the analytic expression of signal-to-noise ratio (SNR) for the fundamental harmonic, which is defined as the ratio of the power density of signal to the noise background, is pDf 2 q2 DU SNR ¼ pffiffiffi 1 exp ð19Þ ½1 k1 2D 2 2 coshðrqÞ with k¼
2D2 f12 q2 exp ðDU=DÞ : pðX2 þ 2D2 expðDU=DÞcosh2 ðrqÞ=p2 Þ
ð20Þ
Similar to above analysis, in the case of weak modulation force X1, X2 W+0 + W0 the expressions of SNR for higher harmonics are obtained pf 2 f 2 q6 D3 DU exp ð21Þ SNR1 ðX ¼ X1 þ X2 Þ ¼ pffiffiffi1 2 ½1 k1 tanh2 ðrqÞ; 2D 2 coshðrqÞ pffiffiffi 2 4 8 3 DU 2pf1 f2 q D ð22Þ exp SNR2 ðX ¼ X1 þ 2X2 Þ ¼ ½1 k1 tanh2 ½1 þ 3tanh2 ðrqÞ: coshðrqÞ 2D
3. Discussion and conclusion Up to now, the expressions of the SNR for fundamental harmonic and higher harmonics are derived by using the two-state theory. The SNR for the fundamental harmonic has been discussed [24] and the SR has been detected. Now we only discuss the influence of multiplicative noise intensity D, additive noise intensity Q, the bias of the mono-stable system r on the SNR for higher harmonics and draw some conclusions. In Figs. 1 and 2, we plot the curves of SNR as a function of multiplicative noise intensity D with different values of r for higher harmonics. There is a single peak in the curve of SNR with increasing D and SR phenomenon appears. The
Fig. 4. SNR2 as a function of additive noise intensity Q with different values of r (a = 1, f1 = 0.3, f2 = 0.1, X1 = 0.3, X2 = 0.1, D = 0.4).
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B. Zhou, W. Xu / Chaos, Solitons and Fractals 40 (2009) 401–407
values of SNR1 in Fig. 1 are larger than those of SNR2 in Fig. 2 when the other parameters choose the same values, which is consistent with the result obtained in Ref. [25]. Therefore, the higher the order of mixed harmonics is, the smaller the SNR value is. Moreover, the position of the resonance peak moves to the left with increasing r. Figs. 3 and 4 show the SNR as a function of additive noise intensity Q with different values of r for higher harmonics. There is a pronounced single peak in the curve and SR displays for this case, which is agree with the result of Refs. [24,25]. We can see there is a strong suppression of higher harmonics at certain values of the noise level. Meanwhile, the position of the peak moves to the right with increasing r, which is opposite to the results obtained in Figs. 1 and 2. The curves of SNR versus the bias of the mono-stable system r with varied D are shown in Figs. 5 and 6 for higher harmonics. It is an interesting phenomenon there is a peak in the curve and SR appears, which is opposite to the result obtained in an asymmetric bistable system. Moreover, the result in Fig. 5 explains the phenomenon in Fig. 1.
Fig. 5. SNR1 as a function of r with different values of D (a = 1, f1 = 0.3, f2 = 0.1, X1 = 0.3, X2 = 0.1, Q = 0.2).
Fig. 6. SNR2 as a function of r with different values of D (a = 1, f1 = 0.3, f2 = 0.1, X1 = 0.3, X2 = 0.1, Q = 0.2).
B. Zhou, W. Xu / Chaos, Solitons and Fractals 40 (2009) 401–407
407
Acknowledgement This work is supported by National Natural Science Foundation of China (Grant nos. 10472091 and 10332030).
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]
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