Theory of frequency and phase synchronization in a rocked bistable stochastic system

Theory of frequency and phase synchronization in a rocked bistable stochastic system Jesús Casado-Pascual,*José Gómez-Ordóñez, and Manuel Morillo Física Teórica, Universidad de Sevilla, Apartado de Correos 1065, Sevilla 41080, Spain Jörg Lehmann,†Igor Goychuk, and Peter Hänggi Institut für Physik, Universität Augsburg, Universitätsstraße 1, D-86135 Augsburg, Germany (Received 20 September 2004; published 5 January 2005) We investigate the role of noise in the phenomenon of stochastic synchronization of switching events in a rocked, overdamped bistable potential driven by white Gaussian noise, the archetype description of stochastic resonance. We present an approach to the stochastic counting process of noise-induced switching events: starting from the Markovian dynamics of the nonstationary, continuous particle dynamics, one finds upon contraction onto two states a non-Markovian renewal dynamics. A proper definition of an output discrete phase is given, and the time rate of change of its noise average determines the corresponding output frequency. The phenomenon of noise-assisted phase synchronization is investigated in terms of an effective, instantaneous phase diffusion. The theory is applied to rectangular-shaped rocking signals versus increasing input-noise strengths. In this case, for an appropriate choice of the parameter values, the system exhibits a noise-induced frequency locking accompanied by a very pronounced suppression of the phase diffusion of the output signal. Precise numerical simulations corroborate very favorably our analytical results. The novel theoretical findings are also compared with prior ones. DOI: 10.1103/PhysRevE.71.011101 PACS number(s): 05.40.⫺a, 05.45.Xt, 05.10.Gg, 02.50.⫺r I. INTRODUCTION The theme of synchronization has widespread applications, covering a plenitude of phenomena [1–4]. Some characteristic examples are the entrainment of a system by external, time-dependent forcing, or the generalization of the synchronization concept for systems that exhibit a chaotic dynamics [5], lag synchronization [6], and also phase synchronization [7]. Synchronization phenomena play not only a key role for diverse technological applications, but increasingly as well for the description, the control, and even for the therapy of selected medical disorders [8]. Due to the interaction with a surrounding environment or with internal degrees of freedom, noise is present in many physical systems. This being so, its role cannot be ignored when investigating synchronization phenomena. In recent years, it has turned out that noise can actually play a constructive role in many physical situations. In particular, noise can boost the transduction of information by means of the phenomenon of stochastic resonance [9]in an ample number of metastable physical and biological systems [10,11]. Furthermore, noise enables Brownian motors to do work against external load forces [12], or to induce phase transitions far away from thermal equilibrium [13,14]. Our focus here is on the role of phase synchronization in stochastic overdamped systems driven by white Gaussian noise. In these cases, the velocity of the dynamics is not a measurable quantity because the stochastic trajectories are neither differentiable nor of finite variation, see, e.g., Ref. [15]. A recently proposed method for measuring the average phase velocity or frequency which is based on the generalization of a Rice rate formula for threshold crossings is consequently not a suitable method [16]. An alternative approach is based on the so-called “Hilbert phase” dynamics, as pioneered by Gabor [17]for deterministic systems. In the present work, we shall take a closer look at the synchronization phenomenon in a periodically driven bistable system. Then it is advantageous to introduce a discrete phase dynamics, as recently proposed by Schimansky-Geier and collaborators [4,14,18,19]. In order to extract this discrete phase dynamics from the underlying continuous process, we shall consider the stochastic counting process of the noise-induced switches between the two potential minima. It turns out that this counting process is in fact a nonstationary renewal process [20]. The outline of the present work is as follows: First, we introduce a dichotomic process associated to the original stochastic process by filtering out the fluctuations around the potential minima. This dichotomic process possesses a clear interpretation in terms of a discrete phase. By contrast to the underlying stochastic process, this two-state process, however, is no longer Markovian. Subsequently, in Sec. III, we analyze in detail the statistical properties of the random switching times associated to the dichotomic process. The one-time statistical properties of the discrete phase are then studied in Sec. IV. Based on these results, exact analytical expressions for the instantaneous output frequency and the phase diffusion are derived. Approximate expressions, valid in the weak-noise limit and for a slow external driving, are then obtained. Finally, our analytical findings will be applied to the case of a symmetric bistable potential driven by a periodic rectangular input signal. To corroborate our analytical results, we compare them with those obtained from a numerical simulation of the original stochastic process. *Email address: [email protected]; http://numerix.us.es †Present address: Department für Physik und Astronomie, Universität Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland. PHYSICAL REVIEW E 71, 011101 (2005) 1539-3755/2005/71(1)/011101(9)/$23.00 ©2005 The American Physical Society011101-1 II. DESCRIPTION OF THE MODEL AND DEFINITION OF THE DISCRETE PHASE To start, we consider a stochastic dynamics characterized by a single degree of freedom x共t兲, whose dynamics (in dimensionless units)is described by the stochastic differential equation x ˙共t兲=−U⬘„x共t兲,t…+ ␰ 共t兲,共1兲 where ␰ 共t兲is a Gaussian white noise of zero mean with autocorrelation 具 ␰ 共t兲 ␰ 共s兲典=2D ␦ 共t−s兲, and U⬘共x,t兲is the derivative with respect to xof the bistable quartic potential, U共x,t兲=x4 4−x2 2−F共t兲x,共2兲 F共t兲representing a periodic forcing with period T. Our focus is on subthreshold signals; more precisely, we will assume that, for any instant of time, 兩F共t兲兩⬍Ath=2/冑27, where Ath is the static threshold value (the dynamical threshold value always exceeds this adiabatic threshold Ath). In this case, the potential possesses two minima at q−1共t兲⬍0 and q+1共t兲⬎0, and a maximum at qM共t兲. Introducing the function ␯ 共t兲 =arccos关F共t兲/Ath兴, with arccos ybeing the principal value of the arc cosine of y(i.e., the value in the interval 关0, ␲ 兴), and ␩ n共t兲=2 冑3cos 冋 ␯ 共t兲+2 ␲ n 3 册 ,共3兲 then ␩ 0共t兲yields the location of the minimum to the right of the barrier [i.e., q+1共t兲= ␩ 0共t兲], ␩ 1共t兲yields the location of the minimum to the left of the barrier [i.e., q−1共t兲= ␩ 1共t兲], and ␩ 2共t兲yields the location of the maximum [i.e., qM共t兲= ␩ 2共t兲]. From now on, we will assume that at an initial instant of time t0the system is placed at one of the minima of the potential q ␣ 0共t0兲, with ␣ 0=+1 or⫺1. The long-time behavior of the quantities of interest can be obtained by taking the limit as t0→−⬁at the end of the calculations. Henceforth, we will make explicit the dependence of all the quantities on the initial preparation by the superscript ␣ 0,t0; see also the discussion after Eq. (26). Thus, for instance, we will write x ␣ 0,t0共t兲instead of x共t兲, meaning that x ␣ 0,t0共t0兲=q ␣ 0共t0兲. To analyze the synchronization phenomenon in this stochastic bistable system, it is convenient to introduce a discrete phase associated to the continuous stochastic process xa0,t0共t兲. In order to do so, first we will proceed to filter out the fluctuations around the minima of the stochastic process x ␣ 0,t0共t兲to obtain a two-state stochastic process ␹ ␣ 0,t0共t兲 which only takes the values ⫹1or⫺1. The procedure used is as follows: At the initial instant of time t0we set ␹ ␣ 0,t0共t0兲 = ␣ 0. A switch of state from ± ␣ 0to ⫿ ␣ 0occurs whenever the system, having started in one of the minima, reaches the other minimum for the first time. The instant of time at which the nth switch of state takes place is a random variable which will be denoted by Tn ␣ 0,t0, with n=1,2,…. Formally, these random variables can be defined recursively as Tn ␣ 0,t0= min 关 t:t⬎Tn−1 ␣ 0,t0and x ␣ 0,t0共t兲=q ␣ n共t兲 兴 ,共4兲 where T0 ␣ 0,t0=t0and ␣ n=共−1兲n ␣ 0. Thus, if we introduce the stochastic process N ␣ 0,t0共t兲= max 关 n:Tn ␣ 0,t0艋t 兴 ,共5兲 which counts the number of switches of state in the interval 共t0,t兴, then the two-state stochastic process ␹ ␣ 0,t0共t兲can be expressed as ␹ ␣ 0,t0共t兲= ␣ 0cos 关 ␲ N ␣ 0,t0共t兲 兴 .共6兲 By analogy with the case of a sinusoidal signal, we will define the discrete phase ␸ ␣ 0,t0共t兲associated with x ␣ 0,t0共t兲as the stochastic process ␸ ␣ 0,t0共t兲= ␲ N ␣ 0,t0共t兲.共7兲 In Fig. 1, we illustrate the procedure just described in the particular case of a rectangular signal [see Eq. (65)and text below]with amplitude A=0.25 and frequency ␻ =2 ␲ /T =0.01, and a noise strength D=0.02. The initial instant of time has been chosen to be t0=0, and the system has been initially placed at q+1共0兲, so that ␣ 0= +1. In the upper panel, we have sketched the rectangular periodic signal F共t兲, whereas in the middle and lower panels, we have depicted a random trajectory of the stochastic process x+1,0共t兲and the corresponding realization of the process ␹ +1,0共t兲, respectively. III. STATISTICAL CHARACTERIZATION OF THE SWITCHING TIMES Tn ␣ 0,t0 According to Eqs. (5)and (7), the statistical properties of the discrete phase ␸ ␣ 0,t0共t兲are closely related to those of the FIG. 1. Illustration of the procedure used to define the stochastic process ␹ ␣ 0,t0共t兲in the particular case of a rectangular signal [see Eq. (65)and text below]with amplitude A=0.25 and frequency ⍀ =2 ␲ /T=0.01, and a noise strength D=0.02. The initial instant of time has been chosen to be t0=0, and the system has been initially placed at q+1共0兲, so that ␣ 0= +1. In the upper panel, we have sketched the rectangular periodic signal F共t兲, whereas in the middle and lower panels, we have depicted a random trajectory of the stochastic process x+1,0共t兲and the corresponding realization of the process ␹ +1,0共t兲, respectively. CASADO-PASCUAL et al. PHYSICAL REVIEW E 71, 011101 (2005) 011101-2 random switching times Tn ␣ 0,t0. The aim of this section is to provide a detailed description of the statistical characterization of these random variables previous to the analysis of ␸ ␣ 0,t0共t兲, which will be postponed to the next section. The connection between the statistical properties of the switching times and the original stochastic process x ␣ 0,t0共t兲will be also analyzed in this section. Following the approach presented in Ref. [20], the random variable Tn ␣ 0,t0can be characterized statistically by its probability density function gn ␣ 0,t0共t兲= lim ⌬t→0+ Prob关t⬍Tn ␣ 0,t0艋t+⌬t兴 ⌬t.共8兲 Besides this probability distribution function, it is also convenient to introduce the (cumulative)distribution function Gn ␣ 0,t0共t兲= Prob关Tn ␣ 0,t0艋t兴= 冕 t0 tdt⬘gn ␣ 0,t0共t⬘兲,共9兲 as well as its complementary, Gn ␣ 0,t0共t兲= Prob关Tn ␣ 0,t0⬎t兴=1−Gn ␣ 0,t0共t兲.共10兲 In the particular case n=1, these functions can be directly determined from the solution of the Fokker-Planck equation (FPE) ⳵ ⳵ tP共x,t兲= ⳵ ⳵ x 冋 D ⳵ ⳵ x+U⬘共x,t兲 册 P共x,t兲,共11兲 with initial condition P共x,t0兲= ␦ „ x−q ␣ 0共t0兲 … ,共12兲 and absorbing boundary condition at q− ␣ 0共t兲, i.e., P关q− ␣ 0共t兲,t兴= 0 for all t艌t0.共13兲 Denoting by P ␣ 0,t0共x,t兲the solution of the above problem, it follows from the definition of G1 ␣ 0,t0共t兲in Eq. (10), with n =1, that G1 ␣ 0,t0共t兲= ␦ ␣ 0,+1 冕 q−1共t兲 +⬁ dx P+1,t0共x,t兲 + ␦ ␣ 0,−1 冕 −⬁ q+1共t兲 dx P−1,t0共x,t兲,共14兲 for t艌t0, and G1 ␣ 0,t0共t兲=1 for t⬍t0. The function G1 ␣ 0,t0共t兲is the conditional survival probability of the discrete state ␣ 0. The function G1 ␣ 0,t0共t兲is then given by Eq. (10)with n=1, whereas g1 ␣ 0,t0共t兲=−G ˙1 ␣ 0,t0共t兲is the corresponding conditional residence time distribution (RTD). The dot indicates the derivative with respect to time t. The knowledge of either the conditional survival probabilities G1 ␣ 0,t0共t兲,or(equivalently) the conditional RTDs g1 ␣ 0,t0共t兲, is sufficient to specify a driven two-state non-Markovian renewal process [21]. These functions can be found from the underlying continuous-state Markovian dynamics by solving Eqs. (11)–(14). For n⬎1, the functions Gn ␣ 0,t0共t兲,Gn ␣ 0,t0共t兲, and gn ␣ 0,t0共t兲can be obtained iteratively from the ones corresponding to the case n=1 by making use of three integral equations. To obtain the first integral equation, let us consider the consistency condition Gn+1 ␣ 0,t0共t兲= 冕 −⬁ +⬁ dt⬘Prob关Tn+1 ␣ 0,t0⬎t兩Tn ␣ 0,t0=t⬘兴gn ␣ 0,t0共t⬘兲. 共15兲 By use of the definition of the switching times in Eq. (4),as well as the Markovian character of the original stochastic process x ␣ 0,t0共t兲, it is straightforward to verify that Prob关Tn+1 ␣ 0,t0⬎t兩Tn ␣ 0,t0=t⬘兴=G1 ␣ n,t⬘共t兲.共16兲 Inserting the above expression into Eq. (15)and taking into account that G1 ␣ n,t⬘共t兲=1 for t⬍t⬘and gn ␣ 0,t0共t兲=0 for t⬍t0, we obtain Gn+1 ␣ 0,t0共t兲=Gn ␣ 0,t0共t兲+ 冕 t0 tdt⬘G1 ␣ n,t⬘共t兲gn ␣ 0,t0共t⬘兲共17兲 for n艌1. The interpretation of this result is straightforward: The probability that the 共n+1兲th switch of state occurs after the time tis equal to the probability that the nth switch occurs after that instant of time plus the probability that the nth switch has happened at any time t⬘before twith the next switch taking place after t. Similar interpretations hold for the integral equations; i.e., Gn+1 ␣ 0,t0共t兲= 冕 t0 tdt⬘G1 ␣ n,t⬘共t兲gn ␣ 0,t0共t⬘兲,共18兲 gn+1 ␣ 0,t0共t兲= 冕 t0 tdt⬘g1 ␣ n,t⬘共t兲gn ␣ 0,t0共t⬘兲,共19兲 which are obtained from Eq. (17)by using Eq. (10)and gn ␣ 0,t0共t兲=−G ˙n ␣ 0,t0共t兲, respectively. A formal solution for gn ␣ 0,t0共tn兲is obtained by solving iteratively the integral equation (19). The result is gn ␣ 0,t0共tn兲= 冕 t0 tndtn−1¯ 冕 t0 t2dt1兿 j=0 n−1 g1 ␣ j,tj共tj+1兲共20兲 for n艌2. Thus, the probability distribution corresponding to the first switch of state, g1 ␣ 0,t0共t兲, which can be obtained from the solution of the FPE (11)with Eqs. (12)and (13), determines completely the statistical properties of the rest of the switching times. IV. ONE-TIME STATISTICAL PROPERTIES OF THE DISCRETE PHASE: THE OUTPUT FREQUENCY AND THE PHASE DISPERSION The one-time statistical properties of the discrete phase ␸ ␣ 0,t0共t兲can be evaluated by making use of the probability distribution of the number of switches of state ␳ n ␣ 0,t0共t兲= Prob 关 N ␣ 0,t0共t兲=n 兴 ,共21兲 with n=0,1,2,…. From the definition of N ␣ 0,t0共t兲in Eq. (5) it follows that THEORY OF FREQUENCY AND PHASE …PHYSICAL REVIEW E 71, 011101 (2005) 011101-3 Prob 关 N ␣ 0,t0共t兲艌n 兴 =Gn ␣ 0,t0共t兲,共22兲 with G0 ␣ 0,t0共t兲=1. Consequently, the probability distribution of the number of switches of state and its derivative with respect to tcan be expressed, respectively, as ␳ n ␣ 0,t0共t兲=Gn ␣ 0,t0共t兲−Gn+1 ␣ 0,t0共t兲=Gn+1 ␣ 0,t0共t兲−Gn ␣ 0,t0共t兲,共23兲 and ␳ ˙n ␣ 0,t0共t兲=gn ␣ 0,t0共t兲−gn+1 ␣ 0,t0共t兲,共24兲 with G0 ␣ 0,t0共t兲=g0 ␣ 0,t0共t兲=0. The average of an arbitrary onetime function of N ␣ 0,t0共t兲,K关N ␣ 0,t0共t兲兴, is obviously given by 具 K 关 N ␣ 0,t0共t兲 兴典 =兺 n=0 ⬁ K共n兲 ␳ n ␣ 0,t0共t兲.共25兲 Equations (23)and (24)can be written in a more transparent form by introducing the probability of an almost immediate switch of state after nswitches, ⌫n ␣ 0,t0共t兲= lim ⌬t→0+ Prob关t⬍Tn+1 ␣ 0,t0艋t+⌬t兩N ␣ 0,t0共t兲=n兴 ⌬t. 共26兲 Note that if the process ␹ ␣ 0,t0共t兲were Markovian, these probabilities could only depend on the state ␣ n. The explicit dependence on the number of jumps nand on the initial preparation ␣ 0at time t0is a consequence of the non-Markovian character of the process. Another fingerprint of the nonMarkovian nature of the dichotomic process is the fact that these probabilities depend on the time teven in the absence of the external driving. In order to clarify this point, let us consider, e.g., the particular case n=0. Then, while initially, right after the particle has been prepared at one of the minima, the distribution function P ␣ 0,t0共x,t兲is still very sharply peaked around q ␣ 0共t0兲, it becomes smeared out around the minimum after the intrawell relaxation time. Consequently, the probability of an immediate switch will be different before and after this relaxation time, even without an external driving. Multiplying and dividing the right-hand side of the above expression by ␳ n ␣ 0,t0共t兲and taking into account that lim ⌬t→0+ Prob 关 t⬍Tn+1 ␣ 0,t0艋t+⌬tand N ␣ 0,t0共t兲=n 兴 ⌬t=gn+1 ␣ 0,t0共t兲, 共27兲 it is readily seen that ⌫n ␣ 0,t0共t兲=gn+1 ␣ 0,t0共t兲 ␳ n ␣ 0,t0共t兲.共28兲 Then, it follows from Eqs. (17),(23), and (28)that ␳ n ␣ 0,t0共t兲= 冕 t0 tdt⬘G1 ␣ n,t⬘共t兲⌫n−1 ␣ 0,t0共t⬘兲 ␳ n−1 ␣ 0,t0共t⬘兲共29兲 for n艌1, with ␳ 0 ␣ 0,t0共t兲=G1 ␣ 0,t0共t兲. Analogously, Eq. (24)leads to the following hierarchy of differential equations: ␳ ˙n ␣ 0,t0共t兲=⌫n−1 ␣ 0,t0共t兲 ␳ n−1 ␣ 0,t0共t兲−⌫n ␣ 0,t0共t兲 ␳ n ␣ 0,t0共t兲共30兲 for n艌1, and ␳ ˙0 ␣ 0,t0共t兲=−⌫0 ␣ 0,t0共t兲 ␳ 0 ␣ 0,t0共t兲,共31兲 which must be solved with the initial condition ␳ n ␣ 0,t0共t0兲 = ␦ n,0. We will also introduce the conditional probability for ␹ ␣ 0,t0共t兲to take the value ␤ = ±1 at time t, provided that it took the value ␣ 0with probability 1 at the initial instant of time t0, p ␤ ␣ 0,t0共t兲= Prob 关 ␹ ␣ 0,t0共t兲= ␤ 兴 .共32兲 Noting that after an even number of switches of state the system ends up in the same state as it was initially, whereas for an odd number of switches the system ends up in the other state, it is clear that the events 兵 ␹ ␣ 0,t0共t兲= ␣ 0其and 兵 ␹ ␣ 0,t0共t兲=− ␣ 0其are, respectively, equivalent to the events 兵N ␣ 0,t0共t兲is even其and 兵N ␣ 0,t0共t兲is odd其, and consequently p ␤ ␣ 0,t0共t兲= ␦ ␣ 0, ␤ 兺 n=0 ⬁ ␳ 2n ␣ 0,t0共t兲+ ␦ − ␣ 0, ␤ 兺 n=0 ⬁ ␳ 2n+1 ␣ 0,t0共t兲.共33兲 Besides the probability distribution of the number of switches of state, later we will also use the probability distribution of the number of switches of state conditioned to the value of ␹ ␣ 0,t0共t兲, ␳ n ␣ 0,t0共t兩 ␤ 兲= Prob关N ␣ 0,t0共t兲=n兩 ␹ ␣ 0,t0共t兲= ␤ 兴.共34兲 Multiplying and dividing the right-hand side of the above expression by p ␤ ␣ 0,t0共t兲and taking into account that Prob关N ␣ 0,t0共t兲=nand ␹ ␣ 0,t0共t兲= ␤ 兴= ␳ n ␣ 0,t0共t兲 ␦ ␣ n, ␤ , it results that ␳ n ␣ 0,t0共t兩 ␤ 兲= ␳ n ␣ 0,t0共t兲 p ␤ ␣ 0,t0共t兲 ␦ ␣ n, ␤ .共35兲 The average of an arbitrary one-time function of N ␣ 0,t0共t兲, K关N ␣ 0,t0共t兲兴, conditioned to the event 兵 ␹ ␣ 0,t0共t兲= ␤ 其will be denoted by 具K关N ␣ 0,t0共t兲兴典 ␤ , with ␤ = +1 or −1, and it is given by 具 K关N ␣ 0,t0共t兲兴 典 ␤ =兺 n=0 ⬁ K共n兲 ␳ n ␣ 0,t0共t兩 ␤ 兲 = ␦ ␣ 0, ␤ 兺 n=0 ⬁ K共2n兲 ␳ 2n ␣ 0,t0共t兲 p ␣ 0 ␣ 0,t0共t兲 + ␦ − ␣ 0, ␤ 兺 n=0 ⬁ K共2n+1兲 ␳ 2n+1 ␣ 0,t0共t兲 p− ␣ 0 ␣ 0,t0共t兲.共36兲 From the above expression and Eq. (25), it follows that 具 K关N ␣ 0,t0共t兲兴 典 = 具 K关N ␣ 0,t0共t兲兴 典 +1p+1 ␣ 0,t0共t兲 + 具 K关N ␣ 0,t0共t兲兴 典 −1p−1 ␣ 0,t0共t兲.共37兲 Another interesting quantity which will be useful later is the probability of an almost immediate switch from state ␤ , defined as CASADO-PASCUAL et al. PHYSICAL REVIEW E 71, 011101 (2005) 011101-4 ␥ ␤ ␣ 0,t0共t兲= lim ⌬t→0+ Prob 关 t⬍TN ␣ 0,t0共t兲+1 ␣ 0,t0艋t+⌬t兩 ␹ ␣ 0,t0共t兲= ␤ 兴 ⌬t. 共38兲 Notice that in the Markovian limit, these probabilities are independent of the initial preparation ␣ 0at time t0and cannot be distinguished from the probabilities defined in Eq. (26). Multiplying and dividing the right-hand side of the above expression by p ␤ ␣ 0,t0共t兲and taking into account the equivalence of the event 兵 ␹ ␣ 0,t0共t兲= ␤ 其with the event 兵N ␣ 0,t0共t兲is even其 ␦ ␣ 0, ␤ +兵N ␣ 0,t0共t兲is odd其 ␦ − ␣ 0, ␤ , as well as Eqs. (26)and (36), it is easy to see that ␥ ␤ ␣ 0,t0共t兲= 具 ⌫N ␣ 0,t0共t兲 ␣ 0,t0共t兲 典 ␤ .共39兲 Differentiating Eq. (33)with respect to tand taking into account Eqs. (30),(36), and (39), it is straightforward to obtain that p ˙ ␤ ␣ 0,t0共t兲=− ␥ ␤ ␣ 0,t0共t兲p ␤ ␣ 0,t0共t兲+ ␥ − ␤ ␣ 0,t0共t兲p− ␤ ␣ 0,t0共t兲共40兲 for ␤ =1 and ⫺1. Equation (40)is a non-Markovian master equation for the conditional probabilities p ␤ ␣ 0,t0共t兲.Itisofthe time-convolutionless form [22,23]. The rate parameters ␥ ␤ ␣ 0,t0共t兲entering this equation are time-dependent quantities even in the absence of time-dependent driving. Such a time dependence reflects primarily a nonexponential distribution of the residence times of the renewal two-state nonMarkovian process [20,24], which results from the projection of a continuous-state Markovian stochastic dynamics onto the two discrete states ␤ = ±1. A time-dependent driving introduces an additional time dependence into ␥ ␤ ␣ 0,t0共t兲which is present also in the driven Markovian case. In this latter case, ␥ ␤ ␣ 0,t0共t兲becomes a time-dependent rate and this rate depends neither on ␣ 0nor on t0(see below). Two other forms are possible to describe the evolution of conditional probabilities p ␤ ␣ 0,t0共t兲. One is given by the generalized master equations (GMEs)with the memory kernels expressed via the corresponding RTDs. In the driven case, the kernels of corresponding GMEs will become functionals of the driving and will depend on both time arguments. Alternatively, integral equations for the conditional probabilities p ␤ ␣ 0,t0共t兲can be derived for the driven two-state renewal process in terms of (conditional)RTDs g1 ␣ 0,t0共t兲[21]. Such integral equations present a generalization of the integral renewal equations of Ref. [20]to the driven case. We apply in this work a timeconvolutionless description of non-Markovian dynamics [22,23]to the synchronization problem. After these rather formal considerations, we shall now apply these results to the evaluation of two important quantities in the study of the synchronization phenomenon: The instantaneous output frequency and phase diffusion. A. The instantaneous output frequency The instantaneous output frequency is defined as the derivative with respect to tof the averaged discrete phase [18], i.e., ⍀out ␣ 0,t0共t兲= ⳵ ⳵ t 具 ␸ ␣ 0,t0共t兲 典 = ␲ ⳵ ⳵ t 具 N ␣ 0,t0共t兲 典 .共41兲 Multiplying Eq. (30)by n, summing up the series 兺n=1 ⬁n ␳ ˙n ␣ 0,t0共t兲, and taking into account Eq. (25), it is easy to obtain that ⳵ ⳵ t具N ␣ 0,t0共t兲典 = 具 ⌫N ␣ 0,t0共t兲 ␣ 0,t0共t兲 典 .共42兲 Consequently, from Eqs. (37),(39), and (41), it results ⍀out ␣ 0,t0共t兲= ␲ 关 ␥ +1 ␣ 0,t0共t兲p+1 ␣ 0,t0共t兲+ ␥ −1 ␣ 0,t0共t兲p−1 ␣ 0,t0共t兲兴.共43兲 This finding for the averaged frequency of the discrete phase dynamics constitutes a first main finding of this work. B. The instantaneous phase diffusion Let us now proceed to the evaluation of the instantaneous phase diffusion Dout ␣ 0,t0共t兲, which can be defined as [18] Dout ␣ 0,t0共t兲= ⳵ ⳵ t 兵 具关 ␸ ␣ 0,t0共t兲 兴 2 典 − 具 ␸ ␣ 0,t0共t兲 典 2 其 = ␲ 2 ⳵ ⳵ t 兵 具关 N ␣ 0,t0共t兲 兴 2 典 − 具 N ␣ 0,t0共t兲 典 2 其 .共44兲 Multiplying Eq. (30)by n2, summing up the series 兺n=1 ⬁n2 ␳ ˙n ␣ 0,t0共t兲, and taking into account Eq. (25),itis straightforward to see that ⳵ ⳵ t 具关 N ␣ 0,t0共t兲 兴 2 典 =2 具 N ␣ 0,t0共t兲⌫N ␣ 0,t0共t兲 ␣ 0,t0共t兲 典 + 具 ⌫N ␣ 0,t0共t兲 ␣ 0,t0共t兲 典 . 共45兲 Replacing the above expression into Eq. (44)and taking into account Eqs. (41)and (42), it results that Dout ␣ 0,t0共t兲= ␲ ⍀out ␣ 0,t0共t兲+2 ␲ 2 关具 N ␣ 0,t0共t兲⌫N ␣ 0,t0共t兲 ␣ 0,t0共t兲 典 −具N ␣ 0,t0共t兲典 具 ⌫N ␣ 0,t0共t兲 ␣ 0,t0共t兲 典兴 .共46兲 Equation (46)can be expressed in a more convenient form by writing all the averages 具¯典in terms of the conditional averages 具¯典 ␤ , according to Eq. (37). Then, after some simplifications one obtains Dout ␣ 0,t0共t兲= ␲ ⍀out ␣ 0,t0共t兲+2 ␲ 2⌬ ␥ ␣ 0,t0共t兲⌿ ␣ 0,t0共t兲 +2 ␲ 2兺 ␤ =±1 C ␤ ␣ 0,t0共t兲p ␤ ␣ 0,t0共t兲,共47兲 where ⌬ ␥ ␣ 0,t0共t兲= ␥ +1 ␣ 0,t0共t兲− ␥ −1 ␣ 0,t0共t兲,共48兲 ⌿ ␣ 0,t0共t兲= 关 具N ␣ 0,t0共t兲典+1 −具N ␣ 0,t0共t兲典−1 兴 p+1 ␣ 0,t0共t兲p−1 ␣ 0,t0共t兲, 共49兲 and we have introduced the conditional covariance C ␤ ␣ 0,t0共t兲= 具 关N ␣ 0,t0共t兲−具N ␣ 0,t0共t兲典 ␤ 兴 ⫻关⌫N ␣ 0,t0共t兲 ␣ 0,t0共t兲− 具 ⌫N ␣ 0,t0共t兲 ␣ 0,t0共t兲 典 ␤ 兴 典 ␤ .共50兲 THEORY OF FREQUENCY AND PHASE …PHYSICAL REVIEW E 71, 011101 (2005) 011101-5 Making use of Eqs. (30)and (40)and after some lengthy calculations, it is possible to prove that ⌿ ␣ 0,t0共t兲satisfies the differential equation ⌿ ˙ ␣ 0,t0共t兲=− ␥ ␣ 0,t0共t兲⌿ ␣ 0,t0共t兲 −兺 ␤ =±1 ␤ 关C ␤ ␣ 0,t0共t兲+ ␥ ␤ ␣ 0,t0共t兲p ␤ ␣ 0,t0共t兲兴p ␤ ␣ 0,t0共t兲, 共51兲 where ␥ ␣ 0,t0共t兲= ␥ +1 ␣ 0,t0共t兲+ ␥ −1 ␣ 0,t0共t兲.共52兲 Equation (51)can be formally solved taking into account that, as it follows from the definition (49),⌿ ␣ 0,t0共t0兲=0. The result is ⌿ ␣ 0,t0共t兲=− 兺 ␤ =±1 ␤ 冕 t0 t dt⬘ 关 C ␤ ␣ 0,t0共t⬘兲+ ␥ ␤ ␣ 0,t0共t⬘兲p ␤ ␣ 0,t0共t⬘兲 兴 ⫻p ␤ ␣ 0,t0共t⬘兲e−兰t⬘ tdt⬙ ␥ ␣ 0,t0共t⬙兲.共53兲 Replacing the above expression into Eq. (47), one obtains Dout ␣ 0,t0共t兲= ␲ ⍀out ␣ 0,t0共t兲+Q ␣ 0,t0共t兲−2 ␲ 2⌬ ␥ ␣ 0,t0共t兲 ⫻兺 ␤ =±1 ␤ 冕 t0 t dt⬘ ␥ ␤ ␣ 0,t0共t⬘兲 关 p ␤ ␣ 0,t0共t⬘兲 兴 2e−兰t⬘ tdt⬙ ␥ ␣ 0,t0共t⬙兲, 共54兲 where all the dependence on the conditional covariance C ␤ ␣ 0,t0共t兲has been included in the function Q ␣ 0,t0共t兲=2 ␲ 2兺 ␤ =±1 C ␤ ␣ 0,t0共t兲p ␤ ␣ 0,t0共t兲−2 ␲ 2⌬ ␥ ␣ 0,t0共t兲 ⫻兺 ␤ =±1 ␤ 冕 t0 t dt⬘C ␤ ␣ 0,t0共t⬘兲p ␤ ␣ 0,t0共t⬘兲e−兰t⬘ tdt⬙ ␥ ␣ 0,t0共t⬙兲. 共55兲 The expression (54)for the instantaneous phase diffusion presents a second main result of this work. C. The weak-noise and low-frequency limit Throughout the following, we will assume that the noise strength Dis sufficiently small so that the intrawell relaxation time scale is negligible compared with the time scale associated to the interwell transitions and, as well, the driving time scale T. In this case, for t−t0much larger than the characteristic intrawell relaxation time, the probability of an almost immediate switch of state after 0 switches, ⌫0 ␤ ,t0共t兲, can be approximated by the Kramers rate of escape [25]from the state ␤ at time t, i.e., ⌫0 ␤ ,t0共t兲⬇ ␥ ␤ K共t兲= ␻ ␤ 共t兲 ␻ M共t兲 2 ␲ ⫻exp 再 −U关qM共t兲,t兴−U关q ␤ 共t兲,t兴 D 冎 ,共56兲 where ␻ ␤ 共t兲=冑U⬙关q ␤ 共t兲,t兴=冑3关q ␤ 共t兲兴2−1 and ␻ M共t兲 =冑兩U⬙关qM共t兲,t兴兩=冑1−3关qM共t兲兴2. Furthermore, from Eqs. (17),(19),(23), and (28)it follows that, within this approximation, we also have that ⌫n ␤ ,t0共t兲⬇ ␥ ␤ K共t兲for n艌1 and, consequently, ␥ ␤ ␣ 0,t0共t兲⬇ ␥ ␤ K共t兲and C ␤ ␣ 0,t0共t兲⬇0. In this case, the conditional survival propabilities and the residence time distributions in state ␣ 0read G1 ␣ 0,t0共t兲= exp 冋 − 冕 t0 t ␥ ␣ 0 K共t⬘兲dt⬘ 册 共57兲 and g1 ␣ 0,t0共t兲= ␥ ␣ 0 K共t兲exp 冋 − 冕 t0 t ␥ ␣ 0 K共t⬘兲dt⬘ 册 ,共58兲 respectively. This corresponds to a two-state Markovian process with rates ␥ ±1 K共t兲. In this Markovian limit for the reduced, two-state dynamics, the instantaneous output frequency and phase diffusion become ⍀out ␣ 0,t0共t兲= ␲ 关 ␥ +1 K共t兲p+1 ␣ 0,t0共t兲+ ␥ −1 K共t兲p−1 ␣ 0,t0共t兲 兴 共59兲 and Dout ␣ 0,t0共t兲= ␲ ⍀out ␣ 0,t0共t兲 −2 ␲ 2⌬ ␥ K共t兲兺 ␤ =±1 ␤ 冕 t0 t dt⬘ ␥ ␤ K共t⬘兲 关 p ␤ ␣ 0,t0共t⬘兲 兴 2 ⫻e−兰t⬘ tdt⬙ ␥ K共t⬙兲,共60兲 respectively. Here, ␥ K共t兲= ␥ +1 K共t兲+ ␥ −1 K共t兲,⌬ ␥ K共t兲= ␥ +1 K共t兲 − ␥ −1 K共t兲, and p ␤ ␣ 0,t0共t兲is obtained by solving the master equation p ˙ ␤ ␣ 0,t0共t兲=− ␥ ␤ K共t兲p ␤ ␣ 0,t0共t兲+ ␥ − ␤ K共t兲p− ␤ ␣ 0,t0共t兲,共61兲 with initial condition p ␤ ␣ 0,t0共t0兲= ␦ ␣ 0, ␤ . In order to obtain expressions independent of the initial preparation, it is necessary to take the limit t0→−⬁of Eqs. (59)and (60). In this limit, it can be shown that the functions ⍀out共t兲=limt0→−⬁⍀out ␣ 0,t0共t兲and Dout共t兲=limt0→−⬁Dout ␣ 0,t0共t兲are periodic functions of the time t. Then, one can perform a cycle average and define the averaged output frequency ⍀out =1 T 冕 0 T dt ⍀out共t兲= ␲ T 冕 0 T dt 关 ␥ +1 K共t兲p+1共t兲+ ␥ −1 K共t兲p−1共t兲 兴 , 共62兲 where p ␤ 共t兲is the periodic long-time solution, p ␤ 共t兲 =limt0→−⬁p ␤ ␣ 0,t0共t兲,ofEq.(61). After some lengthy calculations, it is also possible to show from Eq. (60)that the averaged phase diffusion is given by CASADO-PASCUAL et al. PHYSICAL REVIEW E 71, 011101 (2005) 011101-6 Dout =1 T 冕 0 T dtDout共t兲= ␲ ⍀out − ␲ 2 Tcsch 冉 ␥ KT 2 冊 ⫻兺 ␤ =±1 ␤ 冕 0 T dt 冕 0 T dt⬘⌬ ␥ K共t兲 ␥ ␤ K共t⬘兲关p ␤ 共t⬘兲兴2 ⫻exp 冋 sgn共t-t⬘兲 ␥ KT 2− 冕 t⬘ tdt⬙ ␥ K共t⬙兲 册 ,共63兲 with ␥ K=1 T 冕 0 T dt ␥ K共t兲.共64兲 In the next section we will consider the case of a rectangular input signal. In this case, explicit analytical evaluations of the integrals in Eqs. (62)and (63)can be carried out. V. PERIODIC RECTANGULAR INPUT SIGNAL As an example of the use of Eqs. (62)and (63)which is amenable to analytical treatment, we will consider the case of the periodic rectangular driving force F共t兲=共−1兲n共t兲A,共65兲 where n共t兲=b2t/Tc,bzcbeing the floor function of z, i.e., the greatest integer less than or equal to z. In other words, F共t兲 =A关F共t兲=−A兴if t僆关nT/2,共n+1兲T/2兴with neven (odd). Because the potential fulfills the symmetry property U共x,t+T/2兲=U共−x,t兲, we have qM共t兲=共−1兲n共t兲qM共0兲, and q ␤ 共t兲= ␤ ⌬q共0兲 2−共−1兲n共t兲qM共0兲 2,共66兲 where ⌬q共0兲=q+1共0兲−q−1共0兲. Here, we have taken into account Vieta’s formula q+1共t兲+q−1共t兲+qM共t兲=0. According to the above mentioned symmetry property of the potential, ␥ ␤ K共t兲can be expressed in the form ␥ ␤ K共t兲= ␥ 2关1−共−1兲n共t兲 ␤ ⌬peq共0兲兴,共67兲 where ␥ = ␥ +1 K共0兲+ ␥ −1 K共0兲, and ⌬peq共0兲=p+1 eq共0兲−p−1 eq共0兲, p ␤ eq共0兲being the equilibrium population of the state ␤ corresponding to the rates taken at time t=0, i.e., p ␤ eq共0兲 =关 ␦ ␤ ,−1 ␥ +1 K共0兲+ ␦ ␤ ,+1 ␥ −1 K共0兲兴/ ␥ . Notice that for the rectangular input signal in Eq. (65), ␥ K共t兲= ␥ K= ␥ . We can also write ⌬ ␥ K共t兲= ␥ +1 K共t兲− ␥ −1 K共t兲=−共−1兲n共t兲 ␥ ⌬peq共0兲.共68兲 As shown in Ref. [26], the long-time probabilities p±1共t兲are given by p−1共t兲=1 2关1−共−1兲n共t兲⌬peq共0兲兴 +共−1兲n共t兲⌬peq共0兲e− ␥ 兵t−关n共t兲T/2兴其 1+e− ␥ T/2 ,共69兲 and p+1共t兲=1−p−1共t兲. Replacing the above expressions into Eqs. (62)and (63), one obtains after some lengthy simplifications that ⍀out = ␲ ␥ 2 冦 1−关⌬peq共0兲兴2 冤 1− 4 tanh 冉 ␥ T 4 冊 ␥ T 冥 冧 ,共70兲 and Dout = ␲ ⍀out −2 ␲ 2 T关⌬peq共0兲兴4 冋 tanh 冉 ␥ T 4 冊 册 3 − ␲ 2 2T关⌬peq共0兲兴2兵1−关⌬peq共0兲兴2其 冠 12 tanh 冉 ␥ T 4 冊 − ␥ T 再 1+2 冋 sech 冉 ␥ T 4 冊 册 2 冎 冡 .共71兲 In the next subsection, we will compare these analytical results for the averaged output frequency and phase diffusion with results obtained from a numerical solution of the stochastic differential equation (1). Comparison with numerical results Following the algorithm developed by Greenside and Helfand [27,28](consult also the Appendix in Ref. [29]),we have integrated Eq. (1)for a large number of noise realizations, M, starting from one of the minima q ␣ 0共0兲. From the initial instant of time, which we set equal to zero, we start monitoring the switches of states and recording the instants of time at which those switches occur, according to Eq. (4). We will denote by tn,i ␣ 0,0 the instant of time of the nth switch of state in the ith trajectory. From the switching times tn,i ␣ 0,0, the realization of N ␣ 0,0共t兲corresponding to the ith trajectory, Ni ␣ 0,0共t兲, can be easily calculated using Eq. (5), and the corresponding realization of the discrete phase, ␸ i ␣ 0,0共t兲,byEq. (7). The noise-averaged phase is then obtained by 具 ␸ ␣ 0,0共t兲典 =1 M兺 i=1 M ␸ i ␣ 0,0共t兲,共72兲 and the phase variance by ␷ ␣ 0,0共t兲= 具 关 ␸ ␣ 0,0共t兲兴2 典 −具 ␸ ␣ 0,0共t兲典2=1 M兺 i=1 M 关 ␸ i ␣ 0,0共t兲兴2 −1 M2 冋 兺 i=1 M ␸ i ␣ 0,0共t兲 册 2 .共73兲 After a sufficiently long number of periods Lfor the system to “forget” the initial preparation, the averaged output frequency is calculated from the expression ⍀out =具 ␸ ␣ 0,0关共L+1兲T兴典 −具 ␸ ␣ 0,0共LT兲典 T,共74兲 and the averaged phase diffusion from Dout = ␷ ␣ 0,0关共L+1兲T兴− ␷ ␣ 0,0共LT兲 T.共75兲 Figures 2–4 show the results of the numerical solution just described for a periodic rectangular input signal with THEORY OF FREQUENCY AND PHASE …PHYSICAL REVIEW E 71, 011101 (2005) 011101-7 angular frequency ⍀=2 ␲ /T=0.01 and three different values of the driving amplitude: A=0.14 (see Fig. 2),A=0.25 (see Fig. 3), and A=0.3 (see Fig. 4). As reported previously in Ref. [18]and observed experimentally in Ref. [30], for high enough, but still subthreshold, driving amplitudes, the present system exhibits a noise-induced frequency locking, i.e., starting from a nonzero value of the noise strength D, the frequency of the output signal matches the frequency of the input signal, until, for strong noise, the output signal becomes desynchronized again. This effect is accompanied by a very pronounced suppression of the phase diffusion of the output signal, i.e., a noise-induced phase locking. For the relevant values of the noise strength D, our analytical estimates Eq. (70)and Eq. (71)agree very well with the results obtained from the numerical solution. Only for rather strong noise can a noticeable deviation be observed. In this regime, the Kramers rates (56)are no longer valid. We have also plotted the results of the previous work [18], using the rates (56). We note that while the improvement of our analytical estimates for the frequency synchronization is moderate only, our prediction for the phase diffusion is strongly improved. VI. CONCLUSIONS With this work, we have investigated in detail the phenomenon of frequency and phase synchronization in bistable, periodically driven stochastic systems. This objective is not only of foremost interest for the well known phenomenon of stochastic resonance [9,10]and the topic of rocked Brownian motors [12], but also carries great potential for the study of driven stochastic neuronal dynamics and driven excitable sytems per se [31]. Our approach takes a new look at this prominent problem. Starting out from a driven, Markovian continuous dynamics, we derived in great detail the stochastic renewal dynamics of the noise-induced switching events. This contraction of the full Markovian dynamics in state space onto the discrete counting process of subsequent switches between the metastable states implies a nonMarkovian dynamics for the switching times and the corresponding phase dynamics whose explicit time evolution depends on initial preparation effects. The resulting nonMarkovian expressions still contain the full information of the driven dynamics in the relevant state space and thus are not readily accessible for analytical estimates. In contrast, for weak noise and slow external driving, the dynamics of the underlying process simplifies considerably. Consequently, in the long time limit the phase dynamics now assumes again a Markovian nature. In this regime, we put forward new results for the phase diffusion and the frequency synchronization. In doing so, we have employed rectangular-shaped periodic driving signals. This choice entails two distinct advantages, namely (i)it allows a convenient analytic analysis of the corresponding synchronization quantities and (ii)its two-state character is also known to optimize the efficiency for the synchronization features. The same optimization feature holds true for related effects such as the achievements of optimizing the gain for stochastic resonance [32]or the enhancements of energy transduction in driven chemical reactions [33]. FIG. 4. Like Fig. 2 but for a driving amplitude A=0.30. FIG. 2. Averaged output frequency (upper panel)and averaged phase diffusion (lower panel)as a function of the noise strength D for a periodic rectangular input signal [see Eq. (65)and text below] with amplitude A=0.14 and angular frequency ⍀=2 ␲ /T=0.01. Solid line: Analytical results obtained from Eq. (70)(upper panel) and Eq. (71)(lower panel), respectively. Dashed line: Theoretical result from Ref. [18]. Crosses: Precise numerical results. In the upper panel, a horizontal dotted line indicates the frequency of the input signal. FIG. 3. Like Fig. 2 but for a driving amplitude A=0.25. CASADO-PASCUAL et al. PHYSICAL REVIEW E 71, 011101 (2005) 011101-8 Our analysis is in the spirit of prior works [18,19]; our novel estimates, however, quantitatively supersede in accuracy those prior results, cf. the detailed comparison performed above. While the improvement for the frequency synchronization is moderate only, the novel estimates present a sizable improvement for the role of the phase diffusion. In the weak-noise regime and for slow external driving, the Markovian theory provides a very good agreement with numerical precise simulations. This being so, we are confident that the new insight gained into the complexity of nonMarkovian, driven switching time dynamics together with its Markovian simplification obtained at weak noise and slow driving will prove useful for modeling and interpreting stochastic synchronization phenomena in driven metastable and excitable dynamics. ACKNOWLEDGMENTS We acknowledge the support of the Dirección General de Enseñanza Superior of Spain (BFM2002-03822), the Junta de Andalucía, the DAAD program “Acciones Integradas” (P.H., M.M.), the Sonderforschungsbereich 486 (project A10)of the Deutsche Forschungsgemeinschaft, and the Volkswagen-Stiftung (P.H.)under Grant No. I/77 217. [1]A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization: A Universal Concept in Nonlinear Science (Cambridge University Press, Cambridge, 2001). [2]J. Kurths, S. Boccaletti, C. Grebogi, and Y. C. Lai, Chaos 13, 126 (2003). [3]L. A. Vainstein and D. E. 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