Universal asymptotic behavior in nonlinear systems driven by a two-frequency forcing

PHYSICAL REVIEW E 88, 062919 (2013) Universal asymptotic behavior in nonlinear systems driven by a two-frequency forcing Jes´ us Casado-Pascual,1,*David Cubero,2,†and Ferruccio Renzoni3,‡ 1F´ ısica Te´ orica, Universidad de Sevilla, Apartado de Correos 1065, 41080 Sevilla, Spain 2Departamento de F´ ısica Aplicada I, EUP, Universidad de Sevilla, Calle Virgen de ´ Africa 7, 41011 Sevilla, Spain 3Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, United Kingdom (Received 7 August 2013; published 20 December 2013) We examine the time-dependent behavior of a nonlinear system driven by a two-frequency forcing. By using a nonperturbative approach, we are able to derive an asymptotic expression, valid in the long-time limit, for the time average of the output variable which describes the response of the system. We identify several universal features of the asymptotic response of the system, which are independent of the details of the model. In particular, we determine an asymptotic expression for the width of the resonance observed by keeping one frequency fixed and varying the other one. We show that this width is smaller than the usually assumed Fourier width by a factor determined by the two driving frequencies, and independent of the model system parameters. Additional general features can also be identified depending on the specific symmetry properties of the system. Our results find direct application in the study of sub-Fourier signal processing with nonlinear systems. DOI: 10.1103/PhysRevE.88.062919 PACS number(s): 05.45.−a,05.60.Cd,05.40.−a I. INTRODUCTION Nonlinear systems driven by time-dependent forcings give rise to a wealth of intriguing phenomena, from chaos to the stochastic resonance phenomenon and the ratchet effect [1]. Given the complexity of the systems, a complete analysis of these phenomena is usually only possible via a numerical treatment, which is inherently model dependent. A more general analysis, less model dependent, is usually possible only for the stationary state of the system, in the presence of system symmetries. These approaches thus do not easily allow the determination of the asymptotic scaling properties of the systems, whose identification is essential for the understanding of unique features of nonlinear systems, as the recently highlighted sub-Fourier dynamics which may open the possibility of sub-Fourier information processing [2,3]. In this work, we study theoretically the time-dependent behavior of a generic nonlinear system driven by a twofrequency forcing. To specify the notation, we refer to a singleparticle model, and assume that the average particle velocity is the quantity of interest for the study of the response of the system. This is for example the case relevant to the study of directed motion in driven systems. By using a nonperturbative approach, we are able to derive an asymptotic expression, valid in the long-time limit, for the average particle velocity. This expression allows us to identify several universal features of the asymptotic response of the system, which are independent of the details of the model. Additional general features can also be identified depending on the specific symmetry properties of the system. This paper is organized as follows. In Sec. II, we introduce the system of interest and state the problem under consideration. In Sec. III, we establish some general asymptotic properties of the nonlinear system, valid in the long-time limit. In Sec. IV, we illustrate our theoretical results with *[email protected] †[email protected] ‡[email protected] the help of a one-dimensional model consisting of a Brownian particle, moving in a periodic potential, under the influence of a biharmonic force. We also investigate how the specific symmetries of the evolution equation affect our asymptotic results. Finally, in Sec. V, we present conclusions for the main findings of our work. II. STATEMENT OF THE PROBLEM Let us consider a generic single-particle model in which the time evolution of the particle’s position vector r(t)is described by a differential equation, which may be stochastic or deterministic depending on the presence or absence of noise sources. From the solution of this differential equation with an appropriate set of initial conditions at an arbitrary time t0, one can determine the particle’s position vector at any later time tf>t 0. The average velocity between these two times is then defined as ¯ vt0,tf=1 Tetf t0 dt ˙ r(t)=r(tf)−r(t0) Te ,(1) where Te=tf−t0is the elapsed time, the overdot represents time derivative, and ...the ensemble average with respect to the noise, if present. The infinite-time average velocity is obtained by taking the limit ¯ V=lim Te→+∞ ¯ vt0,t0+Te.(2) In most dissipative systems this limit exists and is independent of the initial time t0and of the initial conditions. In simulations, or real experiments, the variable r(t)is known only in a finite time-interval. Thus, the limit in Eq. (2) can be calculated only approximately by choosing a sufficiently large value for the elapsed time Te.Tobemore precise, Temust be chosen to be much larger than all other relevant time scales in the problem. In particular, if the system is subjected to a periodic time-dependent forcing function with period T, then a necessary condition is that TeT. The main focus of the present work is to determine scaling properties with the finite-time Te. Our results will be applicable 1539-3755/2013/88(6)/062919(5) 062919-1 ©2013 American Physical Society CASADO-PASCUAL, CUBERO, AND RENZONI PHYSICAL REVIEW E 88, 062919 (2013) to any system, with a well-defined infinite-time average (2), containing time-dependent forcing functions of the form F(t)=(ω1t,ω2t).(3) In the above expression, (θ1,θ2) is a vector function [4] periodic with respect to both arguments with period 2π[i.e., (θ1+2π,θ2)=(θ1,θ2+2π)=(θ1,θ2)∀(θ1,θ2)∈R2], and ω1and ω2are two positive parameters with the dimensions of frequency. Bifrequency forcing functions of this type have been extensively used in the literature (e.g., [3,5,6]). If ω1 and ω2are commensurate, i.e., ω2=pω1/q with pand q coprimes, F(t) is periodic of period T=2πq/ω1. To be more specific, we consider the following situation: one of the frequencies, say ω1, is kept fixed, while the other frequency, ω2, is scanned around a fixed value, ω2,0, which is commensurate with ω1. This corresponds to quantumoptical setups of recent experiments [3]. We will indicate the ratio between ω2,0and ω1with ω2,0/ω1=p0/q0, with p0 and q0coprimes. We aim to derive the general asymptotic properties of the function ¯ vt0,t0+Te(ω2) in an interval around ω2,0. As a first step, we analyze to which extent the infinitetime average ¯ V(ω2) can be approximated by the finite-time average ¯ vt0,t0+Te(ω2). This analysis will reveal a general scaling property which will allow us to derive the asymptotic limit. As discussed above, a necessary condition to approximate the infinite-time limit ¯ V(ω2,0) by the finite-time average ¯ vt0,t0+Te(ω2,0)isthatTeT0=2πq0/ω1; i.e., the elapsed time should be significantly larger than the period of the driving. However, as we now show, no matter how large Teis chosen, there always exists a frequency interval centered at ω2,0within which it is not appropriate to approximate the average velocity ¯ V(ω2) by the finite-time average ¯ vt0,t0+Te(ω2), except for the value ω2=ω2,0itself. Indeed, let us consider the frequency interval defined by the inequality |ω2−ω2,0|<2π/(q0Te), and choose any value ω2,1= ω2,0 in this interval of the form ω2,1=p1ω1/q1, with p1and q1 being coprime integers. Taking into account that there are infinitely many rational numbers within any real interval of nonzero length, the number of possible choices for ω2,1is infinity. Since |p1q0−p0q1|⩾1, it follows that ω1/(q0q1)⩽ |ω2,1−ω2,0|<2π/(q0Te). Thus, T1=2πq1/ω1>T eand, consequently, for the chosen value of Te, it is not appropriate to approximate ¯ V(ω2,1)by¯ vt0,t0+Te(ω2,1). The same conclusion can be drawn if ω2,1/ω1is irrational, since any irrational number can be approximated to any degree of accuracy by rational numbers. The above result shows that, given an averaging time Te, there is a frequency interval of length 4π/(q0Te) centered at ω2,0in which the behavior of the infinite-time limit ¯ V(ω2) cannot be inferred from that of the finite-time average ¯ vt0,t0+Te(ω2). It is precisely the scaling with Teof the length of such an interval which allows us to derive the asymptotic behavior of the function ¯ vt0,t0+Te(ω2), the quantity that is directly measurable in a simulation or experiment, within the above-mentioned range of ω2values. III. DERIVATION OF THE ASYMPTOTIC PROPERTIES In this section, we analyze the asymptotic behavior of the average velocity ¯ vt0,t0+Te(ω2)forω2values such that |ω2−ω2,0|is of the same order of magnitude as 2π/(q0Te), and sufficiently large values of Te. To this purpose, we make a change of variables from the variable δω2=ω2−ω2,0 to a dimensionless variable ˜ω2, which is expressed in terms of δω2according to ˜ω2=q0δω2Te/(2π). The average velocity as a function of this new variable will be given by ¯ νt0,t0+Te ω2,0(˜ω2)=¯ vt0,t0+Te[ω2,0+2π˜ω2/(q0Te)], where the subscript ω2,0indicates the midpoint of the region under study. In order to enforce the correct scaling derived in the previous section, we will consider the limit as Te→+∞and δω2→0, so that ˜ω2is held constant. Then, the leading-order of the aforesaid asymptotic behavior can be obtained from the limit ¯ Vω2,0(˜ω2)=lim Te→+∞ ¯ νt0,t0+Te ω2,0(˜ω2),(4) while keeping ˜ω2fixed. As a fact useful in the following, we notice that the time-dependent forcing function in Eq. (3) can be rewritten in the form F(t)=[ω1t,ω2,0t+ϕ(t)],(5) where ϕ(t)=δω2t=2π˜ω2t/(q0Te). In order to find a more explicit expression for the asymptotic result ¯ Vω2,0(˜ω2), it is convenient to introduce an auxiliary function of Te,N(Te), with the following three properties [7]: (i) N(Te) is dimensionless, integer, and positive for all Te>0, (ii) limTe→+∞ N(Te)=+∞, and (iii) limTe→+∞ N(Te)/Te= 0. By using this function, one can divide the interval [t0,tf] into N(Te) subintervals of equal length τ(Te)=Te/N(Te). Then, taking into account the definition in Eq. (1), it is possible to rewrite Eq. (4)as ¯ Vω2,0(˜ω2)=lim Te→+∞ τ(Te) Te N(Te)  j=1 ¯ ντj−1,τj ω2,0(˜ω2),(6) where τj=τj(Te)=t0+jτ(Te). We now use two properties that follow from the definition of the function N(Te). First, from (iii) it follows that limTe→+∞ τ(Te)=+∞, i.e., the length of the abovementioned subintervals tends to infinity as Te→+∞. Second, from (ii) it follows that limTe→+∞ ϕ(Te)=0, where ϕ(Te)=2π˜ω2τ(Te)/(q0Te) is the increment in ϕ(t) corresponding to an increment τ(Te)int. In the calculation of this last limit, the variable ˜ω2is held constant. Thus, within each of these subintervals the function ϕ(t)inEq.(5) can be approximated by a constant (for instance, by its value at the left end point of the subinterval). This last approximation becomes exact in the limit Te→+∞. From these two properties and Eq. (2), it follows that ¯ Vω2,0(˜ω2)=q0 2π˜ω2 lim Te→+∞ N(Te)  j=1 ¯ V(ω2,0,ϕj−1)ϕ, (7) where ϕ =ϕ(Te), ϕj=ϕj(Te)=ϕ[τj(Te)] = 2π˜ω2t0/(q0Te)+jϕ(Te), and ¯ V(ω2,0,ϕj−1)isthe infinite-time average velocity corresponding to a forcing function of the form (5) but with a time-independent phase shift equal to ϕj−1. Finally, taking into account that the sum in Eq. (7) is a Riemann sum, as well as that limTe→+∞ ϕ0(Te)=0 062919-2 UNIVERSAL ASYMPTOTIC BEHAVIOR IN NONLINEAR . . . PHYSICAL REVIEW E 88, 062919 (2013) and limTe→+∞ ϕN(Te)(Te)=2π˜ω2/q0, one obtains ¯ Vω2,0(˜ω2)=q0 2π˜ω22π˜ω2/q0 0 dϕ ¯ V(ω2,0,ϕ).(8) This expression is one of the central results of the present work. It shows that the asymptotic behavior ¯ Vω2,0(˜ω2)is completely determined by ¯ V(ω2,0,ϕ), that is, by the infinitetime average velocity corresponding to a forcing function F0(t,ϕ)=(ω1t,ω2,0t+ϕ), considered as a function of the time-independent phase shift ϕ. Equation (8) allows us to draw conclusions about the behavior of the asymptotic result ¯ Vω2,0(˜ω2) from the specific features of the function ¯ V(ω2,0,ϕ). For instance, it is easy to show that, under quite general assumptions, ¯ V(ω2,0,ϕ)isa periodic function in ϕwith period 2π/q0; i.e., ¯ V(ω2,0,ϕ)=¯ V(ω2,0,ϕ +2π/q0)(9) for all ϕ. Indeed, since p0and q0are coprime, from the B´ ezout identity [8] it follows that there exist integers land nsuch that lp0+nq0=1. From this equality, one can readily check that F0(t+2πl/ω1,ϕ)=F0(t,ϕ +2π/q0). Therefore, by assuming that all the time-dependent noise sources (if any) can be described by stationary stochastic processes, one has that ¯ vt 0,t 0+Te(ω2,0,ϕ)=¯ vt0,t0+Te(ω2,0,ϕ +2π/q0), with t 0=t0+2πl/ω1. Equation (9) then follows from the independence of the limit in Eq. (2) on the initial conditions. Notice that, according to Eq. (8), the average of ¯ V(ω2,0,ϕ) over a period 2π/q0is simply given by ¯ Vω2,0(1). Taking into account Eqs. (8) and (9), it is easy to verify that ¯ Vω2,0(k)=¯ Vω2,0(1) (10) for any nonzero integer k. Similarly, one can check that ¯ Vω2,0(−k/2)−¯ Vω2,0(1) =¯ Vω2,0(1) −¯ Vω2,0(k/2)(11) for any odd integer k. Equations (10) and (11) reveal two remarkable properties of the asymptotic result ¯ Vω2,0(˜ω2) which follow directly from the periodicity of the infinite-time average velocity expressed in Eq. (9). In order to gain some insight into the qualitative features of the function ¯ Vω2,0(˜ω2), let us consider first its behavior in the vicinity of the point ˜ω2=0. From Eqs. (4) and (2), it is clear that ¯ Vω2,0(0) =¯ V(ω2,0,0). Furthermore, by applying L’Hˆ opital’s rule, one can easily show from Eq. (8) that lim˜ω2→0¯ Vω2,0(˜ω2)=¯ V(ω2,0,0). Thus, ¯ Vω2,0(˜ω2)is continuous at ˜ω2=0 and approaches the value ¯ V(ω2,0,0) as ˜ω2tends to zero. To understand what happens when |˜ω2|increases, it is convenient to introduce the function ¯ Dω2,0(˜ω2)=2π˜ω2[¯ Vω2,0(˜ω2)−¯ Vω2,0(1)]/q0, so that ¯ Vω2,0(˜ω2)=¯ Vω2,0(1) +q0¯ Dω2,0(˜ω2) 2π˜ω2 .(12) From the definition of ¯ Dω2,0(˜ω2) and Eq. (8), it is straightforward to see that ¯ Dω2,0(˜ω2)=2π˜ω2/q0 0dϕ [¯ V(ω2,0,ϕ)− ¯ Vω2,0(1)]. Therefore, given that ¯ V(ω2,0,ϕ) is a periodic function in ϕwith period 2π/q0and average value ¯ Vω2,0(1), it follows that ¯ Dω2,0(˜ω2) is a periodic function of ˜ω2with period 1. Consequently, according to Eq. (12), the asymptotic result ¯ Vω2,0(˜ω2) approaches the value ¯ Vω2,0(1) in an oscillatory manner as |˜ω2|increases, i.e., as ω2moves away from the value ω2,0. The result given in Eq. (10) is a further indication of this oscillatory behavior. IV. MODEL-SPECIFIC EXAMPLES To illustrate our results, we will use a one-dimensional model consisting of a Brownian particle, with mass mand position x, moving in a periodic potential U(x), with period λ, under the influence of a biharmonic force of the form F(t)=A[cos(ω1t)+cos(ω2t)]. The dynamics of this system is described by the stochastic differential equation m¨ x(t)=−α˙ x(t)−U[x(t)]+F(t)+√2ξ(t),(13) where U(x) is the derivative with respect to xof U(x), α is the friction coefficient, ξ(t) is a Gaussian white noise of zero mean and autocorrelation ξ(t)ξ(s)=δ(t−s), and is the noise strength. The periodic potential U(x) can always be expressed as U(x)=U0(2πx/λ)/2, where U0is a constant with the dimensions of energy and (θ) is a dimensionless periodic function with period 2π. By expressing Eq. (13)in terms of the dimensionless quantities ˆ t=ω1tand ˆ x=πx/λ, the model is completely determined by five dimensionless parameters, namely, ˆ m=mω 1/α,ˆ U0=π2U0/(λ2αω 1), ˆ A= πA/(λαω 1), ˆ =π2/(λ2α2ω1), and ˆω2=ω2/ω1. In Fig. 1we depict the dependence of the dimensionless average velocity ¯ν0,Te ω2,0(˜ω2)/v0on the variable ˜ω2, where we have introduced the typical velocity v0=λω 1/π. Since the model is one-dimensional, the boldface vector notation has been dropped. For the sake of simplicity, in this first figure we have considered the overdamped deterministic case ( ˆ m=ˆ = 0). Furthermore, in order to avoid the presence of symmetries -4 -3 -2 -1 0 1 2 3 4-0.2 -0.18 -0.16 -0.14 -0.12 -0.1 -0.08 -0.06 -0.04 -0.02 0 0 3 2 -0.15 -0.1 -0.05 0 Δ˜ω2 ¯ν0,T e ω 2,0 (Δ˜ω2)/v0 ϕ ¯ V(ω 2,0 ,ϕ)/v 0 FIG. 1. (Color online) Dependence of the dimensionless average velocity ¯ν0,Te ω2,0(˜ω2)/v0on the dimensionless variable ˜ω2 for the model given by Eq. (13), with U(x)=U0[cos(2πx/λ)+ 1/4cos(4πx/λ +π/3)]/2. The value ˜ω2=0 corresponds to ˆω2= ˆω2,0=p0/q0, with p0=2andq0=3. The other parameters are ˆ m= ˆ =0, ˆ U0=ˆ A=1, Te=104ω−1 1(circles), and Te=105ω−1 1(red crosses). The solid line indicates the asymptotic result ¯ Vω2,0(˜ω2)/v0 obtained by numerically evaluating the integral in Eq. (8). The dotted line shows the value ¯ Vω2,0(1)/v0≈−0.0633. The inset depicts the dimensionless infinite-time average velocity ¯ V(ω2,0,ϕ)/v0as a function of the phase shift ϕ. The infinite-time limit has been approximately calculated by evaluating the average velocity ¯v0,Te(ω2,0,ϕ) for Te=105ω−1 1. 062919-3 CASADO-PASCUAL, CUBERO, AND RENZONI PHYSICAL REVIEW E 88, 062919 (2013) which are discussed later, we have chosen a highly asymmetric periodic potential (see figure caption for details). The results shown in Fig. 1clearly demonstrate that, for the elapsed time values considered (i.e., for Te=104ω−1 1and Te=105ω−1 1), the average velocity ¯ν0,Te ω2,0(˜ω2) has essentially reached its asymptotic limit ¯ Vω2,0(˜ω2). Consequently, ¯ν0,Te ω2,0(˜ω2)displays the typical oscillatory behavior predicted by Eq. (12), and satisfies the conditions imposed by Eqs. (10) and (11). It is to be noted that, in this figure, ¯ Vω2,0(0) =¯ V(ω2,0,0) =λ/T0, where T0=2πq0/ω1=6π/ω1is the period of the biharmonic force (see Ref. [9] for an explanation of this fact). We return now to the general case to investigate how the specific symmetries of the evolution equation affect the shape of ¯ Vω2,0(˜ω2). As a first example, let us assume that the differential equation describing the evolution of r(t) in the presence of the forcing function F0(t,ϕ)is invariant under the transformation S1:{r,˙ r,¨ r,t,ϕ}→{r1− r,−˙ r,−¨ r,t +t1,ϕ−π/q0}, where r1and t1are constant parameters. Then, if r(t) is a solution of this differential equation corresponding to a given value ϕof the phase shift, R(t)=r1−r(t+t1) is also a solution corresponding to the displaced value ϕ+π/q0. Since ˙ R(t)=−˙ r(t+t1)for all t, it follows from Eq. (2) that ¯ V(ω2,0,ϕ)=−¯ V(ω2,0,ϕ +π/q0) for all ϕ. Thus, from Eq. (8), it is clear that ¯ Vω2,0(1) =0. Consequently, in this case, Eqs. (10) and (11) can be rewritten, respectively, as ¯ Vω2,0(k)=0 for any nonzero integer k, and ¯ Vω2,0(−k/2) =−¯ Vω2,0(k/2) for any odd integer k. According to the first of these last two equations it follows that, assuming that there exists a directed motion for ω2=ω2,0, this current can be suppressed by slightly shifting the frequency ω2from the value ω2,0to the nearby values ω2,0±2π/(q0Te). This offers the possibility to discriminate two neighboring values of the frequency ω2differing in a quantity 2π/(q0Te), which is q0times smaller than the usually assumed Fourier width 2π/Te[2]. As a second example, let us assume now that the aforesaid differential equation is invariant under the transformation S2:{r,˙ r,¨ r,t,ϕ}→{r2−r,˙ r,−¨ r,t 2−t, −ϕ}, with r2and t2being constants. Then, if r(t)isasolution corresponding to a given value ϕof the phase shift, R(t)=r2−r(t2−t) is also a solution corresponding to the opposite value −ϕ. Thus, following a similar reasoning as above, it is easy to show that ¯ V(ω2,0,− ϕ)=¯ V(ω2,0,ϕ) for all ϕ. Consequently, from Eq. (8) it readily follows that ¯ Vω2,0(˜ω2)isanevenfunctionof˜ω2; i.e., ¯ Vω2,0(−˜ω2)=¯ Vω2,0(˜ω2) for all ˜ω2. In this case, Eq. (11) can be rewritten in the form ¯ Vω2,0(k/2) =¯ Vω2,0(1) for any odd integer kor, if the differential equation is also invariant under the transformation S1,intheform ¯ Vω2,0(k/2) =0. In Fig. 2we depict again the dependence of the dimensionless average velocity ¯ν0,Te ω2,0(˜ω2)/v0on the variable ˜ω2for the same model as in Fig. 1, given by Eq. (13), but with a symmetric potential and a nonzero noise strength (see figure caption for details). We consider both the overdamped case (top panel) and the underdamped case (bottom panel). In both cases, it is easy to see that the corresponding evolution equations are invariant under the one-dimensional version of the transformation S1with, e.g., t1=−π/ω1and x1= 0[10]. Consequently, in both panels ¯ Vω2,0(1) =0(seethe -4 -3 -2 -1 0 1 2 3 4-0.08 -0.06 -0.04 -0.02 0 0.02 -0.08 -0.06 -0.04 -0.02 0 0.02 0 -0.1 0 0.1 0 -0.1 0 0.1 Δ˜ω2 ¯ν0,Te ω2,0(Δ˜ω2)/v0 ¯ν0,Te ω2,0(Δ˜ω2)/v0 ϕ ϕ ¯ V(ω2,0,ϕ)/v0 ¯ V(ω2,0,ϕ)/v0 FIG. 2. (Color online) The same as in Fig. 1but for a symmetric potential U(x)=U0cos(2πx/λ)/2, a dimensionless noise strength ˆ =0.05, and two values of the dimensionless mass, namely, ˆ m=0 (top panel) and ˆ m=1 (bottom panel). The rest of the parameters are the same as in Fig. 1. The dashed lines indicate the results obtained fromEqs.(14) (insets) and (15) (main figures). The model-dependent parameters are χ=π/3and ¯ V(ω2,0,χ)/v0=0.073 (top panel), and χ=0.963 and ¯ V(ω2,0,χ)/v0=0.067 (bottom panel). These dashed lines are indistinguishable from the corresponding solid lines. The dotted lines show the value ¯ Vω2,0(1)/v0=0. dotted lines). However, only in the case of the overdamped dynamics, there exist values for t2and x2such that the evolution equation is invariant under the one-dimensional version of the transformation S2, e.g., t2=0 and x2=λ/2. Thus, only the top panel shows the features associated with this transformation. At this point, it is important to remember that, in many onedimensional systems driven by biharmonic forces, ¯ V(ω2,0,ϕ) can be expressed approximately as ¯ V(ω2,0,ϕ)=¯ V(ω2,0,χ) cos [q0(ϕ−χ)],(14) where χis the value of ϕin the interval [0,2π/q0)forwhich ¯ V(ω2,0,ϕ) attains its maximum value ¯ V(ω2,0,χ). As shown in Ref. [11], the functional form in Eq. (14) is a consequence solely of the system symmetries, being independent of the interaction details. By introducing Eq. (14)inEq.(8), one obtains ¯ Vω2,0(˜ω2)=¯ V(ω2,0,χ)sin (2π˜ω2−q0χ)+sin(q0χ) 2π˜ω2 . (15) This expression is, in fact, a special case of Eq. (12) where ¯ Vω2,0(1) =0 and ¯ Dω2,0(˜ω2)=¯ V(ω2,0,χ)[sin(2π˜ω2− q0χ)+sin(q0χ)]/q0. The significance of Eq. (15) lies in the fact that it is an explicit expression for the asymptotic result ¯ Vω2,0(˜ω2) with only two model-dependent parameters, namely, χand ¯ V(ω2,0,χ). Particular cases of the sub-Fourier scaling with 2π/(q0Te) evidenced by Eq. (15) have been proposed in the literature based on heuristic arguments [3,12], as well as observed in cold-atom experiments [3]. The results obtained from Eqs. (14) and (15), with an appropriate choice of the 062919-4 UNIVERSAL ASYMPTOTIC BEHAVIOR IN NONLINEAR . . . PHYSICAL REVIEW E 88, 062919 (2013) model parameters, are indicated in Fig. 2with dashed lines. The agreement is so good that it is nearly impossible to distinguish between the solid and the dashed lines. By contrast, a glance at the inset of Fig. 1reveals that it is not reasonable to approximate that curve by a function of the form given by Eq. (14). V. CONCLUSIONS In conclusion, we determined a universal asymptotic behavior of the average of a variable over a finite time interval. Our main results, expressed by Eqs. (8)–(12), are applicable to any system driven by a bifrequency forcing function and with a well-defined infinite-time average, independent of the initial conditions. With the help of Eq. (8) and some symmetry considerations, we have obtained some qualitative and quantitative information about the aforesaid asymptotic behavior. In particular, we have derived an asymptotic expression for the width of the resonance observed by keeping one frequency fixed, and varying the other one. We have showed that this width is smaller than the one predicted by the Fourier inequality [2] by a factor determined by the two driving frequencies, and independent of the model system parameters. The nonperturbative approach presented here to determine the asymptotic scaling finds direct application in the identification of nonlinear systems displaying sub-Fourier resonances [2,3]. ACKNOWLEDGMENTS We acknowledge financial support from the Leverhulme Trust and the Ministerio de Ciencia e Innovaci´ on of Spain FIS2008-02873 (J.C.-P. and D.C.). [1] L. Gammaitoni, P. H¨ anggi, P. Jung, and F. Marchesoni, Rev. Mod. Phys. 70,223 (1998); R. D. Astumian and P. H¨ anggi, Phys. Today 55(11),33 (2002); P. Reimann, Phys. Rep. 361, 57 (2002); P. H¨ anggi, F. Marchesoni, and F. Nori, Ann. Phys. (Berlin) 14,51 (2005); P. H¨ anggi and F. Marchesoni, Rev. Mod. Phys. 81,387 (2009). [2] P. Szriftgiser, J. Ringot, D. Delande, and J. C. Garreau, Phys. Rev. Lett. 89,224101 (2002). [3] R. Gommers, M. Brown, and F. Renzoni, Phys. Rev. A 75, 053406 (2007). [4] The number of components of the vector (θ1,θ2) may not coincide with the spatial dimension of the model. For instance, in the one-dimensional system considered in Ref. [6] it has two components. [5] S. Flach, O. Yevtushenko, and Y. Zolotaryuk, Phys.Rev.Lett. 84,2358 (2000); O. Yevtushenko, S. Flach, Y. Zolotaryuk, and A. A. Ovchinnikov, Europhys. Lett. 54,141 (2001); M. Schiavoni, L. Sanchez-Palencia, F. Renzoni, and G. Grynberg, Phys.Rev.Lett.90,094101 (2003); P. H. Jones, M. Goonasekera, and F. Renzoni, ibid. 93,073904 (2004); R. Gommers, P. Douglas, S. Bergamini, M. Goonasekera, P. H. Jones, and F. Renzoni, ibid. 94,143001 (2005); R. Gommers, S. Bergamini, and F. Renzoni, ibid. 95,073003 (2005); R. Gommers, S. Denisov, and F. Renzoni, ibid. 96, 240604 (2006); A. Wickenbrock, P. C. Holz, N. A. Abdul Wahab, P. Phoonthong, D. Cubero, and F. Renzoni, ibid. 108,020603 (2012). [6] R. Gommers, V. Lebedev, M. Brown, and F. Renzoni, Phys. Rev. Lett. 100,040603 (2008). [7] An example of such a function would be N(Te)= √Te/T0, with zbeing the floor function of z, i.e., the greatest integer less than or equal to z. [8] G. A. Jones and J. M. Jones, Elementary Number Theory (Springer Verlag, London, 1998). [9] A. Ajdari, D. Mukamel, L. Peliti, and J. Prost, J. Phys. I (France) 4,1551 (1994). [10] Notice that the stochastic processes ξ(t), −ξ(t+t1), and ξ(t2− t) are completely equivalent. [11] N. R. Quintero, J. A. Cuesta, and R. Alvarez-Nodarse, Phys. Rev. E 81,030102(R) (2010). [12] D. Cubero and F. Renzoni, Phys. Rev. E 86,056201 (2012). 062919-5