PHYSICAL REVIEW E 81, 031105 ͑2010͒
1Õf noise from nonlinear stochastic differential equations
J. Ruseckas* and B. Kaulakys Institute of Theoretical Physics and Astronomy, Vilnius University, A. Goštauto 12, LT-01108 Vilnius, Lithuania ͑Received 20 October 2009; published 8 March 2010͒ We consider a class of nonlinear stochastic differential equations, giving the power-law behavior of the power spectral density in any desirably wide range of frequency. Such equations were obtained starting from the point process models of 1/ f noise. In this article the power-law behavior of spectrum is derived directly from the stochastic differential equations, without using the point process models. The analysis reveals that the power spectrum may be represented as a sum of the Lorentzian spectra. Such a derivation provides additional justification of equations, expands the class of equations generating 1/ f noise, and provides further insights into the origin of 1/ f noise.
DOI: 10.1103/PhysRevE.81.031105 PACS number͑s͒: 05.40.Ϫa, 72.70.ϩm, 89.75.Da
I. INTRODUCTION signals with 1/ f noise were obtained in Refs. ͓29,30͔͑see ͓ ͔͒ Power-law distributions of spectra of signals, including also recent papers 5,31 , starting from the point process / ͓ ͔ 1/ f noise ͑also known as 1/ f fluctuations, flicker noise, and model of 1 f noise 27,32–39 . pink noise͒, as well as scaling behavior in general, are ubiq- The purpose of this article is to derive the behavior of the uitous in physics and in many other fields, including natural power spectral density directly from the SDE, without using phenomena, human activities, traffics in computer networks, the point process model. Such a derivation offers additional and financial markets. This subject has been a hot research justification of the proposed SDE and provides further in- / topic for many decades ͑see, e.g., a bibliographic list of pa- sights into the origin of 1 f noise. pers by Li ͓1͔, and a short review in Scholarpedia ͓2͔͒. II. PROPOSED STOCHASTIC DIFFERENTIAL Despite the numerous models and theories proposed since EQUATIONS its discovery more than 80 years ago ͓3,4͔, the intrinsic ori- gin of 1/ f noise still remains an open question. There is no Starting from the point process model, proposed and ana- conventional picture of the phenomenon and the mechanism lyzed in Refs. ͓27,32–38͔, the nonlinear stochastic differen- leading to 1/ f fluctuations are not often clear. Most of the tial equations are derived ͓5,29,30͔. The general expression models and theories have restricted validity because of the for the SDE is assumptions specific to the problem under consideration. A / short categorization of the theories and models of 1 f noise dx = 2ͩ − ͪx2−1dt + xdW. ͑1͒ is presented in the introduction of the paper ͓5͔. 2 Until recently, probably the most general and common / Here, x is the signal, is the exponent of the multiplicative models, theories and explanations of 1 f noise have been based on some formal mathematical description such as frac- noise, defines the behavior of stationary probability distri- tional Brownian motion, the half-integral of the white noise, bution, and W is a standard Wiener process. SDE ͑1͒ has the simplest form of the multiplicative noise or some algorithms for generation of signals with scaled properties ͓6–14͔ and the popular modeling of 1/ f noise as term, x dW. Multiplicative equations with the drift coeffi- the superposition of independent elementary processes with cient proportional to the Stratonovich drift correction for the Lorentzian spectra and a proper distribution of relaxation transformation from the Stratonovich to the Itô stochastic / ͓ ͔ equation ͓40͔ generate signals with the power-law distribu- times, e.g., a 1 relax distribution 15–21 . The weakness of ͓ ͔ ͑ ͒ the latter approach is that the simulation of 1/ f noise with tions 5 . Equation 1 is of such type and has probability ͑ ͒ϳ − the desirable slope  requires finding the special distribu- distribution of the power-law form P x x . Because of the divergence of the power-law distribution and the requirement tions of parameters of the system under consideration; at ͑ ͒ least a wide range of relaxation time constants should be of the stationarity of the process, the SDE 1 should be assumed in order to allow correlation with experiments analyzed together with the appropriate restrictions of the dif- ͓22–28͔. fusion in some finite interval. For simplicity, in this article, Nonlinear stochastic differential equation with linear we will adopt reflective boundary conditions at x=xmin and noise and nonlinear drift, was considered in Ref. ͓9͔.Itwas x=xmax. However, other forms of restrictions are possible. For example, exponential restriction of the diffusion can be found that if the damping is decreasing with increase in the ͑ ͒ absolute value of the stochastic variable, then the solution of obtained by introducing additional terms in Eq. 1 , such a nonlinear stochastic differential equation ͑SDE͒ has m m m xmin m x long correlation time. Recently nonlinear SDEs generating dx = 2ͩ − + ͩ ͪ − ͩ ͪ ͪx2 −1dt + x dW. 2 2 x 2 xmax ͑2͒ *[email protected] Here, m is some parameter.
1539-3755/2010/81͑3͒/031105͑7͒ 031105-1 ©2010 The American Physical Society J. RUSECKAS AND B. KAULAKYS PHYSICAL REVIEW E 81, 031105 ͑2010͒
͑ ͒ ץ Equation 1 with the reflective boundary condition at xmin 1 S͑x,t͒ = ͩ − ͪx2−1P − x2P. ͑10͒ ץ and xmax can be rewritten in a form that does not contain 2 2 x parameters and xmin. Introducing the scaled stochastic vari- → / →2 2−2 able x x xmin and scaled time t xmin t one transforms At the reflective boundaries xmin=1 and xmax= the probabil- Eq. ͑1͒ to ity current S͑x,t͒ should vanish, and, therefore, the boundary conditions for Eq. ͑8͒ are 2−1 dx = ͩ − ͪx dt + x dW. ͑3͒ ͑ ͒ ͑ ͒ ͑ ͒ 2 S 1,t =0, S ,t =0. 11 The scaled Eq. ͑3͒ has a boundary at x=1 and at A. Eigenfunction expansion xmax = . ͑4͒ ͑ ͒ x We solve Eq. 8 using the method of eigenfunctions. An min ansatz of the form Further, we will consider Eq. ͑3͒ only. In order to obtain 1/ f ͑ ͒ ͑ ͒ −t ͑ ͒ noise we require that the region of diffusion of the stochastic P x,t = P x e 12 ӷ variable x should be large. Therefore, we assume that 1. leads to the equation 2ץ 1 ץ III. POWER SPECTRAL DENSITY FROM ͩ ͪ 2−1 2 ͑ ͒ ͑ ͒ THE FOKKER-PLANCK EQUATION − − x P + x P =− P x , 13 x2ץ x 2ץ 2 According to Wiener-Khintchine relations, the power ͑ ͒ Ն spectral density is where P x are the eigenfunctions and 0 are the corre- sponding eigenvalues. The eigenfunctions P͑x͒ obey the or- ϱ ϱ thonormality relation ͓42͔ S͑f͒ =2͵ C͑t͒eitdt =4͵ C͑t͒cos͑t͒dt, ͑5͒ −ϱ 0 ͵ ⌽͑x͒ ͑ ͒ ͑ ͒ ␦ ͑ ͒ e P x PЈ x dx = ,Ј, 14 where =2f and C͑t͒ is the autocorrelation function. For 1 the stationary process, the autocorrelation function can be ⌽͑ ͒ ͑ ͒ expressed as an average over realizations of the stochastic where x is the potential, associated with Eq. 8 , process, ⌽͑ ͒ ͑ ͒ ͑ ͒ x =−lnP0 x . 15 ͑ ͒ ͗ ͑ Ј͒ ͑ Ј ͒͘ ͑ ͒ C t = x t x t + t . 6 It should be noted that the restriction of diffusion of the This average can be written as variable x by xmin and xmax ensures that the eigenvalue spec- trum is discrete. Expansion of the transition probability den- ͑ ͒ ͵ ͵ Ј Ј ͑ ͒ ͑ Ј ͉ ͒ ͑ ͒ sity in a series of the eigenfunctions has the form ͓42͔ C t = dx dx xx P0 x Px x ,t x,0 , 7 ⌽͑x͒ −t P ͑xЈ,t͉x,0͒ = ͚ P͑xЈ͒e P͑x͒e . ͑16͒ ͑ ͒ x where P0 x is the steady-state probability distribution func- tion and P ͑xЈ,t͉x,0͒ is the transition probability ͑the condi- x ͑ ͒ ͑ ͒ tional probability that at time t the signal has value xЈ with Substituting Eq. 16 into Eq. 7 we get the autocorrelation the condition that at time t=0 the signal had the value x͒. function The transition probability can be obtained from the solution −t 2 C͑t͒ = ͚ e X. ͑17͒ of the Fokker-Planck equation with the initial condition ͑ ͉ ͒ ␦͑ ͒ Px xЈ,0 x,0 = xЈ−x . Therefore, for the calculation of the power spectral den- Here, sity of the signal x we will use the Fokker-Planck equation instead of stochastic differential Eq. ͑3͒. The Fokker-Planck ͵ ͑ ͒ ͑ ͒ X = xP x dx 18 equation corresponding to the Itô solution of Eq. ͑3͒ is 1 ͓41,42͔ is the first moment of the stochastic variable x evaluated with 2 -the -th eigenfunction P͑x͒. Such an expression for the au ץ 1 ץ ץ P =−ͩ − ͪ x2−1P + x2P. ͑8͒ x2 tocorrelation function has been obtained in Ref. ͓43͔. Usingץ x 2ץ t 2ץ Eqs. ͑5͒ and ͑17͒ we obtain the power spectral density The steady-state solution of Eq. ͑8͒ has the form ͑ ͒ 2 ͑ ͒ −1 − S f =4͚ 2 2 X. 19 x , 1, + 1−1− P ͑x͒ = ͑9͒ 0 Ά 1 · This expression for the power spectral density resembles the −1 x , =1. models of 1/ f noise using the sum of the Lorentzian spectra ln ͓15–18,27,28,44,45͔. Here, we see that the Lorentzians can The boundary conditions for Eq. ͑8͒ can be expressed using arise from the single nonlinear stochastic differential equa- the probability current ͓42͔ tion. Similar expression for the spectrum has been obtained
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ץ in Ref. ͓14͔ where reversible Markov chains on finite state 1 ͑−͒/͑−1͒ ͓ ͑ ͒ ͓ ͔ ␥ S͑z͒ = ͑ −1͒z u͑z͒. ͑24͒ zץ spaces were considered Eq. 34 in Ref. 14 with − k,m 2 playing the role of ͔. /  A pure 1 f power spectrum is physically impossible be- ͑ ͒ cause the total power would be infinity. It should be noted Therefore, the boundary conditions for Eq. 21 , according to ͑ ͒ Ј͑ ͒ Ј͑1−͒ Ј͑ ͒ ͑ ͒ /  Eq. 11 are u 1 =0 and u =0. Here, u z is the de- that the spectrum of signal x, obeying SDE 3 , has 1 f ͑ ͒ behavior only in some intermediate region of frequencies, rivative of the function u z . The orthonormality relation Ӷ Ӷ Ӷ ͑14͒ yields the orthonormality relation for functions u͑z͒, fmin f fmax, whereas for small frequencies f fmin the spectrum is bounded. The behavior of spectrum at frequen- 1− cies f Ӷ f Ӷ f is connected with the behavior of the au- 1−1− min max ͵ ͑−͒/͑1−͒ ͑ ͒ ͑ ͒ ␦ z u z u z dz = . tocorrelation function at times 1/ f ӶtӶ1/ f . Often ͑ ͒͑ ͒ Ј , Ј max min −1 1− 1 1/ f noise is described by a long-memory process, charac- terized by S͑f͒ϳ1/ f as f →0. An Abelian-Tauberian theo- ͑25͒ rem relating regularly varying tails shows that this long- range dependence property is equivalent to similar behavior The expression ͑18͒ for the first moment X of the stochastic of autocorrelation function C͑t͒ as t→ϱ ͓46͔. However, this variable x evaluated with the -th eigenfunction becomes behavior of the autocorrelation function is not necessary for obtaining required form of the power spectrum in a finite 1 1− ͵ 1+−/1− ͑ ͒ ͑ ͒ X = z u z dz. 26 interval of the frequencies which does not include zero ͓47–49͔. 1− 1 From Eq. ͑19͒, it follows that if the terms with small dominate the sum, then one obtains 1/ f2 behavior of the spectrum for large frequencies f. If the terms with 1/ f ͑with C. Solution of the equation for eigenfunctions Ͻ2͒ are present in Eq. ͑19͒, then at sufficiently large fre- The solutions of Eq. ͑21͒ are ͓50͔ quencies those terms will dominate over the terms with 1/ f2 behavior. Since the terms with small lead to 1/ f2 behavior  ͑ ͒ ␣͓ ͑ ͒ ͑ ͔͒ ͑ ͒ of the spectrum, we can expect to obtain 1/ f spectrum in a u z = z c1J␣ z + c2Y␣ z , 27 frequency region where the main contribution to the sum in Eqs. ͑17͒ and ͑19͒ is from the large values of . Thus we where J␣͑z͒ and Y␣͑z͒ are the Bessel functions of the first ͑ ͒ need to determine the behavior of the eigenfunctions P x and second kind, respectively. The coefficients c1 and c2 for large . The conditions when eigenvalue can be con- needs to be determined from the boundary and normalization sidered as large will be investigated below. conditions for function u͑z͒. The asymptotic expression for the function u͑z͒ is B. Eigenfunctions of the Fokker-Planck equation ␣−1/2 −1/2 u͑z͒Ϸcz cos͑z + a͒, z ӷ 1. ͑28͒ For 1, it is convenient to solve Eq. ͑13͒ by writing the eigenfunctions P͑x͒ in the form Here, a is a constant to be determined from the boundary − 1− conditions and c is the constant to be determined from the P͑x͒ = x u͑x ͒. ͑20͒ normalization ͑25͒. 1− The behavior of the power spectral density in Eq. ͑19͒ as The functions u͑z͒ with z=x obey the equation 1/ f can be only due to terms with large . Therefore, we will consider the values of for which at least the product d2 1 d ͑ ͒ ͑ ␣ ͒ ͑ ͒ 2 ͑ ͒ ͑ ͒ z is large, z ӷ1. The first moment of the variable x 2 u z − 2 −1 u z =− u z , 21 max max dz z dz in the expression for the autocorrelation function ͑17͒ is ex- pressed via integral ͑26͒. If the condition zӷ1 is satisfied ␣ where the coefficients and are for all z then the function u͑z͒ has frequent oscillations in ͱ all the region of the integration, and the integral is almost −1 2 zero. Consequently, the biggest contribution to the sum in ␣ =1+ , = . ͑22͒ 2͑1−͒ ͉ −1͉ Eq. ͑17͒ makes the terms corresponding to those values of , for which the condition zӷ1 is not satisfied for all values The area of diffusion of the variable z=x1− is restricted by of z between zmin and zmax. Therefore, we will restrict the values of by the condition z Ӷ1. Thus we will consider the minimum and maximum values zmin and zmax, min eigenvalues satisfying the conditions 1−, Ͼ 1, 1, Ͼ 1, ͭ ͮ ͭ ͮ ͑ ͒ zmin = zmax = 23 / Ӷ Ӷ / ͑ ͒ 1, Ͻ 1, 1− , Ͻ 1. 1 zmax 1 zmin. 29
−1 The probability current S͑x͒, Eq. ͑10͒, rewritten in terms of Explicitly, we have conditions 1ӶӶ if Ͼ1 and 1− functions u,is 1/ ӶӶ1ifϽ1.
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The derivative of the function u͑z͒, Eq. ͑27͒,is −3  =1+ ͑33͒ Ј͑ ͒ ␣͓ ͑ ͒ ͑ ͔͒ ͑ ͒ 2͑ −1͒ u z = z c1J␣−1 z + c2Y␣−1 z . 30 ␣Ͼ ␣Ͻ Since we consider the case z Ӷ1, then, using Eq. ͑30͒, and “+” sign is for 1, while “−” is for 1. min Ϯ␣ Ͼ Ӷ Ј͑ ͒ If + 0, taking into account that zmin 1, we can instead of the boundary condition u zmin =0 we can ap- proximately take the condition replace the lower limit of the integration by 0, X c 1 zmax −1 Ϸ  ͐ y JϮ␣͑y͒dy. We get that the integral in the ͓ ͑ ͒ ͑ ͔͒ ͉1−͉ 0 lim c1J␣−1 y + c2Y␣−1 y =0. → expression for X approximately does not depend on the y 0 lower limit of integration zmin. We can integrate the integral ␣Ͼ ␣Ͻ ͑␣͒ If 1, then we get c2 =0; if 1 then c2 =−c1 tan . by parts and use the properties of the Bessel functions to Using those values of the coefficient c2, we obtain the solu- obtain tions of Eq. ͑21͒ c ␣ −1 z X Ϸ ͩϯy JϮ͑␣ ͒͑y͉͒ max cz J ␣͑z͒, ␣ Ͻ 1,  −1 z ͑ ͒Ϸͭ − ͮ ͑ ͒ ͉1−͉ min u z 31 ␣ ͑ ͒ ␣ Ͼ cz J␣ z , 1. zmax ͑ ͒ Ϯ ͑ ␣ ͒͵ −2 ͑ ͒ ͑ ͒ + −2 y JϮ͑␣ ͒ y dyͪ. 34 From approximate solution 31 , using asymptotic expres- −1 sion for the Bessel functions, we can determine the param- zmin ͑ ͒ eter a in the asymptotic expression 28 . We obtain that the Using expression ͑31͒ for the function u͑z͒, the boundary parameter a depends on ␣ and does not depend on . Ј͑ ͒ Ј͑ ͒ conditions u zmin =0 and u zmax =0 leads to ͑ ͒ ͑ ͒ ͑ ͒ D. Normalization JϮ͑␣−1͒ zmin =0, JϮ͑␣−1͒ zmax =0. 35 Taking =Ј from Eq. ͑25͒ we get the normalization con- ͑ ͒ ͑ ͒ Therefore, the first term in the expression 34 for X is zero. dition. Using Eq. 31 , we have Ͻ 5 ӷ If 2 , taking into account that zmax 1, we can extend 1−1− 1− the upper limit of integration to +ϱ. We get that the integral c2 ͵ zJ2 ͑z͒dz ͑ ͒͑ ͒ Ϯ␣ for X approximately does not depend on the upper limit of −1 1− 1 integration zmax. 1−1− c2 zmax Therefore, the first moment X of the variable x is propor- ͵ 2 ͑ ͒ Ϸ = 2 yJϮ␣ y dy 1. tional to the expression ͑ −1͒ ͉1−͉ zmin c 1 Taking into account the condition z Ӷ1 and replacing the . ͑36͒ min ͉ ͉  lower limit of integration by 0, we obtain that the integral is 1− approximately equal to Now we are ready to estimate the power spectral density. zmax z ͵ yJ2 ͑y͒dy Ϸ max . Ϯ␣ 0 B. Power spectral density ӷ ӷ Ј͑ ͒ Here, we assumed that zmax 1. Therefore, the normaliza- Since zmax 1, from the boundary condition u zmax tion constant c is =0 using the asymptotic expression ͑28͒ for the function ͑ ͒ ͑ ͒ u z we obtain the condition sin zmax+a =0 and zmax ͱ͉1−͉ −1 =n−a. Then c Ϸ . ͑32͒ z 1−1− max ͑1−͒2 Ϸ ͑n − a͒2. ͑37͒ n 2z2 IV. CALCULATION OF THE POWER max SPECTRAL DENSITY Equation ͑37͒ shows that the density of eigenvalues D͑͒ is proportional to 1/ͱ. Since the parameter a does not depend A. Estimation of the first moment X of the stochastic variable on , it follows that the density of eigenvalues and, conse- x quently, the autocorrelation function do not depend on the The expression ͑17͒ for the autocorrelation function con- parameter a. tains the first moment X of the variable x, expressed as an In order to estimate the sum in expression ͑17͒ for the integral of the function u, Eq. ͑26͒. Using Eq. ͑31͒,weget autocorrelation function, we replace summation by the inte- 1− gration, c X Ϸ ͵ z−1J ͑z͒dz Ϯ␣ 1− 1 ͑ ͒Ϸ͵ −t 2 ͑͒ ͑ ͒ C t e XD d 38 c zmax −1 ͑ ͒ =  ͵ y JϮ␣ y dy. Such a replacement is valid when z ӷ1. Using the ap- ͉1−͉ max zmin proximate expressions ͑32͒ and ͑36͒, we get the expression Here, for the autocorrelation function
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102 100 1 ͑ ͒ 10 FIG. 1. Color online Probability distribution ͑ ͒͑ ͒ 10-4 100 function P x left and power spectral density ͑ ͒͑ ͒ 10-1 S f right for the stochastic process defined by P(x) 10-8 S(f) ͑ ͒ 10-2 the stochastic differential Eq. 44 . Dashed green ͑ ͒ -12 10-3 lines are analytical expression 9 for the steady- 10 ͑ ͒ 10-4 state distribution function P0 x on the left and -16 -5 the slope 1/ f on the right. Parameters used are 10 10 -1 0 1 2 3 4 -3 -2 -1 0 1 2 3 4 2 10 10 10 10 10 10 10 10 10 10 10 10 10 10 xmin=1, xmax=10 , and =1. (a) x (b) f
 z−2 − ,  Ͻ 2, ͑ ͒ϳ͵ min − −t ͑ ͒ϳͭ ͮ ͑ ͒ C t e d S f −2 43 −2 ,  Ն 2. zmax −1͓⌫͑  −2 ͒ ⌫͑  −2 ͔͒ ͑ ͒ The second term in the expansion is proportional to −2.In = t 1− ,zmaxt − 1− ,zmint . 39 the case of Ͻ2, the term with − becomes larger than the ⌫͑ ͒ ͐ϱ a−1 −t −2 −2 Ӷ /  Here, a,z = z t e dt is the incomplete Gamma function. term with when zmax . Therefore, we obtain 1 f 2 Ӷ Ӷ 2 ӶӶ2͑−1͒ Ͼ When zmin t zmax we have the following lowest powers in spectrum in the frequency interval 1 if 1 the expansion of the approximate expression ͑39͒ for the and the frequency interval 1/2͑1−͒ ӶӶ1ifϽ1. It autocorrelation function in the power series of t: should be noted that time t and frequency in our analysis are dimensionless. ͑ ͒ ͑ ͒ z2 −1 tz2 −2 Equation ͑41͒ shows that the shape of the power spectrum max − max ,  Ͼ 2  −1  −2 depends on the behavior of the eigenfunctions and the eigen- 2 2 −2 values in terms of the function XD͑͒. This function z + ͑␥ −1͒t + t ln͑z t͒,  =2   max max X2D͑͒ should be proportional to − in order to obtain 1/ f ͑ ͒ z2 −1 behavior. Similar condition has been obtained in Ref. ͓14͔. C͑t͒ϳ max + t−1⌫͑1−͒, 1 Ͻ  Ͻ 2 .  −1 Equation ͑13͒ for discrete time process and the unnumbered ͑ ͒ ␥ ͑ −2 ͒  equation after Eq. 34 for a continuous time process in Ref. Ά− −ln zmaxt , =1 ͓ ͔ 2 ͑͒ϳ− 14 are analogous to the condition XD since the 1 density of eigenvalues in Ref. ͓14͔ is proportional to ⌫͑1−͒,  Ͻ 1 t1− 1/͉␥Ј͑x͉͒. Equations ͑39͒ and ͑42͒ for the power spectral density and autocorrelation function resembles those ob- ͑40͒ tained from the sum of Lorentzian signals with appropriate ͓ ͔ Here, ␥Ϸ0.577216 is the Euler’s constant. Similar first weights in Ref. 27 . terms in the expansion of the autocorrelation function in the power series of time t has been obtained in Ref. ͓5͔. V. NUMERICAL EXAMPLES Similarly, when z ӷ1, replacing in Eq. ͑19͒ the sum- max  ͑ ͒ mation by the integration, we obtain the power spectral den- If =3, we get that =1 and stochastic differential Eq. 1 / sity should give signal exhibiting 1 f noise. We will solve nu- 5 Ͼ 1 Ͻ merically two cases: = 2 1 and =−2 1. For the nu- merical solution, we use Euler-Marujama approximation, 2 S͑f͒Ϸ4͵ XD͑͒d. ͑41͒ transforming differential equations to difference equations. 2 + 2 Equation ͑44͒ with =5/2 was solved using variable step of ͑ ͒ / Equation, similar to Eq. ͑41͒ has been obtained in Ref. ͓8͔ by integration, solution Eq. 45 with =−1 2 was performed considering a relaxing linear system driven by white noise using a fixed step of integration. 5 ͑ ͒ 2 4 ͓Eq. ͑27͒ in Ref. ͓8͔͔. Similar equation also has been ob- When = 2 and =3 then Eq. 1 is dx= x dt 5/2 tained in Ref. ͓14͔ where reversible Markov chains on finite +x dW. Using exponential restriction of the diffusion re- state spaces were considered. In both Refs. ͓8,14͔ the power gion, we have the equation spectral density is expressed as a sum or an integral over the 1 x 1 x eigenvalues of a matrix describing transitions in the system. dx = 2ͩ1+ min − ͪx4dt + x5/2dW. ͑44͒ Using the approximate expressions ͑32͒ and ͑36͒,weget 2 x 2 xmax the equation The equation was solved using the variable step of integra- ⌬ 2 / 3 Ӷ −2 tion, tk = xk, with 1 being a small parameter. The zmin 1 1 ͑ ͒ ͑ ͒ϳ͵ ͑ ͒ steady-state probability distribution function P0 x and the S f −1 2 2 d . 42 −2 + power spectral density S͑f͒ are presented in Fig. 1. We see a zmax good agreement of the numerical results with the analytical −2 ӶӶ −2 / When zmax zmin then the leading term in the expansion expressions. The 1 f interval in the power spectral density in Ϸ ϫ −1 of the approximate expression ͑42͒ for the power spectral Fig. 1 is approximately between fmin 2 10 and fmax density in the power series of is Ϸ2ϫ102. The width of this region is much narrower than
031105-5 J. RUSECKAS AND B. KAULAKYS PHYSICAL REVIEW E 81, 031105 ͑2010͒
107 0 6 FIG. 2. ͑Color online͒ Probability distribution 10 10 5 ͑ ͒͑ ͒ 10 function P x left and power spectral density -2 ͑ ͒͑ ͒ 10 104 S f right for the stochastic process defined by P(x) S(f) ͑ ͒ 103 the stochastic differential Eq. 45 . Dashed green 10-4 ͑ ͒ 102 lines are analytical expression 9 for the steady- 1 state distribution function P ͑x͒ on the left and 10-6 10 0 the slope 1/ f on the right. Parameters used are 100 -6 -5 -4 -3 -2 -1 2 1 10 100 10 10 10 10 10 10 xmin=1, xmax=10 , and =1. (a) x (b) f the width of the region 1Ӷ2f Ӷ106 ͑=102͒ predicted in multiplicative noise greater than 1 has been used. Here, we the previous section. showed that it is possible to obtain 1/ f noise from the same When =−1/2 and =3 then Eq. ͑1͒ is nonlinear SDE for Ͻ1, as well. The analysis reveals that the power spectrum may be represented as a sum of the 2 Lorentzian spectra with the coefficients proportional to the dx =−2 dt + dW. ͑45͒ x2 ͱx squared first moments of the stochastic variable evaluated with the appropriate eigenfunctions of the corresponding We used reflective boundary conditions at xmin=1 and xmax Fokker-Planck equation. Nonlinear SDE, corresponding to a =100. The equation was solved with a constant step of inte- particular case of Eq. ͑1͒ with =0, i.e., with linear noise gration. The steady-state probability distribution function and nonlinear drift, was considered in Ref. ͓9͔. It was found ͑ ͒ ͑ ͒ P0 x and the power spectral density S f are presented in that if the damping is decreasing with increase of ͉x͉, then the Fig. 2. The 1/ f interval in the power spectral density in Fig. solution of such a nonlinear SDE has long correlation time. Ϸ −6 Ϸ ϫ −4 2 is approximately between fmin 10 and fmax 2 10 . As Eq. ͑41͒ shows, the shape of the power spectrum de- The width of this region is much narrower than the width of pends on the behavior of the eigenfunctions and the eigen- −6Ӷ Ӷ ͑ 2͒ 2 the region 10 2 f 1 =10 predicted in the previous values in terms of the function XD͑͒, where D͑͒ is the section. density of eigenvalues. The SDE ͑3͒ considered in this article Numerical solution of the equations confirms the presence gives the density of eigenvalues D͑͒ proportional to 1/ͱ. of the frequency region for which the power spectral density One obtains 1/ f behavior of the power spectrum when this /  2 − has 1 f dependence. The width of this region can be in- function XD͑͒ is proportional to for a wide range of creased by increasing the ratio between minimum and maxi- eigenvalues , as is the case for SDE ͑3͒. Similar condition mum values of the stochastic variable x. In addition, the has been obtained in Ref. ͓14͔. region in the power spectral density with the power-law be- One of the reasons for the appearance of the 1/ f spec- havior depends on the exponent :if=1 then this width trum is the scaling property of the stochastic differential Eq. is zero; the width increases with increasing the difference ͑1͒: changing the stochastic variable from x to xЈ=ax ͉−1͉. However, the estimation of the width of the region, changes the time-scale of the equation to tЈ=a2͑1−͒t, leaving obtained in the previous section, is too broad, the width ob- the form of the equation invariant. From this property it fol- tained in numerical solutions is narrower. Such a discrepancy lows that it is possible to eliminate the eigenvalue in Eq. can be explained as the result of various approximations, ͑13͒ by changing the variable from x to z=1/2͑1−͒x. The made in the derivation. dependence of the eigenfunction on eigenvalue then enters only via the boundary conditions. Such scaling properties VI. DISCUSSION were used estimating the norm of the eigenfunction and the first moment X of the stochastic variable x evaluated with In summary, we derived the behavior of the power spec- the -th eigenfunction. Other factor in obtaining the power- tral density from the nonlinear stochastic differential equa- law spectrum is wide range of the region of diffusion of the tion. In Refs. ͓29,30͔ only the values of the exponent of the stochastic variable x.
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