Effects of additive noise on vibrational resonance in a bistable system

Effects of additive noise on vibrational resonance in a bistable system J. Casado-Pascual* Universidad de Sevilla, Física Teórica, Apartado de Correos 1065, Sevilla 41080, Spain J. P. Baltanás† Departamento de Matemáticas y Física Aplicadas y Ciencias de la Naturaleza, Universidad Rey Juan Carlos, Tulipán s/n, Móstoles 28933, Madrid, Spain (Received 16 September 2003; published 26 April 2004; publisher error corrected 12 May 2004) We study the overdamped motion of a particle in a bistable potential subject to the action of a bichromatic force and additive noise, within the context of the vibrational resonance phenomenon. Under appropriate conditions, we obtain analytical expressions for the relevant observables which quantifies this phenomenon. The theoretical results are compared with those obtained by the numerical solution of the stochastic differential equation which describes the dynamics of the system. The limits of validity of the theoretical approach are also discussed. DOI: 10.1103/PhysRevE.69.046108 PACS number(s): 05.90.⫹m, 05.40.⫺a, 05.45.⫺a I. INTRODUCTION During the last three decades, a large amount of work has been devoted to the study of nonlinear systems subject to noise and time-dependent forces. In the course of these studies it has become clear that, in contrast to the role played by stochastic forces in linear systems, noise can drastically alter the response of nonlinear dynamical systems to the external forcing under some particular circumstances. A particularly interesting example of the effects of noise within the framework of signal processing by nonlinear systems is stochastic resonance (SR), i.e., the amplification of a weak input signal by the concerted actions of noise and the nonlinearity of the system. Although discussed initially within the context of dynamical systems with bistable potentials [1], the phenomenon of SR has also been found in other dynamical systems [2,3], including systems of biological interest (see, e.g., Ref. [4]and references therein). Many authors have studied the occurrence of SR not only in the processing of harmonic signals but also of aperiodic signals [5–9], and in the presence of both white and colored noise [10–13]. Furthermore, similar effects have been also found when a chaotic signal is used instead of noise [14,15]. The paradigmatic model in the study of SR is the overdamped motion of a particle in a symmetric double-well potential driven by a periodic signal and noise. Its dynamics is described by the stochastic differential equation (SDE) x ˙共t兲=−U⬘关x共t兲兴 +F共t兲+ ␰ 共t兲,共1兲 where U⬘共x兲is the derivative with respect to xof the symmetric double-well potential U共x兲,F共t兲represents the periodic forcing, and ␰ 共t兲is a zero-average Gaussian white noise with autocorrelation function 具 ␰ 共t兲 ␰ 共s兲典=2D ␦ 共t−s兲.SRin this model can be understood in terms of the synchronization of two time scales: the Kramers time, characterizing the noise-induced interwell transitions, and the time scale associated to the external driving. Roughly speaking, when Kramers’ time matches half the period of the external driving for a given noise strength D, the amplification of the weak signal F共t兲is optimal. Recently, an analogous phenomenon named vibrational resonance (VR)has been shown to occur when the noise is replaced by a high-frequency periodic force of varying amplitude. Originally described by Landa and McClintock [16], its study has also been addressed by other authors and from different points of view, in excitable [17], spatially extended [18], and bistable systems [19,20]. The interplay of two monochromatic driving forces has been also considered in the context of controlling stochastic resonance [21,22].In very many physical situations, noise is present and its effects cannot be avoided (for instance, thermal noise or noise arising from the environment). The interplay between noise and nonlinearity can give rise to new aspects of the VR phenomenon, absent in a purely deterministic dynamics. In Ref. [20], a brief numerical study of the effects of additive noise on VR has been presented, but a more detailed investigation of this topic is still lacking. In this paper, we undertake an analytical study of the problem that will provide explicit expressions for the relevant observables. Our analytical results are compared with numerical simulations performed on the model described in Ref. [20]. II. DESCRIPTION OF THE MODEL AND CHARACTERIZATION OF VR We consider a system described by the SDE (1), with U共x兲being the symmetric quartic potential in dimensionless form, U共x兲=x4 4−x2 2,共2兲 and F共t兲the bichromatic force *Electronic address: [email protected] †Electronic address: [email protected] PHYSICAL REVIEW E 69, 046108 (2004) 1539-3755/2004/69(4)/046108(7)/$22.50 ©2004 The American Physical Society69 046108-1 F共t兲=Acos共⍀t兲+N⍀rcos共N⍀t+ ␸ 兲.共3兲 Here, for reasons that will become clear in the following section, we have introduced in the second monochromatic force an arbitrary initial dephasing ␸ . The parameter Nis chosen to be a positive integer, so that F共t兲is periodic with the same period T=2 ␲ /⍀as the first monochromatic force. By rwe have denoted the ratio of the amplitude of the second force to its frequency. This ratio ris assumed to be of the same order as the parameters characterizing the potential U共x兲. We are interested in situations in which the parameters N⍀rand N⍀appearing in the second monochromatic force are much larger than the rest of the parameters in the problem. In this sense, we will say that N⍀rcos共N⍀t+ ␸ 兲is a strong, high-frequency monochromatic force. This situation can be formally achieved by taking the limit N→⬁, with the ratio rkept fixed. The corresponding Fokker-Planck equation (FPE)for the probability density P共x,t兲reads ⳵ ⳵ tP共x,t兲= ⳵ ⳵ x 冋 D ⳵ ⳵ x+U⬘共x兲−F共t兲 册 P共x,t兲.共4兲 The analysis of this equation is simplified by making use of two important theorems: the Htheorem, which ensures the existence of a unique long time distribution function P⬁共x,t兲 关23,24兴, and the Floquet theorem, which guarantees that P⬁共x,t兲is periodic in time with the same period T,asthe external force 关25兴. Henceforth, we will assume that the relaxation transient stage has ended and, consequently, the long time distribution P⬁共x,t兲has been reached. The first moment of the probability distribution can be used to characterize the system response to the external driving. As a consequence of the above mentioned theorems, its long time limit 具x共t兲典⬁is a periodic function of time with period Tand, therefore, it can be expanded in Fourier series as follows: 具x共t兲典⬁=兺 n=−⬁ ⬁ Xnein⍀t=X0+2兺 n=1 ⬁ 兩Xn兩cos共n⍀t− ␾ n兲,共5兲 where Xn=1 T 冕 0 Tdt具x共t兲典⬁e−in⍀t,共6兲 and the phases ␾ nhave been chosen so that cos ␾ n=Re 冉 Xn 兩Xn兩 冊 ,共7兲 sin ␾ n=−Im 冉 Xn 兩Xn兩 冊 .共8兲 The observable that has been usually considered to quantify the VR phenomenon is the parameter Q=2兩X1兩 A.共9兲 This parameter represents the ratio of the amplitude associated to the first harmonic to A, and it is directly related to the spectral amplification ␩ through the expression Q=冑 ␩ 共see Ref. 关2兴兲. Notice that the above definition of the parameter Qdiffers by a factor of 2 from the definition used in the theoretical approach in Ref. 关20兴, and by a factor A−1 from the one used in the numerical results appearing in the same reference. In the deterministic case, D=0,ithas been shown that Qgoes through a maximum as the intensity of the high-frequency force is increased. This maximum is approximately localized around the value r =冑2/3 of the ratio of the amplitude of the second force to its frequency 关19,20兴. This behavior resembles that of SR, with the noise strength Dplaying the role of the intensity of the high-frequency force. When white noise is added, the situation is different. More precisely, the two main effects of increasing the value of the noise strength Dare that the value at the maximum of the resonance curve decreases and, at the same time, its location is shifted towards lower values of the high-frequency amplitude. Even more, for large enough values of D, the VR phenomenon completely disappears [20]. These effects can be qualitatively understood as a consequence of the fact that the white noise provides an input to the system with contributions to all the frequencies. On the one hand, the fraction corresponding to the high-frequency region advances the appearance of the maximum. On the other hand, the remaining contribution masks the high-frequency force, decreasing its relative importance [20]. To shed some more light on this question, an extension of the theoretical approach put forward in Ref. [20]to the case in which white noise is present would be desirable. This is the main aim of the following section. III. THEORETICAL APPROACH In this section, we will obtain analytical expressions for the parameter Qbased on three approximations of different nature. The first two ones are simply generalizations to the noisy problem of those carried out in the theoretical approach developed in the absence of noise [20]. The third approximation is specific to the noisy problem. For clarity in the exposition, as well as in the discussion of their validity conditions, we will present them separately. A. Derivation of the Effective Dynamics As a consequence of Eqs. (3)and (4), the time derivative of the probability density P共x,t兲diverges as Nin the limit N→⬁. To avoid this divergence, it is convenient to extract the fast dependence from x共t兲, and define the new stochastic process y共t兲=x共t兲−rsin共N⍀t+ ␸ 兲.共10兲 The Fourier components of 具x共t兲典⬁and 具y共t兲典⬁are related according to J. CASADO-PASCUAL AND J. P. BALTANÁS PHYSICAL REVIEW E 69, 046108 (2004) 046108-2 Yn=Xn+ir 2共ei ␸ ␦ n,N−e−i ␸ ␦ n,−N兲,共11兲 where Ynis defined from Eq. 共6兲, by replacing 具x共t兲典⬁by 具y共t兲典⬁. Therefore, as N⫽±1, we can substitute the coefficient X1by Y1in the definition of the parameter Q关see Eq. 共9兲兴. The advantage of using the process y共t兲instead of x共t兲 becomes clear after writing the FPE for its probability density, which reads ⳵ ⳵ tP共y,t, ␸ 兲= ⳵ ⳵ y 冋 D ⳵ ⳵ y+U⬘共y,t, ␸ 兲−Acos共⍀t兲 册 P共y,t, ␸ 兲. 共12兲 Here, we have introduced the new potential U共y,t, ␸ 兲 ªU关y+rsin共N⍀t+ ␸ 兲兴, and the dependence of the functions on the phase ␸ has been written explicitly. From Eq. 共12兲and the definition of U共y,t, ␸ 兲it follows that the time derivative of P共y,t, ␸ 兲is at most of order 1 as N→⬁. Therefore, a large number of oscillations of the function rsin共N⍀t+ ␸ 兲appearing in the definition of U共y,t, ␸ 兲 takes place before a significant change in P共y,t, ␸ 兲occurs. As a consequence, it is to be expected that, for NⰇ1, P共y,t, ␸ 兲is almost independent of the phase ␸ .Ifwedefine the phase average of an arbitrary function f共y,t, ␸ 兲as f ¯ 共y,t兲=f共y,t, ␸ 兲:= 1 2 ␲ 冕 0 2 ␲ d ␸ f共y,t, ␸ 兲,共13兲 then, for NⰇ1 and any value of ␸ 苸关0,2 ␲ 兴, one can approximate P共y,t, ␸ 兲⬇P ¯ 共y,t兲. Furthermore, the decoupling approximation f共y,t, ␸ 兲P共y,t, ␸ 兲⬇f ¯ 共y,t兲P ¯ 共y,t兲also holds. Carrying out the phase average in Eq. 共12兲and using the decoupling approximation, one obtains ⳵ ⳵ tP ¯ 共y,t兲= ⳵ ⳵ y 冋 D ⳵ ⳵ y+Ueff ⬘共y兲−Acos共⍀t兲 册 P ¯ 共y,t兲, 共14兲 where we have introduced the effective potential Ueff共y兲 ªU共y,t, ␸ 兲, which is clearly time independent. This averaging procedure has been previously used in the study of thermal activation in bistable systems under the influence of a periodic force with moderate to large frequency 关26兴.Ithas also been used in the explanation of the enhancement observed in the tunnel splitting of a quantum bistable system when a very high frequency driving is applied 关27兴. It also follows from the above considerations that, for N Ⰷ1, 具y共t兲典⬁⬇具y共t兲典⬁ 共eff兲:=兰−⬁ ⬁dyyP ¯ ⬁共y,t兲, where P ¯ ⬁共y,t兲is the long time distribution corresponding to the FPE (14).In particular, within this approximation, the parameter Qis given by Q=2兩Y1 共eff兲兩 A,共15兲 where Y1 共eff兲is obtained from Eq. 共6兲with n=1, by replacing 具x共t兲典⬁by 具y共t兲典⬁ 共eff兲. The explicit calculation of Ueff共y兲for the potential in Eq. (2)leads to Ueff共y兲=y4 4−a共r兲y2 2,共16兲 where we have dropped an irrelevant constant and have introduced the quantity a共r兲=1−3r2/2. From this result, it is clear that the stability of this effective potential depends on the ratio r. More precisely, if r⬍冑2/3 the potential is bistable, whereas if r艌冑2/3 it is monostable. Thus, an increase in rleads to a decrease in the effective barrier height and, eventually, to its disappearance. In summary, we have shown that, for NⰇ1, the time evolution of the original stochastic process x共t兲can be approximately described in terms of the dynamics of a Brownian particle moving in the effective potential Ueff共y兲and in the presence of noise and the monochromatic force Acos共⍀t兲. B. Linear response theory Henceforth, following the approach in Ref. [20], we will assume that the amplitude Ais small enough, so that linear response theory (LRT)provides a good description of the effective dynamics obtained in the preceding section. An extensive study of the validity conditions of LRT can be found in Refs. [28–30]. Within LRT the parameter Qin Eq. (15)is given [2]by Q=兩 ␹ ˆeff共⍀兲兩,共17兲 where ␹ ˆeff共⍀兲= 冕 0 ⬁ dt ␹ eff共t兲e−i⍀t共18兲 is the value at ␻ =⍀of the Fourier transform of the response function, ␹ eff共t兲, corresponding to an overdamped Brownian particle moving in the effective potential Ueff共y兲. This response function obeys the well-known fluctuationdissipation theorem 关31–33兴, ␹ eff共t兲=− ␩ 共t兲 D d dtKeff共t兲,共19兲 where ␩ 共t兲is the Heaviside step function and Keff共t兲 =具y共t兲y共0兲典eq 共eff兲is the equilibrium autocorrelation function of the effective system described by the FPE 共14兲in the absence of external driving. Thus, the stochastic process y共t兲appearing in the definition of Keff共t兲is a solution of the SDE y ˙共t兲=−Ueff ⬘关y共t兲兴 + ␰ 共t兲.共20兲 Use of Eq. 共19兲in Eq. 共18兲yields EFFECTS OF ADDITIVE NOISE ON VIBRATIONAL…PHYSICAL REVIEW E 69, 046108 (2004) 046108-3 Re关 ␹ ˆeff共⍀兲兴 =−⍀ D 冋 冕 0 ⬁ dtKeff共t兲sin共⍀t兲−Keff共0兲 ⍀ 册 , 共21兲 Im关 ␹ ˆeff共⍀兲兴 =−⍀ D 冕 0 ⬁ dtKeff共t兲cos共⍀t兲.共22兲 These two equations, together with Eq. 共17兲, allow us to express Q共LRT兲in terms of the equilibrium autocorrelation function Keff共t兲. For the nonlinear potential in Eq. (16), explicit expressions for Keff共t兲are unknown, so that this correlation function must be evaluated either numerically or by resorting to suitable approximations. Before applying to our problem approximate techniques discussed in the literature, it is convenient to reduce the SDE (20)to a more standard form. In order to do so, let us rescale the coordinate and time, and define the new stochastic process y ˜ 共t ˜ 兲=兩a共r兲兩−1/2y关兩a共r兲兩−1t ˜ 兴 (for r⫽冑2/3). Then, from Eq. (20), it is easy to verify that y ˜ 共t ˜ 兲fulfills the SDE y ˜ ˙共t ˜ 兲=−U ˜ ⬘关y ˜ 共t ˜ 兲兴 + ␰ ˜ 共t ˜ 兲,共23兲 where we have introduced the rescaled potential U ˜ 共y ˜ 兲=y ˜ 4 4+ sgn关a共r兲兴y ˜ 2 2,共24兲 and the rescaled Gaussian white noise ␰ ˜ 共t ˜ 兲with zero average and autocorrelation function 具 ␰ ˜ 共t ˜ 兲 ␰ ˜ 共t ˜ ⬘兲典=2兩a共r兲兩−2D ␦ 共t ˜ −t ˜ ⬘兲. From these considerations, it is straightforward to prove that the equilibrium autocorrelation function of the original process, Keff共t,D兲, and that of the rescaled one, K ˜ 共t ˜ ,D ˜ 兲, are related by Keff共t,D兲=兩a共r兲兩K ˜ 冉 兩a共r兲兩t,D 兩a共r兲兩2 冊 .共25兲 Analogously, Eqs. 共21兲,共22兲, and 共25兲lead to the following relation between the Fourier transform of the response function of the original process, 兩 ␹ ˆeff共⍀,D兲兩, and that of the rescaled one, 兩 ␹ ˜ ˆ共⍀ ˜ ,D ˜ 兲兩, 兩 ␹ ˆeff共⍀,D兲兩 =1 兩a共r兲兩冏 ␹ ˜ ˆ 冉 ⍀ 兩a共r兲兩,D 兩a共r兲兩2 冊 冏.共26兲 C. Weak noise approximation The expressions (25)and (26)allow us to evaluate Keff共t,D兲and 兩 ␹ ˆeff共⍀,D兲兩 from the rescaled functions K ˜ 共t ˜ ,D ˜ 兲and 兩 ␹ ˜ ˆ共⍀ ˜ ,D ˜ 兲兩. The asymptotic behaviors of these rescaled functions for small values of the noise strength D ˜ have been widely studied in the literature. To summarize the results, we will consider separately two cases. First, if r⬍冑2/3, then U ˜ 共y ˜ 兲=y ˜ 4/4−y ˜ 2/2 is the archetypal symmetric quartic double-well potential expressed in dimensionless form. For this bistable potential, we can use the two-mode approximation [34]. This approximation is based on the existence of a clear-cut separation between the time scales associated to interwell and intrawell motions, and it is expected to be valid for small values of the noise strength D ˜ . Within this approximation, K ˜ 共t ˜ ,D ˜ 兲and 兩 ␹ ˜ ˆ共⍀ ˜ ,D ˜ 兲兩 are given by K ˜ 共t ˜ ,D ˜ 兲=g1共D ˜ 兲exp关−␭1共D ˜ 兲t ˜ 兴+g2共D ˜ 兲exp关− ␣ t ˜ 兴,共27兲 and 兩 ␹ ˜ ˆ共⍀ ˜ ,D ˜ 兲兩 =1 D ˜ 再 2 ␣ g1共D ˜ 兲g2共D ˜ 兲␭1共D ˜ 兲关 ␣ ␭1共D ˜ 兲+⍀ ˜ 2兴 关␭1 2共D ˜ 兲+⍀ ˜ 2兴共 ␣ 2+⍀ ˜ 2兲 +g1 2共D ˜ 兲␭1 2共D ˜ 兲 ␭1 2共D ˜ 兲+⍀ ˜ 2+g2 2共D ˜ 兲 ␣ 2 ␣ 2+⍀ ˜ 2 冎 1/2 .共28兲 In the above expressions, the parameter ␣ =U ˜ ⬙共±1兲=2 is the curvature at the minima of the potential U ˜ 共y ˜ 兲, and ␭1共D ˜ 兲is the smallest nonvanishing eigenvalue of the Fokker-Planck operator corresponding to the SDE 共23兲. In the steepestdescent approximation, this eigenvalue is given to leading order in D ˜ by ␭1共D ˜ 兲⬇冑2 ␲ exp 冉 −1 4D ˜ 冊 .共29兲 The weights g1共D ˜ 兲and g2共D ˜ 兲read g2共D ˜ 兲=关␭1共D ˜ 兲+1兴m2共D ˜ 兲−m4共D ˜ 兲 ␭1共D ˜ 兲− ␣ ,共30兲 g1共D ˜ 兲=m2共D ˜ 兲−g2共D ˜ 兲,共31兲 where m2n共D ˜ 兲= 冕 −⬁ ⬁ dy ˜ y ˜ 2ne−U ˜ 共y ˜ 兲/D ˜ 冕 −⬁ ⬁ dy ˜ e−U ˜ 共y ˜ 兲/D ˜ =共2D ˜ 兲n/2⌫共n+ 1/2兲D−n−1/2关−共2D ˜ 兲−1/2兴 ⌫共1/2兲D−1/2关−共2D ˜ 兲−1/2兴共32兲 are the nonvanishing moments of the equilibrium distribution. Using the asymptotic expansion of the parabolic cylinder functions for small values of D ˜ , these weights can be approximated by g1共D ˜ 兲⬇1−共 ␣ +1兲D ˜ / ␣ and g2共D ˜ 兲⬇D ˜ / ␣ to first order in D ˜ . Second, if r⬎冑2/3, then U ˜ 共y ˜ 兲=y ˜ 4/4+y ˜ 2/2 is a monostable potential. In this case, for small values of D ˜ , the autocorrelation function K ˜ 共t ˜ ,D ˜ 兲is given [35]by J. CASADO-PASCUAL AND J. P. BALTANÁS PHYSICAL REVIEW E 69, 046108 (2004) 046108-4 K ˜ 共t ˜ ,D ˜ 兲=D ˜ exp关−兩⌳1共D ˜ 兲兩t ˜ 兴,共33兲 with ⌳1共D ˜ 兲=−2+共1−6D ˜ 兲1/2.共34兲 Consequently, the Fourier transform of the response function reads 兩 ␹ ˜ ˆ共⍀ ˜ ,D ˜ 兲兩 =兩⌳1共D ˜ 兲兩 关⌳1 2共D ˜ 兲+⍀ ˜ 2兴1/2 .共35兲 To evaluate Keff共t,D兲and 兩 ␹ ˆeff共⍀,D兲兩 one simply introduces either the expressions 共27兲and 共28兲or expressions 共33兲and 共35兲into Eqs. 共25兲and 共26兲. Finally, within this approximation, the parameter Qwill be evaluated by replacing in Eq. 共17兲the result obtained for 兩 ␹ ˆeff共⍀,D兲兩. It is important to emphasize that the approximate expressions 共27兲,共28兲,共33兲, and 共35兲have been obtained under the assumption that D ˜ is a small parameter. Therefore, the resulting expressions for Keff共t,D兲,兩 ␹ ˆeff共⍀,D兲兩, and Qare expected to be valid only for small values of D兩a共r兲兩−2. Taking into account that, for D⫽0, the parameter D兩a共r兲兩−2 diverges at r=冑2/3, then it is clear that, even for DⰆ1, there exists a region around r =冑2/3 in which the resulting expressions for Keff共t,D兲, 兩 ␹ ˆeff共⍀,D兲兩, and Qare not applicable. In particular, the expression for Qis expected to provide a successful explanation of the VR phenomenon whenever its maximum takes place at a value of routside the above mentioned region. From the above considerations, it is also possible to obtain an expression for the location of the maximum characterizing the VR phenomenon in the presence of white noise. According to the numerical results put forward in Ref. [20], this maximum is located at a value rMsuch that rM⬍冑2/3. Then, to evaluate the parameter Qin the neighborhood of rM, one has to consider Eq. (28)and follow the procedure described in the above paragraph. As the expression thus obtained is rather cumbersome, it is convenient to introduce a further approximation. More precisely, we will assume that, around rM, the contribution due to the intrawell motion is negligible, so that just the second term in Eq. (28)with g1共D ˜ 兲=1 is present. Therefore, the parameter Qas a function of robtained from Eqs. (17)and (26)is given by Q共r兲=a共r兲 D ␭1关D/a2共r兲兴 兵␭1 2关D/a2共r兲兴 +关⍀/a共r兲兴2其1/2 共36兲 for r⬍冑2/3. The location of the maximum, rM, is readily determined from the equation Q⬘共rM兲=0, where Q⬘共r兲represents the derivative of Q共r兲with respect to r. Consequently, from Eq. 共36兲it follows that rMobeys the transcendental equation ␭1 2 冋 D a2共rM兲 册 =2 冋 ⍀ a共rM兲 册 2 冋 a2共rM兲 4D−1 册 .共37兲 After making the replacements D/a2共rM兲→Dand ⍀/a共rM兲 →⍀in the above equation, the resulting expression resembles, except for a factor of 2, the transcendental equation obtained within the two-state model to determine the noise intensity that maximizes the spectral amplification 共see, e.g., Ref. 关2兴and references therein兲. The difference in the factor of 2 is due to the fact that, in that case, only the noise intensity is varied, whereas in our case, by varying r,weare simultaneously modifying the frequency and the noise intensity. The transcendental equation (37)provides an explanation for the numerically observed behavior of the location of the maximum as a function of D(see Sec. II). For a fixed value of D, the function ␭1 2关D/a2共r兲兴 is a monotonically increasing function of rin the interval 关0,冑2/3兲. Also, for fixed values of Dand ⍀,2关⍀/a共r兲兴2关a2共r兲/共4D兲−1兴is a monotonically decreasing function of rin the same interval. Thus, the existence of a solution of Eq. (37)in the interval 共0,冑2/3兲 requires the inequality ␭1 2关D/a2共0兲兴=␭1 2共D兲⬍2关⍀/ a共0兲兴2关a2共0兲/共4D兲−1兴=2⍀2关1/共4D兲−1兴. Therefore, the same inequality is required for the existence of VR. If we now keep rand ⍀fixed and look at both sides of Eq. (37)as a function of D, we notice that 2关⍀/a共r兲兴2关a2共r兲/共4D兲−1兴is a monotonically decreasing function of D, whereas ␭1 2关D/a2共r兲兴 is a monotonically increasing function of D.We then conclude that the abscissa at the intersection point between these two functions, rM, is shifted towards the left as Dincreases. This is exactly the behavior observed numerically in Ref. [20]. Obviously, VR disappears for values of D greater than the critical value Dcfor which rM=0. This critical value fulfills the transcendental equation ␭1 2共Dc兲=2⍀2 冋 1 4Dc−1 册 .共38兲 IV. COMPARISON WITH THE NUMERICAL SOLUTION In this section, we compare the results obtained for the parameter Qby means of the numerical solution of the SDE (1), with the analytical expressions obtained in the preceding section. In order to evaluate numerically the parameter Q,a large enough number of stochastic trajectories, x共k兲共t兲, has been generated by integrating the SDE (1)for every realization of the white noise ␰ 共t兲, starting from a given initial condition x0. The numerical solution of the SDE (1)has been carried out by using the algorithm put forward in Ref. [36] (see also the Appendix in Ref. [30]). After allowing for a relaxation transient stage, we evaluate the long time average by 具x共t兲典⬁=1 M兺 k=1 M x共k兲共t兲,共39兲 where Mis the number of stochastic trajectories considered. Then, the parameter Qis obtained from Eqs. 共9兲and 共6兲, with n=1, by numerical quadrature. In Fig. 1, we depict the dependence of Qvs. r. The following set of parameter values has been considered: N=50, ␸ =0, ⍀=3.4⫻10−2,A=1.1⫻10−2,D=1.34⫻10−2 in panel EFFECTS OF ADDITIVE NOISE ON VIBRATIONAL…PHYSICAL REVIEW E 69, 046108 (2004) 046108-5 (a),D=6.72⫻10−2 in panel (b),D=1.34⫻10−1 in panel (c), and D=4.03⫻10−1 in panel (d). They correspond to four representative cases appearing in Fig. 6 in Ref. [20]. Note that the parameter values appearing in that reference must be rescaled in order to reduce the bistable potential used there to the standard dimesionless form in Eq. (2). With circles we have plotted the results obtained from the numerical solution of the SDE (1)together with Eqs. (2)and (3). With solid lines we have depicted the analytical values of Qprovided by Eqs. (17)and (26), and either Eqs. (28)–(32), for r⬍冑2/3, or Eqs. (35)and (34), for r⬎冑2/3. The location of the critical value r=冑2/3 is indicated by the vertical dotted lines. In addition, in panels (a)and (b), we have depicted with vertical dashed lines the locations of the maxima predicted by Eq. (37). The critical value of the noise strength at which the VR phenomenon disappears, obtained from Eq. (38),isDc=1.154⫻10−1. The noise strength values in panels (c)and (d)are greater than Dc, so it is to be expected that the VR phenomenon is not present in these cases, as it is confirmed numerically. A glance at Fig. 1 reveals that the analytical expressions provide a good description of the main features of the VR phenomenon in the presence of white noise, at least if the ratio ris far enough away from the critical value r=冑2/3.In particular, the analytical results describe correctly the shift of the location of the maximum towards lower values of r,as well as the decrease of the value of Qat the maximum, as the noise strength increases. It is important to emphasize that, as we mentioned in Sec. III C, the analytical results are not applicable in a neighborhood of the critical value r=冑2/3,as a consequence of the divergence of the effective noise strength D/关a共r兲兴2. The appearance of the vertical asymptote at this critical value makes this failure evident. Note that, in panel (a), the agreement between the numerical and analytical results around the maximum gets worse. This is due to the proximity of the location of this maximum to the critical value r=冑2/3. V. CONCLUSIONS We have studied the motion of a particle in a bistable potential in the presence of white noise and an external bichromatic force, within the context of the VR phenomenon. Analytical expressions for the parameter Qwhich quantifies this phenomenon have been obtained based on three simplifying approximations of different nature. First, the exact dynamics of the system has been reduced to that of a Brownian particle moving in an effective potential and under the influence of white noise and one of the monochromatic forces. In order to make this approximation, we have assumed that the other monochromatic force is a strong, high-frequency field. Second, we have applied LRT to express the parameter Qin terms of the equilibrium autocorrelation function associated to the effective dynamics in the absence of the external driving. Finally, we have obtained analytical expressions for Qwithin the weak noise approximation. To do so, it has been necessary to consider two disjoint regions of values of the ratio rseparated by the critical value r=冑2/3. We have also obtained a transcendental equation for the location of the maximum of Q共r兲which characterizes the VR phenomenon, as well as for the condition under which this phenomenon disappears. It is important to emphasize that our analytical approach allows us to predict the three main effects of additive noise on the VR phenomenon independent of numerical simulations. In particular, we have the following Equation (37)predicts that the location of the maximum characterizing the VR phenomenon shifts to the left as the noise strength increases. The analytical expression for Qpredicts the lowering of the maximum value of the resonance curve as the noise strength increases. Finally, our theoretical results also predict that the VR phenomenon disappears for large enough values of the noise strength. Furthermore, we also give an expression [cf. Eq. (38)] for the highest value of Dbeyond which the VR phenomenon disappears. Comparison with the numerical solution of the original SDE shows that these analytical expressions provide a good quantitative description of the VR phenomenon, at least if it takes place outside a critical region around r=冑2/3. For sufficiently large Dvalues, this limitation is not important, as the VR maximum is well outside this critical rregion. In the very small noise limit, the peak in Qappears around r =冑2/3 and our approach becomes problematic. Nevertheless, in this limit the VR phenomenon is well described by the previously known deterministic treatment. With this work, we have attempted to provide a theoretical background to the numerical studies on the influence of additive noise on VR carried out in Ref. [20]. FIG. 1. Dependence of the parameter Qvs rcorresponding to the noise strength values D=1.34⫻10−2 [panel (a)],D=6.72 ⫻10−2 [panel (b)],D=1.34⫻10−1 [panel (c)], and D=4.03⫻10−1 [panel (d)]. The rest of the parameter values are N=50, ␸ =0, ⍀ =3.4⫻10−2, and A=1.1⫻10−2. With circles we have plotted the results obtained from the numerical solution of the SDE (1)together with Eqs. (2)and (3). With solid lines we have depicted the analytical values of Qprovided by Eqs. (17)and (26), and either Eqs. (28)–(32), for r⬍冑2/3, or Eqs. (35)and (34), for r⬎冑2/3. The dotted lines indicate the location of the critical value r=冑2/3, and the dashed lines the location of the maximum predicted by Eq. (37). J. CASADO-PASCUAL AND J. P. BALTANÁS PHYSICAL REVIEW E 69, 046108 (2004) 046108-6 ACKNOWLEDGMENTS We acknowledge the support of the Dirección General de Enseñanza Superior of Spain by Projects Nos. BFM2002- 03822 (J. C.-P.)and BFM2000-0967 (J. P. B.). We also want to thank Professor M. Morillo and Professor P. Hänggi for their critical reading of the manuscript and fruitful discussions. [1]R. Benzi, A. Sutera, and A. Vulpiani, J. Phys. A 14, L453 (1981). [2]L. Gammaitoni, P. Hänggi, P. Jung, and F. Marchesoni, Rev. Mod. Phys. 70, 223 (1998). [3]V. Anishchenko, A. Neiman, F. Moss, and L. Schimansky- Geier, Phys. Usp. 42,7(1999). [4]P. Hänggi, ChemPhysChem 3, 285 (2002). [5]J. J. Collins, C. C. Chow, and T. T. Imhoff, Phys. Rev. E 52, R3321 (1995). [6]J. J. Collins, C. C. Chow, A. C. Capela, and T. T. Imhoff, Phys. Rev. E 54, 5575 (1996). [7]C. C. Chow, T. T. Imhoff, and J. J. Collins, Chaos 8, 616 (1998). [8]S. Barbay, G. Giacomelli, and F. Marin, Phys. Rev. Lett. 85, 4652 (2000). [9]I. Goychuk and P. Hänggi, Phys. Rev. E 61, 4272 (2000). [10]P. Hänggi, P. Jung, C. Zerbe, and F. Moss, J. Stat. Phys. 70,25 (1993). [11]A. Neiman and L. Schimansky-Geier, Phys. Rev. Lett. 72, 2988 (1994). [12]A. Neiman and W. Sung, Phys. Lett. A 223, 341 (1996). [13]D. Nozaki, D. J. Mar, P. Grigg, and J. J. Collins, Phys. Rev. Lett. 82, 2402 (1999). [14]V. Anishchenko, A. Neiman, and M. Safonova, J. Stat. Phys. 70, 183 (1993). [15]S. Sinha, Physica A 270, 204 (1999). [16]P. S. Landa and P. V. E. McClintock, J. Phys. A 33, L433 (2000). [17]E. Ullner, A. Zaikin, R. Bascones, J. García-Ojalvo, and J. Kurths, Phys. Lett. A 312, 348 (2003). [18]A. A. Zaikin, L. López, J. P. Baltanás, J. Kurths, and M. A. F. Sanjuán, Phys. Rev. E 66, 011106 (2002). [19]I. I. Blekhman and P. S. Landa, Int. J. Non-Linear Mech. 39, 421 (2004). [20]J. P. Baltanás, L. López, I. I. Blechman, P. S. Landa, A. Zaikin, J. Kurths, and M. A. F. Sanjuán, Phys. Rev. E 67, 066119 (2003). [21]L. Gammaitoni et al.,Phys. Rev. Lett. 82, 4574 (1999). [22]M. Löcher et al.,Phys. Rev. E 62, 317 (2000). [23]J. L. Lebowitz and P. G. Bergmann, Ann. Phys. (N.Y.)1,1 (1957). [24]H. Risken, The Fokker-Planck Equation (Springer-Verlag, Berlin, 1984). [25]P. Jung and P. Hänggi, Phys. Rev. A 44, 8032 (1991). [26]P. Jung, Z. Phys. B: Condens. Matter 76, 521 (1989). [27]F. Grossmann, P. Jung, T. Dittrich, and P. Hänggi, Z. Phys. B: Condens. Matter 84, 315 (1991). [28]J. Casado-Pascual, J. Gómez-Ordóñez, M. Morillo, and P. Hänggi, Europhys. Lett. 58, 342 (2002). [29]J. Casado-Pascual, J. Gómez-Ordóñez, M. Morillo, and P. Hänggi, Fluct. Noise Lett. 2, L127 (2002). [30]J. Casado-Pascual, C. Denk, J. Gómez-Ordóñez, M. Morillo, and P. Hänggi, Phys. Rev. E 67, 036109 (2003). [31]R. Kubo, J. Phys. Soc. Jpn. 12, 570 (1957). [32]R. Kubo, Rep. Prog. Phys. 29, 255 (1966). [33]P. Hänggi and H. Thomas, Phys. Rep. 88, 207 (1982). [34]P. Jung and P. Hänggi, Z. Phys. B: Condens. Matter 90, 255 (1993). [35]C. W. Gardiner, Handbook of Stochastic Methods, 2nd ed. (Springer-Verlag, Berlin, 1985). [36]H. S. Greenside and E. Helfand, Bell Syst. Tech. J. 60, 1927 (1981). EFFECTS OF ADDITIVE NOISE ON VIBRATIONAL…PHYSICAL REVIEW E 69, 046108 (2004) 046108-7