Reduced fractal model for quantitative analysis of averaged micromotions in mesoscale: Characterization of blow-like signals

Chaos, Solitons & Fractals 76 (2015) 166–181

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Chaos, Solitons & Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos

Reduced fractal model for quantitative analysis of averaged micromotions in mesoscale: Characterization of blow-like signals Raoul R. Nigmatullin a, Vyacheslav A. Toboev b, Paolo Lino c, Guido Maione c,⇑ a Radioelectronic and Informative Measurements Techniques Department, Kazan National Research Technical University (KNRTU-KAI), 10 Karl Marx str., 420011 Kazan, Tatarstan, Russian Federation b Department of Mathematics, Chuvash State University, Cheboksary, Russian Federation c Department of Electrical and Information Engineering (DEI), Politecnico di Bari, Via E. Orabona, 4, Bari, Italy

a r t i c l e

i n f o

Article history: Received 4 February 2015 Accepted 26 March 2015

a b s t r a c t It has been shown that many micromotions in the mesoscale region are averaged in accordance with their self-similar (geometrical/dynamical) structure. This distinctive feature helps to reduce a wide set of different micromotions describing relaxation/exchange processes to an averaged collective motion, expressed mathematically in a rather general form. This reduction opens new perspectives in description of different blow-like signals (BLS) in many complex systems. The main characteristic of these signals is a finite duration also when the generalized reduced function is used for their quantitative fitting. As an example, we describe quantitatively available signals that are generated by bronchial asthmatic people, songs by queen bees, and car engine valves operating in the idling regime. We develop a special treatment procedure based on the eigen-coordinates (ECs) method that allows to justify the generalized reduced fractal model (RFM) for description of BLS that can propagate in different complex systems. The obtained describing function is based on the self-similar properties of the different considered micromotions. This kind of cooperative model is proposed here for the first time. In spite of the fact that the nature of the dynamic processes that take place in fractal structure on a mesoscale level is not well understood, the parameters of the RFM fitting function can be used for construction of calibration curves, affected by various external/random factors. Then, the calculated set of the fitting parameters of these calibration curves can characterize BLS of different complex systems affected by those factors. Though the method to construct and analyze the calibration curves goes beyond the scope of this paper, this result could benefit future studies that will employ the developed reduced models in diagnosis, prevention, and control of unpredicted and undesired phenomena of some engineering applications that possibly exhibit such BLS. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction Abbreviations: AFR, amplitude–frequency response; BLR, basic linear relationship; BLS, blow-like signal(s); ECs, eigen-coordinates; LLSM, linear least square method; OMA, one-mode approximation; RFM, reduced fractal model. ⇑ Corresponding author. Tel.: +39 080 5963 247; fax: +39 080 5963 410. E-mail address: [email protected] (G. Maione). http://dx.doi.org/10.1016/j.chaos.2015.03.022 0960-0779/Ó 2015 Elsevier Ltd. All rights reserved.

By blow-like signal (BLS) we denote the response of a complex system that provides a signal having finite duration in time or space, starting from zero, achieving maximal/minimal values during a finite time interval, and in the end tending to zero again. Fig. 1(a)–(d) depict typical

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167

Fig. 1. (a) Blow-like fragment of an acoustic signal recorded from a patient suffering from bronchial asthma. As initial treatment, we applied the two steps described in the beginning of Section 4. The ‘‘up’’ and ‘‘down’’ envelopes of this signal are marked by yellow color. They are obtained in accordance with the criterion (20). (b) Blow-like fragment of the generated acoustic signal with finite duration corresponding to the queen bee song. All song consists of 10–15 fragments having approximately the same length as depicted in this figure. For treatment, it is convenient to use arbitrary time units (representing 3500 data points decreased to 1000 data points). The amplitudes of the corresponding BLS are given also in arbitrary units. (c) Fragment of the initial signal from car valves. The spikes correspond to the valve knocks. After elimination of the trend by means of procedure of the optimal linear smoothing (POLS), we obtain the blow-like signals shown in Fig. 4(c). (d) Blow-like signal recorded for an earthquake of small intensity. The ‘‘up’’ and ‘‘down’’ envelopes are marked by cyan color. The correlation coefficient between them achieves only the value 0.7. The absolute values of duration and amplitudes of this earthquake are not essential and so they are given in some arbitrary units. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

smoothed BLS. The formation of BLS and their quantitative description are affected by many controllable and uncontrollable factors and evoke a sincere interest of many researches. Some specific models and prehistory of the general problem are considered in papers cited below. These BLS cover many branches of applied sciences and engineering, including analysis of medical diseases (Fig. 1(a)), acoustics signals identified as a part of a ‘‘song’’ recorded from insects (Fig. 1(b)), car engine noise (Fig. 1(c)), seismology (Fig. 1(d)), etc. The quantitative description of these blow-like signals (in terms of a finite set of fitting parameters) presents itself a difficult problem and many researches use specific models for their

qualitative description, at least, not considering an accurate fit of BLS to some reliable hypothesis or a welljustified model. So, a surprising result occurs that could be generated by many factors: similar signals with finite duration are registered from different phenomena and could be described in the same way by only a relatively small set of fitting parameters. The problem can be formulated as follows. Is it possible to find a general reason justifying the generation of these BLS? Is it possible to describe the BLS quantitatively in terms of a general set of fitting parameters? In this paper, we want to show that there is a general underlying reason generating these types of signals that

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are insensitive to details of numerous microscopic motions. In other words, we want to show that a general averaging procedure exists for describing BLS both at micro and mesoscale levels. This natural conclusion can be understandable on the intuitive level but, as far as we know, the correct mathematical analysis justifying this procedure, is absent in the literature. In this paper, we want to demonstrate this approach, which shows how it becomes possible to reduce a set of micromotions described by a function f ðzÞ to a new reduced function, which keeps only the asymptotic values of the function f ðzÞ on a mesoscale. By mesoscale region we indicate the intermediate level of scales ðk < g < KÞ, where a studied complex system exhibits its self-similar (fractal) properties in space or time. For simple models describing a disordered medium, which can be presented in the form of the self-similar branched fractals/channels, it becomes possible to derive and generalize the well-known Kohlrausch–Williams–Watts function [1,2] and find a new ‘‘universal’’ function (expressed, for some partial cases, in the form of the well-known log-normal distribution), which can be used also for description of collective motions in disordered medium. As a confirmation of validity of the obtained model, we describe three different sets of blow-like signals: 1. The envelopes of BLS corresponding to the bronchial asthma. 2. The envelopes of the acoustic songs generated by the queen bees (Apis mellifera L.). 3. The envelopes of acoustic signals recorded from car engine valves in the idling regime. For additional justification, we have to develop the special fitting procedure based on the generalization of the eigen-coordinates (ECs) method, which enables us to reduce an initial problem of nonlinear fitting to a linear fitting problem [3,4]. The new set of parameters corresponds to the desired global fitting minimum, which follows from the linear least square method (LLSM). This transformation allows to avoid the procedure of an initial guess of the fitting parameters, which is mandatory for any nonlinear fitting program used for finding the desired global minimum corresponding to a true set of the fitting parameters. The calculated set of the fitting parameters describes the envelopes of the BLS with high accuracy and can be sensitive to the influence of some predominant/external factors. This observation should, in future work, help us to construct the so-called calibration curve (not shown here for lack of space), i.e. a dependence of a fitting parameter with respect to some external factor (concentration of an additive, temperature, etc.) without detailed knowledge of microscopic mechanisms underlying in the ground of the complex studied phenomenon. More specifically, the fitting parameters of the reduced fractal model (RFM) play as quantitative indicators that serve to differentiate one curve from another one. The approach is similar to that proposed in [5], where correlation between random sequences of streamed data is analyzed and a reduced set of parameters is determined

to characterize each sequence of data. To sum up, the main results presented in this paper can be summarized as follows. First, a characterization of BLS is provided by a reduced fractal model (RFM) specified by the set of fitting parameters. The reduced model takes advantage of self-similarity in the averaged micromotions, which is a distinctive property considered in the representation of many other complex systems [6,7]. Second, if complex BLS can be described by a finite set of the fitting parameters, then future research definitely opens new perspectives. 2. The reduced fractal model and its realization in the fractal-branched structures For description of time-domain evolution of a BLS through a self-similar medium, we shall try to use as a prototype the collective model of relaxation initially developed and successfully used for description of cooperative behavior of microemulsion droplets near the percolation threshold [8]. We make the following assumptions to obtain the generalized model of collective motion in the mesoscale: A1. An elementary event transferring excitation/relaxation along the channel with length Lj is described by a microscopic relaxation function f ðz=zj Þ, where zj is a dimensional variable, characterizing the jth stage of the whole cooperative process and z is an intensive (not depending on the size of a system) variable characterizing the considered self-similar branching structure. Throughout the paper the basic variable z will coincide with the dimensionless time z ¼ t=s. The parameter s can coincide with a characteristic time necessary for the occurrence of an elementary event of excitation/relaxation. A2. It is assumed that zj ¼ a Lj , where Lj is the ‘effective’ length of a channel of relaxation on the jth stage of similarity and a is a coefficient of proportionality. Possible lengths Lj are distributed by a self-similar law j

Lj ¼ L0 k ;

ð1Þ

where L0 is a minimal scale and k is a scaling factor. A3. The number of neighbors (here one can imply excitation/transfer centers) involved in the local transfer/relaxation process located along the length Lj also obeys to the scaling law j

nj ¼ n0 b ;

ð2Þ

where b is the scaling factor ðb > 1Þ and n0 is the minimal number of the neighbor centers located near the fixed excitation center. In other words, the propagation of a signal through a medium has a branching structure. For description of cooperative relaxation, one can start with the ideas developed in papers [8–12]. In these papers, the authors considered a number of excitations of a donor molecule to an acceptor molecule in various

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heterogeneous media. If a process of an excitation transfer takes place through many parallel channels, then one can obtain the following relationship for this cooperative type of relaxation

UðtÞ ¼

Y

½1  c þ c expðt WðRj ÞÞ;

ð3Þ

j

where UðtÞ is the relaxation function normalized to unity, t is the current time, c is the concentration of donors in the system, Rj is the distance between a donor and an acceptor, located on the jth site, and WðRj Þ is the microscopic relaxation rate of the excitation transfer from the donor to acceptor at distance Rj . The product extends over all structure sites except for the origin. The fractal-branched structures relationship (3) can be essentially generalized and written as

UN ðzÞ ¼

N 1 Y

j

½f ðz n Þ

n0 b j

"

N1 X j ¼ exp n0 b ln½f ðz n j Þ

j¼0

#

j¼0

 exp½SN ðzÞ;

ð4Þ

where z ¼ t=ða L0 Þ; n ¼ 1=k and the value of N refers to the last stage of the self-similarity transfer process taking place in the considered self-similar structure. In general, the process of a BLS propagation can be divided in two parts. The first part after application of external potential is related to the excitation (the act of excitation is described by the function gðzÞ), while the second part is related to the relaxation process and disappearance of the initial excitation because of its dissipation inside the self-similar branching structure. So, it is natural to generalize expression (4) and present the whole process in the form of product of two factors

Y Y UN ðzÞ ¼ EN ðzÞ RN ðzÞ ¼ gðz=zj Þ f ðz=zj Þnj : j

ð5Þ

(6) has been applied for describing the collective relaxation process in the complex system of microemulsion droplets near the percolation threshold [8] and other disordered systems. The parameters c and k of the relaxation function are expressed in the form of integral expressions, that functionally depend on the concrete type of the function f ðzÞ describing the evolution of variety of micromotions on a microscopic level. In this paper, we want to show how to evaluate expression (4) not only in the continuous approximation but also for discrete variables taking into account the influence of log-periodic oscillations. For a wide class of functions f ðzÞ, expression (6) and its general forms (obtained below) can serve as a basic analytical expression for description of many excitation/relaxation phenomena in different disordered systems. The evaluation of expression (4) essentially depends on the asymptotic behavior of the function f ðzÞ and on the interval of location of the scaling parameters n and b. We initially suppose that these scaling parameters satisfy the following inequality (the case n b P 1 is considered in Appendix A):

0 < n b 6 1:

SN ðz nÞ ¼

UðzÞ ¼ A zl expðc zm  k zÞ

ð6Þ

where the power-law exponents l; m depend on the asymptotic behavior of the function f ðzÞ (see expression (9) below and more accurate evaluation of these parameters in Appendix A). The simplified expression

  n0 1 N1 SN ðzÞ þ n0 b ln f ðz nN Þ  ln½f ðzÞ: b b

ð8Þ

We suppose that the asymptotic behavior of the function f ðzÞ is described as follows. For jzj  1 we obtain the power series

f ðzÞ ¼ c0 þ c1 z þ c2 z2 þ   

ð9aÞ

for jzj  1 (with b; r > 0) we obtain

f ðzÞ ¼

A A1 expðr zÞ þ 1þb expð2 r zÞ þ    : zb z

ð9bÞ

The last relationship combines the exponential and powerlaw asymptotic behaviors to consider them together. For b – 0 and r ¼ 0, it describes the power-law asymptotic behavior; the case b ¼ 0 and r – 0 corresponds to the pure exponential asymptotic behavior. In the limit N  1 one can obtain the simplified functional equation for the sum S1 ðzÞ  SðzÞ

Sðz nÞ ¼ 1. For many partial forms imposed on the function f ðtÞ, expression (4) can lead to the stretched-exponential law of relaxation that is widely used for description of cooperative processes in many disordered systems [10–12]. 2. The evaluation of the general formula (4) for N  1 in the continuous approximation (by the Euler summation formula) leads to the expression

ð7Þ

From expression (4) one can obtain that the sum SN ðzÞ satisfies the scaling equation of the type

j

Here we suppose that an elementary or microscopic event of excitation is described by the finite function f ðzÞ; zj as before proportional to Lj , where the effective excitation length is defined by (1). Now it is necessary to add some arguments why expressions (4) and (5) can be selected as the most probable formalizations for description of cooperative excitation/ relaxation process in self-similar complex systems. We see for that the following important reasons:

169

    1 n0 c 1 n0 n0 A SðzÞ   r z þ b lnðzÞ  ln b c0 b c0 b b 1 ð10Þ ¼ SðzÞ þ B z þ C lnðzÞ þ D: b

The last three terms in (10) follow from expressions (9) and the details of their derivation are explained in Appendix A. The limits of the intermediate asymptotic behaviors for jzj  1 and jzj  1 are respectively determined from inequalities

c2 2 jzj  1; c0

A1 exp½2 r jzj  ð1 þ bÞ lnðjzjÞ  1:

ð11Þ

The scaling Eq. (10) can be accurately solved analytically based on the method of variation of a free constant. The details for the case n b P 1 are considered in Appendix A.

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We are giving the closed-form solutions for the function UðzÞ related to SðzÞ by relationship (4). Case A ðb – 1; n b 6 1Þ. Consider

UðzÞ ¼ P0 zl exp½zm rðlnðzÞÞ þ k z:

ð12Þ

Here rðlnðzÞ  lnðnÞÞ ¼ rðlnðzÞÞ is a log-periodic oscillating function with period lnðnÞ. This function can be decomposed into the infinite Fourier series   X   þ1 lnðzÞ lnðzÞ ¼ r C n exp 2pnj lnðnÞ lnðnÞ n¼1 þ1 X   C n expðjXn lnðzÞÞ þ C n expðjXn lnðzÞÞ  C0 þ n¼1

 C0 þ

þ1  X n¼1

    lnðzÞ lnðzÞ Acn cos 2pn þ Asn sin 2pn : lnðnÞ lnðnÞ ð13Þ

In the one-mode approximation (OMA) this function can be written in the form

rðlnðzÞÞ ¼ cðmÞ þ c1 expðjhXi lnðzÞÞ þ c 1 expðjhXi lnðzÞÞ:

ð14Þ

Here the zero Fourier-component ðC 0 ¼ cðmÞÞ should accept negative values because of convergence of expression (12). Other parameters in (12) are defined by expressions

m¼

lnð1bÞ ; lnðnÞ

X0 ¼

k¼

B ; n  1b

D C lnðnÞ  2 ; 1  1b 11

l¼

C ; 1  1b

ð15Þ

P0 ¼ expðX 0 Þ;

b

where parameters B; C; D are defined by expression (10). Further investigations show that, for the case b n > 1; c1 ¼ 0 in expansion (9a); the power-law exponents l and m can be simultaneously negative satisfying the condition l m > 0. The damping constant k can accept positive or negative values. Case B ðb ¼ 1; n < 1Þ. In this case, one can obtain the following solution (details are given in Appendix A)

UðzÞ ¼ exp½rðlnðzÞÞ þ c ðln2 ðzÞÞ þ l ðlnðzÞÞ þ k z

ð16Þ

with parameters

c¼

n0 b ; 2 lnð1=nÞ

l ¼ n0

  lnðA=c0 Þ b c1 =c0  r  ; k ¼ n0 : lnð1=nÞ 2 1n ð17Þ

In (16), as in the previous expression (12), rðlnðzÞÞ determines a log-periodic function again. The important result here, in comparison with evaluations made previously in the continuous approximation [8] and successfully applied for description of dielectric properties of microemulsions and other complex fluids [9], is the appearance of log-periodic oscillations expressed by the function rðlnðzÞ  lnðnÞÞ ¼ rðlnðzÞÞ. So, for the clearly expressed discrete scale invariance effect [13–16] that can be realized in self-similar structures with the definite pronounced scale periodicity, the stretched exponent in expression (12) should exhibit the oscillating properties. It would be important to find confirmation of this effect in

many experimental measurements and different data analysis. Expression (12) generalizes the well-known Kohlrausch–Williams–Watts relaxation law that was suggested many years ago [1,2] for description of nonexponential relaxation phenomena in many disordered systems [10–12]. As it follows from these calculations, the stretched-exponential law can be easily derived from the conjecture that a process of relaxation in heterogeneous medium has a self-similar branching structure determined by expression (4). In this sense, the relaxation laws (12) and (16), in comparison with previously obtained expression (6), should have a general character because the process of excitation/relaxation taking place in the mesoscale region does not depend on the concrete form of the microscopic relaxation function f ðzÞ, which is averaged actually on the mesoscale region. So, we discovered and mathematically confirmed the reduction phenomenon, occurring when a set of micromotions is averaged and transformed into a collective motion on an intermediate level of scales. It is natural to determine the general expressions (12) and (16) as the reduced fractal model (RFM). It is interesting to note that different partial cases (for some concrete forms of f ðtÞ, describing various types of micromotions) leading to the ‘pure’ stretched-exponential dependence in time domain have been considered by many authors (see Section 8 published in the Proceedings of the International Symposium [17]). These non-exponential functions have been applied for description of relaxation phenomena of statistical defects in condensed media, in glasses, etc. In spite of the fact that specific features of transportation mechanisms on some micro-excitation level of a BLS cannot be well understood (i.e. the specific form of the microscopic function f ðzÞ is not known), we may conjecture that the functions (12) and (16) can be considered as the most probable for description of evolution of BLS on the mesoscale region, when details of microscopic excitation/relaxation acts expressed by the concrete function f ðzÞ are not essential for the whole transfer process. Only asymptotic parameters of this function describing some micromotions expressed by (9) are essential in formation of the averaged function that describes the collective process of relaxation in another level of its self-similar organization. Finally note that we derived general functions (12) and (16) from simple expression (4). One can expect some corrections to formulae (12) and (16) if the more general expression (5) is considered. But at present time it is not necessary to consider (5) with additional asymptotic behaviors imposed by another function gðzÞ. The calculations of these corrections can constitute a subject of a separate research, if other experimental measurements not described by (12) and (16) can lead to the necessary corrections. 3. The fitting procedure of functions containing logperiodic oscillations If we take into account the influence of log-periodic oscillations that figure in expressions (12) and (16), then the fitting procedure becomes very complicated and direct application of these expressions to comparison with real

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data becomes impossible. So, it is necessary to develop a special algorithm that can be used for the fitting of many data containing BLS, as depicted in Fig. 1(a) and (b). To construct the desired algorithm, we use ideas of previous papers [18–20]. But before starting we remind some attractive features of the ECs method [3,4] that allows in many cases to reduce the non-linear fit to a routine procedure as the LLSM. These features are described in Appendix B. But this initial fit is not sufficient. It helps only to find the initial fitting vector VðP 0 ; l; c; m; kÞ containing five parameters that follows from general expression (12) if we do not consider the influence of log-periodic function

H0 ðzÞ ¼ P0 zl expðc zm þ k zÞ:

ð18Þ

If the influence of the log-periodic oscillations in (12) is becoming essential     K  X lnðtÞ lnðtÞ rðlnðtÞÞ ¼ c þ Ack cos 2pk þ Ask sin 2pk ; lnðnÞ lnðnÞ k¼1

ð19Þ then it is necessary to take them into account considering this log-periodic function as the fitting function with the additional set of ð2K þ 1Þ parameters ðn; Ack ; Ask ; k ¼ 1; 2; . . . ; KÞ. Here and below the arbitrary variable z will be identified with the temporal dimensionless variable t. The final mode K that enters in (19) is found from the minimization of the fitting error value

RelErrðyðtÞ; Uðt; KÞÞ ¼ min

  StDev ðyðtÞ  Uðt; KÞÞ 100%; meanðjyðtÞjÞ ð20Þ

where yðtÞ is the initial function and Uðt; KÞ coincides with the function (12) containing the log-periodic function (19). To realize the fit of yðtÞ, we take the natural logarithm from (12)

8 K X > > > LðtÞ ffi L0 ðtÞ þ ½Ack yck ðtÞ þ Ask ysk ðtÞ; > > > > k¼1 > > < L0 ðtÞ ¼ Y 0 þ l lnðtÞ  c t m þ k t;

 > lnðtÞ > > yck ðtÞ ¼ expðm lnðtÞÞ cos 2pk lnðnÞ ; > > >

 > > > : ysk ðtÞ ¼ expðm lnðtÞÞ sin 2pk lnðtÞ : lnðnÞ

K X ½Ack yck ðtÞ þ Ask ysk ðtÞ:

ð23Þ where (23) holds true and agrees with the first two terms of (22) because x1 – kx0 ¼ x0 (for k ¼ 1) and x2 – kx0 ¼ 2x0 (for k ¼ 2), even if the frequencies x1 and x2 are close to x0 and 2x0 . In particular, R1 ðtÞ includes a first term similar to R0 ðtÞ but at a slightly different frequency, i.e. x1  x0 , and a second term at a frequency x2  2x1 . These three frequencies are approximate and considered as inoculating frequencies, that are determined by the minimization procedure specified below. Then R0 ðtÞ and R1 ðtÞ represent the initial/inoculating log-periodic functions that appear as the first terms in decomposition (22). They should be taken into account for more reliable evaluation of the nonlinear fitting parameter n. One can also notice that the functions R0 ðtÞ and R1 ðtÞ satisfy, correspondingly, the following differential equations:

(

ðD2 þ x20 Þ R0 ðtÞ ¼ 0; D  t dtd ;

ðD2 þ x21 ÞðD2 þ x22 Þ R1 ðtÞ ¼ 0;

D2  D  D: ð24Þ

The unknown modes x0;1;2 enter in these equations by a linear way. This observation allows applying the ECs method again and use the LLSM for the corresponding BLRs that are given in Appendix D. These calculations allow to find the confidence interval for the inoculating frequencies x0;1;2 and present the desired frequency in the form



xðsÞ ¼ xmin þ sðxmax  xmin Þ; s 2 ½0;1; xmin ¼ minðx0 ; x1 ;0:5 x2 Þ; xmax ¼ maxðx0 ; x1 ;0:5 x2 Þ: ð25Þ

ð21Þ

From expression (25) it becomes possible to minimize the value of the relative error   StDev ðRmðtÞ  Fðt; s;KÞÞ RelErrðRmðtÞ; Fðt; s;KÞÞ ¼ min 100%; meanðjRmðtÞjÞ

ð26Þ

Here Y 0 ¼ lnðP0 Þ and t is supposed to be normalized to the unit value. One can compare the basic linear relationship (BLR, see Appendix C) for the function L0 ðtÞ, then find the nonlinear parameter m, and evaluate the other four unknown fitting parameters ðY 0 ; l; c; kÞ that enter into the zeroth hypothesis L0 ðtÞ. Subtracting this function from LðtÞ provides the remnant function RmðtÞ that still contains the unknown nonlinear scaling parameter n:

RmðtÞ ¼ LðtÞ  L0 ðtÞ ffi

8 RmðtÞ expðm lnðtÞÞ ffi R0 ðtÞ þ R1 ðtÞ; > > > > 2p > > < R0 ðtÞ ¼ Ac0 cosðx0 lnðtÞÞ þ As0 sinðx0 lnðtÞÞ; x0 ¼ lnðnÞ ; R1 ðtÞ ¼ Ac1 cosðx1 lnðtÞÞ þ As1 sinðx1 lnðtÞÞ > > > > þAc2 cosðx2 lnðtÞÞ þ As2 sinðx2 lnðtÞÞ; > > : x0 ffi x1 ffi x2 =2;

ð22Þ

k¼1

To find this desired parameter, we apply the method described in papers [18–20]. The fitting parameter m was initially calculated and so the first terms of the sum in (22) (for k ¼ 1; 2) can be approximately written as

where

8 K X > > > ½Ack yck ðs; tÞ þ Ask ysk ðs; tÞ; > Fðt; s; KÞ ¼ < k¼1

> yck ðs; tÞ ¼ expðm lnðtÞÞ cos ðxðsÞ k lnðtÞÞ; > > > : ysk ðs; tÞ ¼ expðm lnðtÞÞ sin ðxðsÞ k lnðtÞÞ:

ð27Þ

The last expressions provide the optimal value sopt from the interval ½0; 1 and the value of the final mode K opt that obey the fitting error requirement: 1% < RelErrðRmðtÞ; Fðt; sopt ; K opt ÞÞ < 15%. Coming back to the initial function

yðtÞ ffi UðtÞ ¼ exp½L0 ðtÞ þ Fðt; sopt ; K opt Þ;

ð28Þ

one can finish the fitting procedure that takes into account the corrections contributed by the log-periodic function (22).

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The same procedure can be applied for the conjecture expressed by (16). We stress here only the differences that appeared in the treatment procedure for this function. The power-law exponent m is absent and so the zeroth hypothesis is simplified and can be presented in the form 2

L0 ðtÞ ¼ Y 0 ðtÞ þ c ln ðtÞ þ l lnðtÞ þ k t

ð29Þ

with the unknown parameters ðY 0 ; c; l; kÞ that can be found from (29) by the LLSM. Other expressions (22), (23) still remain valid at m ¼ 0 and the procedure becomes similar to the previous one described above. The BLR for the functions R0;1 ðtÞ are given in Appendix C. So, the treatment procedure described for the correct evaluation of log-periodic corrections can be applied for a wide set of blow-like signals. The fitting procedure suggested above can be applied to this type of signals and enables us to identify their possible fractal origin/structure. 4. The application of the original procedure to real data We want to stress here that the identification of the desired model on available data has an illustrative character. The basic aim is to confirm by the available data the existence of this reduced fractal model (RFM). We should omit the speculations about the specific reasons of possible dynamical self-similarity that can arise in each considered specific complex system. This matter is important and deserves some detailed research for each type of the considered signals. We only want to focus the attention of potential researches on the existence of the general scenario that governs the dynamic behavior of many BLS on mesoscale. In preface to this section, we want to stress also the following two treatment steps that we consider as general for all types of the analyzed available BLS. S1. We use the procedure of reduction to three incident points that has been described in [18]. It helps to extract the maximal/minimal values of the studied signal and compresses the initial signal to a less number of data points by keeping the values of the amplitudes in the same interval as for the initial signal containing a large number of data points. S2. For further calculations of the obtained BLS envelopes, we use the procedure of the optimal linear smoothing (POLS) [21,22], which helps to obtain the desired envelope that is used for verification of the RFM described in the previous section. In order to keep the ‘‘up’’ and ‘‘down’’ parts of the smoothed envelope statistically similar to each other, we calculate the value of the smoothing window from the evaluation of the Pearson correlation coefficient [23] satisfying, in turn, the condition

0

1

B EupðtÞ  EdnðtÞ C 0:97 6 @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA 6 1:0: 2 Eup2 ðtÞ Edn ðtÞ

ð30Þ

Here the functions EupðtÞ and EdnðtÞ determine the corresponding envelopes obtained after applying the reduction procedure to three incident points. These steps will be used as obligatory ones in processing the available data considered below.

Consider a BLS generated by an earthquake of small intensity (see Fig. 1(d)). If one compares this signal with the one depicted in Fig. 1(a), one can note that qualitatively they are similar to each other. But they describe different phenomena. We should note here that the available earthquake data (one signal is depicted in Fig. 1(d)) have strong variability in calculations of the desired ‘‘up’’ and ‘‘down’’ envelopes, that are not similar to the envelopes in Fig. 1(a) and (b). The correlation coefficient is less than 0.75. So, this type of BLS, for which ‘‘up’’ and ‘‘down’’ envelopes are not close to each other, should be considered separately and, based on this observation, the BLS for different earthquakes are not considered in this paper. The analysis of model data shows that, in the case of strong variability of the corresponding envelopes obtained by the smoothing procedure and reduction to three incident points, one can consider the averaged envelope representing itself the combination ðEupðtÞ þ EdnðtÞÞ=2. 4.1. The description of the envelopes corresponding to the bronchial asthma disease A brief description of the original acoustic data follows. The available data were recorded by standard methods described in literature [24,25]. The signal (in the form of acoustic pressure) coming from a surface of the human’s body is registered by a microphone button placed in the head of a stethoscope. The Butterworth filter [26,27] of sixth order with cut frequency equal to 130 Hz was used to suppress heart beats from acoustic signals. The asthma data were selected because this disease has the goose breathing phenomenon, which is expressed in the form of BLS. A typical BLS, where this effect is expressed clearly, is shown in Fig. 1(a). We received similar data corresponding to five randomly selected persons. After realization of the two steps explained in the beginning of this section, one can obtain ten full envelopes (including ‘‘up’’ and ‘‘down’’ parts) of the BLS that were used to test the possible guessed expressions (12) and (16). The calculations show that the simple expression (16) is more preferable because it leads to small values of the fitting error defined by (20) and includes less log-periodic components in comparison with the more general guess (12). So, for further calculations, we use the fitting function (16) as the basic one. In Fig. 2(a) we demonstrate the ‘‘up’’ part of the envelope that needs to be fitted to expression (16). For the realization of the final fit, we go through the next steps. 1. We calculate the four unknown fitting parameters ðY 0 ; c; l; kÞ from expression (29). Then, after subtracting this hypothesis from the function LðtÞ ¼ lnðUðtÞÞ, we fit the remnant function RmðtÞ. 2. This remnant function should be fitted with the inoculating functions appearing in expression (23) in order to find three initial frequencies x0;1;2 . To this aim, we use the BLRs given in Appendix D. 3. We determine the limits of a ‘‘true’’ frequency from the interval ½xmin ; xmax  and use (25) for calculation of xopt . This value helps to evaluate the last nonlinear scaling parameter from the expression lnðnÞ ¼ 2p=xopt .

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173

Fig. 2. (a) Final fit realized by the original fitting procedure for envelope corresponding to person #1. The initial envelope is shown in Fig. 1(a). Each of ‘‘up’’ and ‘‘down’’ branches contains 1000 data points. The fit with 15% accuracy contains in total 85 fitting parameters. The basic parameters in (16) are collected in Table 1. (b) Distributions of amplitudes of log-periodic functions that enter in (16) and describe the ‘‘up’’ and ‘‘down’’ branches of the BLS for person #1. (c) Final fit realized by the original fitting procedure for envelope corresponding to person #5. Each of ‘‘up’’ and ‘‘down’’ branches contains 1000 data points again. The fit with accuracy less than 10% contains in total 85 fitting parameters. The basic parameters in (16) are collected in Table 1. (d) Distributions of amplitudes of log-periodic functions that enter in expression (16) and describe ‘‘up’’ and ‘‘down’’ branches of the BLS for person #5.

4. We realize the final fit of the RmðtÞ to expression (22) by using the conventional LLSM. We demonstrate the final fit only for persons #1 (Fig. 2(a) and (b)) and #5 (Fig. 2(c) and (d)), respectively, together with the distributions of the decomposition 1=2

coefficients Ack ; Ask ; Amdk ¼ ½ðAck Þ2 þ ðAsk Þ2  in order not to overload the content by a large number of supplementary figures. Other important parameters corresponding to the fitting function (16) and describing the BLS for persons #1  5 are collected in Table 1. The most interesting facts that follow from this analysis on different persons are: (i) the values of the decomposition coefficients in Fig. 2(b) and (d) are high in comparison with initial values of the envelopes (see previous Fig. 2(a) and (c)); (ii) they correspond to some deterministic law that it is not known but will be interesting to investigate to justify its origin. The analysis of these results is given in the final section.

4.2. The description of the fragments of queen bee acoustic signals The analyzed data represent the successive sounds that are generated by young queen bees, when they are ready to go out from a queen cell. These sounds can be divided into fragments having a finite duration and each fragment repeats a shape of the desired BLS. A typical fragment of this ‘‘song’’ is shown in Fig. 1(b). The recorded acoustic signal is registered by an electrostatic microphone with sensitivity 3.16 mV/Pa. In the range of the analyzed recorded signals (20–1000 Hz), the used microphone had a linear amplitude-frequency response (AFR). The coefficient of the nonlinear distortions (on the frequency 1 Hz) did not exceed the value 0.01%. All signals were recorded in the form of time-series with the discretization frequency equal to 11.025 kHz. Each song contains from 6 to 20 BLS. The first one has, as usual, the maximal duration from 1 to 1.5 s, the next ones are shorter

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Table 1 The additional fitting parameters that correspond to the asthmatic data tested in accordance with hypothesis (16). The maximal number of modes is K ¼ 40. These additional parameters correspond to expression (16) where Y 0 ¼ r0 corresponds to constant value qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi in the log-periodic decomposition. The last column signifies the value RangeðyÞ ¼ maxðyÞ  minðyÞ, where the function y coincides with Amdk ¼ Ac2k þ As2k . Person Person Person Person Person Person Person Person Person Person Person

#1 #1 #2 #2 #3 #3 #4 #4 #5 #5

(up) (dn) (up) (dn) (up) (dn) (up) (dn) (up) (dn)

lnðnÞ

k

l

c

Y0

RelErrð%Þ

RangeðAmdÞ

5.4778 5.6422 5.3062 5.1204 5.3062 5.1204 5.3410 5.3683 5.3651 5.6181

0.1336 0.1184 0.1184 0.1208 0.1184 0.1208 0.1046 0.1045 0.0991 0.0954

3.6988 3.0735 1.7095 1.9871 1.7095 1.9871 1.6618 1.6286 1.9886 1.7112

1.3766 1.1925 0.9368 1.0015 0.9368 1.0015 0.8578 0.8535 0.8498 0.7724

1.8155 2.2281 3.4744 3.2395 3.4744 3.2395 3.0934 3.1516 2.2368 2.2889

13.8981 15.7892 3.1048 3.0173 3.1048 3.0173 5.2208 5.8721 1.9193 1.5235

368,926 260,073 67436.3 309.75 67436.3 309.75 305,180 441,127 183,643 193,356

and occupy a time range of duration between 0.25 and 0.6 s [28–30]. For verification of the RFM, we randomly chose one of the time-series stream recorded from a young queen bee and used two steps for preparing five fragments that remind BLS and are suitable for our analysis. In order to save space, we only show the envelope of the first fragment (Fig. 1b) and ‘‘up’’ and ‘‘down’’ envelopes referring to this fragment are shown in Fig. 3(a). In Fig. 3(b) we observe a similar behavior to that we found for envelopes associated with asthma disease. Again: (i) the values of the decomposition coefficients are high in comparison with initial values of envelopes (see the previous Fig. 1(b)); (ii) their nonrandom behavior leads to some hidden law that governs by showing these behaviors. We should note also that the values of amplitudes associated with the ‘‘up’’ branch are less approximately in 25 times. The same analysis and observations can be repeated for the fifth fragment. The fit of the ‘‘up’’ and ‘‘down’’ branches of the envelope is shown in Fig. 3(c) and the distribution of amplitudes of log-periodic functions is shown in Fig. 3(d). We observe a similar behavior that we found for envelopes associated with fragment #1, then remarks (i) and (ii) still hold true. The values of amplitudes associated with the ‘‘down’’ branch are less approximately in 50 times. Finally, Table 2 shows additional fitting parameters according to (16) and describing the fragments of queen bee songs. As it has been mentioned above, the detailed analysis and recognition of the information hidden in these songs represent a separate problem and they are out of the scope of this research. Our basic aim is the justification of the RFM on different finite signals and confirmation of the conjectures expressed by (12) or (16). In this case, as in the first example considered above, the simple guess (16) is relevant, because it gives the smaller value of the fitting error and requires a relatively small number of the log-periodic components. 4.3. The acoustic data recorded from car valves in the idling regime Another example is the data recorded by the acoustic signals from the car valves. The location for the microphone and its characteristics, together with the car parameters in the idling regime, are explained in Fig. 4(a). The fragment of the initial signal is shown in

Fig. 1(c). After the application of detrending procedure described in [21,22], one can obtain a set of blow-like signals. A typical fragment of the acoustic signals of valve knocks is shown in Fig. 4(b). After obtaining this set, one can select one BLS including a small signal in the back and apply the two treatment steps described in the beginning of this section. Fig. 4(c) shows a selected fragment of the signal in Fig. 4(b) and explains the two treatment steps. Smoothing of the signal yields the desired envelope. The two desired branches (‘‘up’’ and ‘‘down’’) of the envelope are also shown in the small picture on the right-hand side of Fig. 4(c). The treatment procedure is applied to the fitting of these two branches. The fit of these ‘‘up’’ and ‘‘down’’ branches to the function (16) is shown in Fig. 4(d). The arbitrary units (a.u) are considered in the following way. The smoothed signal contains approximately 4000 data points. The total set of points for simplicity is increased in 1000 times and the signals are reduced to the same interval for their comparison. The AFR containing the amplitudes of log-periodic functions, then the distributions of the decomposition coefficients for the branches are shown in Figs. 5(a) and (b), respectively. Note that the decomposition coefficients for the ‘‘down’’ branch are qualitatively similar to the coefficients for the ‘‘up’’ branch. In complete analogy with the fitting of the first BLS, one can fit other eight signals. The basic fitting parameters are collected in Table 3. This interesting approach can be used for characterization of the different car regimes and can serve as an alternative approach [31] that has used for these purposes. 5. Concluding remarks on fundamental results and open questions Based on examples taken from different complex systems, one can notice an important feature: many complex systems demonstrate a common scenario of generation of signals having finite duration. It becomes possible to describe them because many specific features appeared in the microlevel are averaged and, in the result of this reduction, one can obtain the functional dependence of the desired signal expressed in the analytical forms (12) or (16). In order to see these ‘‘smoothed’’ forms of the BLS, we should undertake certain steps and develop a special fitting procedure that help to simplify the random

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Fig. 3. (a) The fit of the ‘‘up’’ and ‘‘down’’ branches of the envelope that corresponds to the first fragment of the queen bee song. The accuracy of the fitting is high and does not exceed 5%. All additional parameters are collected in Table 2. (b) Distributions of amplitudes of log-periodic functions that enter in (16) and describe ‘‘up’’ and ‘‘down’’ branches of the BLS for the fragment #1. (c) The fit of the ‘‘up’’ and ‘‘down’’ branches of the envelope that corresponds to the fifth fragment of the queen bee song. The accuracy of the fitting is high and does not exceed 5%. All additional parameters are collected in Table 2. (d) Distributions of amplitudes of log-periodic functions that enter in (16) and describe ‘‘up’’ and ‘‘down’’ branches of the BLS for the fragment #5.

Table 2 The additional fitting parameters that correspond to description of the fragments of queen bee song in accordance with hypothesis (16). The maximal number of modes is K ¼ 36. The notations are the same as in Table 1. Fragment

lnðnÞ

Frag. Frag. Frag. Frag. Frag. Frag. Frag. Frag. Frag. Frag.

1.4250 1.4485 1.5150 1.5485 1.5392 1.4605 1.4620 1.4441 1.5167 1.4451

1 1 2 2 3 3 4 4 5 5

(up) (dn) (up) (dn) (up) (dn) (up) (dn) (up) (dn)

k 18.8175 16.5962 18.7275 16.7062 13.0280 10.5834 8.0935 10.5475 8.0105 9.9246

l 12.1666 12.5392 13.1666 12.2492 26.5833 26.0651 18.6944 16.8687 18.1395 16.5209

form of the initial registered signal: (a) to apply the POLS [21,22] that eliminates the high-frequency fluctuations; (b) to reduce a cluster of data points to three incident points; (c) to extract a signal with finite duration after

c 32.1816 29.8193 33.1816 30.8193 3.7329 0.5952 21.0758 23.6268 20.8241 22.6192

Y0 27.7608 25.6875 28.6608 24.6875 7.4491 4.4271 17.5249 19.8473 17.0491 18.9114

RelErrð%Þ

RangeðAmdÞ

0.9759 1.1021 0.8749 1.2121 0.5152 0.7522 0.6007 0.6991 1.0762 0.6612

2955.19 67,982 3055.19 69,282 285,306 248,826 83845.4 25,447 964,680 19094.4

elimination of a possible remaining trend. An attentive reader has a right to ask the authors: are there some chances to keep possible fluctuations in the result obtained after applying these simplification steps (a,b,c)? We are

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Fig. 4. (a) Typical noises were recorded in the place shown by red circle. The parameters of the car: ‘‘Toyota Lite ice’’, mileage 180.000 km. The type of the Diesel engine ‘‘3c-te’’ (4 cylinders, 8 valves). Three valves were not adjusted (2 inlet and one outlet valves). The typical valves knocking appeared because of the engine incoordination. The characteristics of the microphone used: microphone button MB 265 (Germany) - sensitivity 50 mV/Pa, frequency range 2– 20,000 Hz, the maximal sound pressure measured level 145 dB. The records of acoustic signals generated by valve knocks correspond to the idling regime: 825 rpm, the duration of the working cycle 145.5 ms. (b) Typical fragment of the initial signal after elimination of the trend contains a set of typical blowlike signals. (c) Treatment of the acoustic signals of valve knocks: a fragment of the signal in Fig. 4(c) is smoothed to get the desired envelope. The ‘‘up’’ and ‘‘down’’ branches of this envelope are shown on the right-hand side. (d) The fit of the ‘‘up’’ and ‘‘down’’ branches to the function (16): results are marked by yellow and white bold lines, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

ready to give a certain answer. The long-range fluctuations are preserved and expressed in the form of log-periodic oscillations that keep their specific memory on the mesoscale level. The spectrum of these long-range fluctuations (expressed in the form of coefficients A0 ; n; Ack ; Ask ; k ¼ 1; 2; . . . ; KÞ helps to preserve the specific features of the studied BLS. Here we want to stress the following important fact: in the result of application of this approach, the number of modes K  N (initial number of data points). This simplification can be used as an important diagnostic feature of the studied complex system. This

spectrum serves as a ‘‘specific piano’’ and each mode expresses ‘‘a specific music’’ of these fluctuations on the mesoscale level. Definitely, this spectrum can be used as a specific indication/sign of the studied complex system if additional information about the specific properties of the system is absent. In conclusion, we stress that this approach is rather general and based only on the single basic assumption, namely, on the branching structure of generation of self-similar signals (see expression (4)) on different micro-levels. May be this assumption can serve as the basic property of different complex systems that

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Ack(up)

Ask(dn)

Ask(up) The decomposition coefficients that enter in the log-periodic function for the "up" branch

0.3

0.0

-0.3

0

10

20

2

30

2 1/2

(Ack+Ask)

2 1/2

The distribution of decomposition coefficients for the down branch

2

(Ack+Ask)

0.3

0.0

-0.3

0

10

1 < k < 30

20

30

1 < k < 30

Fig. 5. (a) The distribution of decomposition coefficients of the log-periodic function rðlnðtÞÞ in expression (16) for the first signal corresponding to the ‘‘up’’ branch. (b) The distribution of decomposition coefficients of the log-periodic function rðlnðtÞÞ in expression (16) for the signal corresponding to the ‘‘down’’ branch.

Table 3 The additional parameters that enter to the fitting function (16) and describe the valve knocks in the idling regime (the reasons for the anomalous range of the amplitudes modulus in the last fragment are not clear and need specific research). Fragment

lnðnÞ

k

l

c

Y0

RelErrð%Þ

RangeðAmdÞ

Frag. Frag. Frag. Frag. Frag. Frag. Frag. Frag. Frag. Frag. Frag. Frag. Frag. Frag. Frag. Frag. Frag. Frag.

1.5932 1.6348 1.5968 1.6099 1.7315 1.6314 1.6182 1.5960 1.6737 1.7043 1.7054 1.8369 1.6422 1.7071 1.7272 1.7763 2.0067 2.0358

4.5431 3.2689 4.7822 3.4574 3.7088 2.1861 2.6812 3.1396 4.2467 1.5963 2.5153 0.1019 0.9586 2.4121 2.6330 2.6961 2.0844 1.8766

1.7983 1.1699 1.7283 1.4261 1.4164 1.0781 1.2501 1.6896 2.3474 1.4435 1.5366 0.5494 1.2752 1.5118 1.8734 1.9192 1.3953 1.3389

5.8107 4.3337 6.2367 4.4928 4.8440 2.8117 3.4326 3.8920 5.1629 1.7937 3.0252 0.2066 0.9534 2.9521 3.0732 3.1529 2.5300 2.3065

8.7942 7.3995 9.1179 7.5608 7.9036 6.1186 6.5877 7.0253 8.2089 5.2635 6.2904 3.3818 4.4899 6.0768 6.3114 6.3441 5.8365 5.5524

0.6085 0.8055 0.8711 0.6388 0.6195 0.9846 0.8409 0.6017 1.1757 0.6136 0.5585 0.6881 1.3074 0.8897 1.0359 0.6083 0.8668 0.6867

0.4490 0.3945 0.4064 0.4089 0.3793 0.4317 0.4411 0.3420 0.3872 0.4159 0.3849 6.0312 0.4589 0.3935 0.4114 0.4244 7177.15 8366.45

1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9

(up) (dn) (up) (dn) (up) (dn) (up) (dn) (up) (dn) (up) (dn) (up) (dn) (up) (dn) (up) (dn)

helps to understand their behavior from a general point of view. Here we want to stress also one important point. If the analyzed signal crosses zero in two points only (in the beginning and in the end of the signal) then it cannot be considered as a BLS. For the BLS defined above, ‘‘zero’’ regions should have a finite duration. These regions are clearly seen from the Fig. 1(a)–1(d). This specific requirement is associated with asymptotic behavior of the microscopic function f ðzÞ in (12) and (16). The assumed conjecture needs to be tested on other available data in order to formulate a more general principle that governs by means of the random behavior of different studied complex systems. Finally, there are engineering applications in which the BLS can arise from complex dynamics that characterize anomalous behaviors, possible faults in systems operation, performance different from specifications, etc. In this case, the characterization of BLS should help identification or prevention of related

undesired phenomena, then design of control strategies by properly operating on the calibration curves. In particular, for any calibration curve, the RFM should help us to understand the monotone changing of some external factor from the respective monotone changing of the fitting parameter. Each BLS can be replaced by a reduced set of the fitting parameters that have different sensitivity to the external factor. Then two signals can be compared only by means of the complete correlation factor in the space of the fractional moments [5]. This interesting possibility will be explored in future papers where data will be treated by considering the variations determined by some external factor or parameter. Appendix A Case A: n b P 1. Solution of the functional Eq. (10) for the case n b P 1

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UN ðzÞ ¼

0 Y

Xðz nÞ ¼ XðzÞ þ B z þ C lnðzÞ þ D:

n0 b j

½f ðz n j Þ

j¼ðN1Þ

"

¼ exp n0

The solution of (A.8) is written in the form

#

N1 X j b ln½f ðz nj Þ  exp½SN ðzÞ

2

ðA:1Þ

Here we use a general variable z instead of t in order to stress a general character of the expressions derived below. Comparing expression (8) with (A.1) one can notice that these cases are equivalent to each other if we realize the following transformations: n ! 1=n; b ! 1=b. So, the case n b 6 1 is considered in complete analogy with the case n b P 1 given below. From expression (A.1) and (8) based on these transformations one can easily obtain

1 N SðzÞ þ n0 ln½f ðz nÞ  n0 b lnðz nNþ1 Þ b

ðA:2Þ

Taking into account the series (9) one can easily see that, N

for jzj  1, the last term becomes negligible if ðn bÞ  1 that is equivalent to the requirement ðn bÞ P 1. So, this functional equation (A.2) is equivalent to Eq. (8) and will have the same solution but with different values of the constants. For further purposes, we present the functional Eq. (10) in the form

1 Xðz nÞ ¼ XðzÞ þ B z þ C lnðzÞ þ D: b

ðA:3Þ

Here z is an arbitrary variable, which can accept real or complex values, n is a scaling factor. This equation can be solved by a free variation constant method. We present the solution of this equation in the form

XðzÞ ¼ zm pm ðlnðzÞÞ þ k z þ l lnðzÞ þ X 0 :

ðA:4Þ

Here pm ðlnðzÞÞ is a log-periodic function with period lnðnÞ, the unknown constants m; k; l, and X 0 are found by the free variation constant method from (A.1). Taking into account the relationship

XðznÞ ¼ zm nm pm ðlnðzÞ þ lnðnÞÞ þ kzn þ l lnðzÞ þ l lnðnÞ þ X 0 ¼ zm nm pm ðlnðzÞÞ þ kzn þ l lnðzÞ þ l lnðnÞ þ X 0 ðA:5Þ and putting formulae (A.4) and (A.5) in (A.3) we obtain the following system of equations for the constants m; k; l and X 0

nm ¼

1 ; b

kn ¼

k þ B; b

XðzÞ ¼ p0 ðlnðzÞÞ þ k z þ c ln ðzÞ þ l lnðzÞ:

ðA:9Þ

Taking into account the relationship

j¼0

Sðz nÞ ¼

ðA:8Þ

l X l ¼ þ C; l lnðnÞ þ X 0 ¼ 0 þ D: b

b

ðA:6Þ From the system (A.6) we obtain finally

lnð1=bÞ B C ; k¼ ; l¼ ; lnðnÞ n  1=b 1  1=b D C lnðnÞ X0 ¼  : 1  1=b ð1  1=bÞ2

m¼

2

Xðz nÞ ¼ p0 ðlnðzÞÞ þ k z n þ c ln ðzÞ þ 2 c lnðnÞ lnðzÞ 2

þ c ln ðnÞ þ l lnðzÞ þ l lnðnÞ

ðA:10Þ

and putting (A.9) and (A.10) in (A.8), we obtain finally

k¼

B ; n1

c¼

C ; 2 lnðnÞ

C 2

l¼ þ

D lnðnÞ

ðA:11Þ

Based on the series (9), the functional equation (A.2) can be considered by analogy with the general functional equation (A.3). In this paper, we are not interested by analyzing the relationships (A.11) that connect the power-law exponents with the parameters figuring in series (9). We are going to calculate only the parameters figuring in equations (A.4) and (A.9). Moreover, the further relationships tightly related with parameters of the corresponding asymptotic behaviors (10), (17) merit further research. Appendix B A short description of the eigen-coordinates (ECs) method and the basic linear relationship (BLR) for the simplified function (12) is provided here. 1. The ECs method can reduce a wide class of functions initially containing a set of non-linear fitting parameters to another presentation of this function, which is defined as BLR

Yðxj Þ ¼ C 1 X 1 ðxj Þ þ    þ C s X s ðxj Þ:

ðB:1Þ

Here C k ðk ¼ 1; 2; . . . ; sÞ is a new set of parameters, which is related to the initial set of parameters by a nonlinear way. The parameter j ¼ 1; 2; . . . ; N coincides with the discrete number of data points. This BLR is obtained for an admissible/competitive hypothesis Fðx; AÞ (x is a current variable, Aða1 ; . . . ; ak Þ is the kth component fitting vector). We suppose that this function, in turn, satisfies a differential equation with new set of parameters C k entering into this equation by a linear way. 2. This differential equation can be transformed into the integral Volterra equation (in order not to increase an initial error by operation of numerical differentiation). It is implied that the initial error always accompanies the admissible hypothesis Fðx; AÞÞ chosen as the fitting function. 3. The BLR (B.1) reminds the decomposition of a ‘wave’ function Yðxj Þ over the finite set of eigenfunctions fX k ðxj Þgk¼1;2;...;s . Using the process of orthogonalization

ðA:7Þ

one can choose the set of orthogonal functions fWk ðxj Þgk¼1;2;...;s and present the initial function Yðxj Þ in

Case B: b ¼ 1. For this case, the functional equation (10) accepts the form

the form of a linear combination of fWk ðxj Þgk . This transformation is realized with the help of the following formula:

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~ 1 W1 ðxj Þ þ    þ C~ s Ws ðxj Þ: Yðxj Þ ¼ C

ðB:2Þ

C 1 ¼ l  m lnðP 0 Þ þ 1; C 2 ¼ kðm  1Þ; C 3 ¼ l m; C 4 ¼ m ðB:9Þ

Here

Wk ðxj Þ ¼ X k ðxj Þ 

k1 X ðWp ; X k Þ p¼1

ðWp ; Wp Þ

Wp ðxj Þ; W1 ðxj Þ ¼ X 1 ðxj Þ ðB:3Þ

for k ¼ 1; 2; . . . ; s. The parentheses in (B.3) determine the value of scalar product in the discrete space of dimension N, where N, as before, is determined by the number of measured points

ðWp ; X k Þ ¼

N X

Wp ðxj Þ X k ðxj Þ:

ðB:4Þ

j¼1

It is easy to notice from (B.3) that the new set of the functions fWk ðxj Þgs is orthogonal, i.e.

ðWp ; Wq Þ ¼ ðWp ; Wp Þ dpq :

ðB:6Þ

The usage of the orthogonal set of functions fWk ðxj Þgk helps to reduce the correlation matrix to the set of matrix elements located only on the main diagonal [21]. The basic advantage of the equivalent presentation (B.1) is that it does not need an initial guess of the parameters a1 ; . . . ; ak forming an initial fitting vector A. The constants CðC k Þ (forming a new fitting vector and appearing in the result of transformation of the initial fitting function Fðx; AÞ into the integral Volterra equation) are calculated with the help of the linear least square method (LLSM) as the values of the slopes of straight lines. The examples of the transformation of the corresponding fitting vectors ðA ! CÞ are given below. This presentation helps to increase the reliability of the hypothesis and always corresponds to the global minimum. The spurious (‘strange’) function Gðx; BÞ which does not satisfy the corresponding differential equation is degenerated to the set of the curves C k ðxj Þ, initially corresponding to a set of straight lines ‘tuned’ for recognition of the ‘native’ function Fðx; AÞ.

H0 ðzÞ ¼ P0 zl expðc zm þ k zÞ:

ðB:7Þ

This function is obtained from expression (12) if we replace the log-periodic function rðlnðzÞÞ by the constant ðcÞ. For further purposes it is convenient to choose the current time t as an independent variable. The initial fitting vector V for this function contains five components, i.e. VðP 0 ; l; c; m; kÞ. The differential equation for the function (B.7) has the form

dH0 ðtÞ t ¼ ðl  m lnðP 0 ÞÞ H0 ðtÞ  k ðm  1Þ t H0 ðtÞ dt  l m lnðtÞ H0 ðtÞ þ m H0 ðtÞ lnðH0 ðtÞÞ:

YðtÞ ¼ C 1 X 1 ðtÞ þ C 2 X 2 ðtÞ þ C 3 X 3 ðtÞ þ C 4 X 4 ðtÞ:

ðB:8Þ

One can notice that this nonlinear differential equation of the first order has a new set of four constants

ðB:10Þ

The new functions obtained after integration of (B.8) are written as

YðtÞ ¼ t H0 ðtÞ  h. . .i; Z t H0 ðuÞdu  h. . .i; X 1 ðtÞ ¼ X 3 ðtÞ ¼

ðB:5Þ

The initial set of constants C k is found from the linear system of equations s X ðWk ; X p Þ ~ k ¼ ðY; Wk Þ ¼ C k þ ~s ¼ Cs: Cp; C C ðWk ; Wk Þ ð Wk ; Wk Þ p¼kþ1

entering into equation (B.8) by a linear way. These relationships help to find four initial fitting parameters VðP0 ; l; m; kÞ, except c. So, the ECs method for this function is applicable and the BLR for the function (B.7) can be presented in the form

¼

t0 Z t

Z

X 2 ðtÞ ¼

Z

t

uH0 ðuÞdu  h. . .i;

t0

lnðuÞH0 ðuÞdu  h. . .i;

X 4 ðtÞ

t0 t

H0 ðuÞ ln½H0 ðuÞdu  h. . .i:

t0

ðB:11Þ In these coordinates the initial function (B.1) can be presented in the form of four straight lines. The last fitting parameter c (figuring as the third fitting parameter in the initial fitting vector V) is found as value of tangent from the following eigen-coordinates

Y c ðtÞ ¼ c X c ðtÞ; Y c ðtÞ ¼ lnðH0 ðtÞÞ  lnðP0 Þ  l lnðtÞ  k t; X c ðtÞ ¼ t m :

ðB:12Þ

The bracket h. . .i figuring in (B.11) coincides with the corresponding mean value

h. . .i ¼

N 1X ð. . .Þ N j¼1

ðB:13Þ

and should be subtracted. This operation is necessary to eliminate possible constants from equation (B.10) and provide the basic requirement for the linear least square method (LLSM), i.e. hei ¼ 0 for the error e [21]. The value of the error that should be minimized is defined from (B.10) as

e ¼ YðtÞ 

4 X

C s X s ðtÞ:

ðB:14Þ

s¼1

The constants C 1 ; . . . ; C 4 are found by applying the LLSM to (B.10). The initial vector of the fitting parameters VðA; l; c; m; kÞ is found from the nonlinear system of algebraic equations (B.9) and relationship (B.12). No needless to say that relationships (B.8)–(B.11) are initially verified for model data created for the function (B.1) with a certain level of initial error

ymod ðtÞ ¼ yðtÞ ð1 þ errðtÞÞ:

ðB:15Þ

Here the function errðtÞ imitated an error of measurements. The model experiments show that the recognition procedure remains stable if the relative value of the standard deviation

RelErr ¼

StDev ðyðtÞ  yf ðtÞÞ 100%; meanðjyðtÞjÞ

ðB:16Þ

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R.R. Nigmatullin et al. / Chaos, Solitons & Fractals 76 (2015) 166–181

does not exceed the 10–15% value. Here yf ðtÞ coincides with the fitting function. As it follows from numerical experiments that are initially realized for a variety of model files, one can only say that the ECs are expected to work reliably for the files having a large number of experimental points (in average more than 40–50 data points) and accompanied by an error with a relative dispersion less than 10–15%.

as the BLR and the value of the desired frequency x0 can be found from (D.2) by the LLSM. The differential equation for R1 ðtÞ from (23) is written in the form

h

i D4 þ ðx21 þ x22 Þ D2 þ x21 x22 R1 ðtÞ ¼ 0;

L0 ðtÞ ¼ eðtÞ þ l lnðtÞ þ k x þ Y 0 ;

eðtÞ ¼ c t m :

ðC:1Þ

The power-law function eðtÞ satisfies to the simplest differential equation of the type

deðtÞ m ¼ eðtÞ ! eðtÞ ¼ m dt t

Z

t

eðuÞ

t0

du : u

DL0 ðtÞ ¼ mD

Z

t

L0 ðuÞ t0



du 2 þ C 2 D½ln ðxÞ þ C 1 D½lnðxÞ þ C 0 D½x; u

DFðxÞ ¼ FðxÞ  h. . .i ¼ FðxÞ 

N 1X Fðxj Þ; N j¼1

ðC:3Þ which provides the BLR for the finding of the desired power-law exponent m by the LLSM. Here we replaced the function eðtÞ figuring in (C.2) from equation (C.1) and made necessary transforms of the corresponding integrands in order to obtain the desired BLR for L0 ðtÞ. Other constants C 0;1;2 entering in (C.3) are not essential and can be omitted. Appendix D Here we provide the BLR for the inoculating functions R0;1 ðtÞ from (23). The differential equation for the function R0 ðtÞ is written in the form

ðD2 þ x20 Þ R0 ðtÞ ¼ 0;

Dt

d : dt

Y 1 ðtÞ ¼

5 X

ðD:1Þ

After double integration and subtraction of the inessential constants, this equation with respect to the unknown frequency x0 can be transformed to the following form

Y 0 ðtÞ ¼ C 1 X 1 ðtÞ þ C 2 X 2 ðtÞ; Y 0 ðtÞ ¼ R0 ðtÞ  h. . .i; Z t pffiffiffiffiffiffiffiffiffi du ðlnðtÞ  lnðuÞÞ R0 ðuÞ  h. . .i; x0 ¼ C 1 X 1 ðtÞ ¼ u t0 X 2 ðtÞ ¼ lnðtÞ  h. . .i ðD:2Þ The value of the constant C 2 is not essential for further calculations and can be omitted. Expression (D.2) is used

ðD:3Þ

C s X s ðtÞ;

s¼1

Y 1 ðtÞ ¼ R1 ðtÞ  hR1 ðtÞi Z t du X 1 ðtÞ ¼ ½lnðtÞ  lnðuÞR1 ðuÞ  h. . .i; C 1 ¼ ðx21 þ x22 Þ; u t0 Z 1 t du 3 ½lnðtÞ  lnðuÞ R1 ðuÞ  h. . .i; C 2 ¼ ðx21 x22 Þ; X 2 ðtÞ ¼ 6 t0 u 3

2

X 3 ðtÞ ¼ ln ðtÞ  h. . .i; X 4 ðtÞ ¼ ln ðtÞ  h. . .i; X 5 ðtÞ ¼ lnðtÞ  h. . .i: ðD:4Þ

ðC:2Þ

Substituting in the last expression the function eðtÞ from (C.1) and making necessary integrations we finally obtain

d : dt

After fourfold integration this equation can be transformed into the BLR of the type

Appendix C Here we show the calculation of the power-law exponent m in the zeroth hypothesis L0 ðtÞ from (21). To this aim, we use the ECs method again. We present the function L0 ðtÞ in the form

Dt

The desired frequencies can be calculated from the quadratic equation

k2  C 1 k  C 2 ¼ 0;

x1;2 ¼

pffiffiffiffiffiffiffiffi k1;2 :

ðD:5Þ

The BRL of the type (D.4) is used for calculation of the nonlinear fitting parameters x1;2 in both cases. Other parameters C 3;4;5 figuring in the first row of expression (D.4) are not essential for calculation and hence they can be omitted. The symbol h. . .i, as before, is defined by expression (B.13) in Appendix B. References [1] Kohlrausch R. Ann Phys (Leipzig) 1847;12:393. [2] Williams G, Watts DC. Non-symmetrical dielectric relaxation behaviour arising from a simple empirical decay function. Trans Faraday Soc 1970;66:80–5. [3] Nigmatullin RR. Eigen-coordinates: new method of analytical functions identification in experimental measurements. J Appl Magn Reson 1998;14(4):601–33. [4] Nigmatullin RR. Recognition of nonextensive statistical distributions by the eigencoordinates method. Physica A Stat Mech Appl 2000;285(3):547–65. [5] Nigmatullin RR, Ceglie C, Maione G, Striccoli D. Reduced fractional modeling of 3D video streams: the FERMA approach. Nonlinear Dyn 2014. http://dx.doi.org/10.1007/s11071-014-1792-4. [6] West BJ, Bologna M, Grigolini P. Physics of fractal operators. New York, NY: Springer-Verlag; 2003. [7] Mainardi F. Fractional calculus and waves in linear viscoelasticity: an introduction to mathematical models. London, UK: Imperial College Press; 2010. [8] Feldman Yu, Kozlovich N, Alexandrov Yu, Nigmatullin R, Ryabov Ya. Mechanism of the cooperative relaxation in microemulsions near the percolation threshold. Phys Rev E 1996;54(5):5420–7. [9] Feldman Yu, Puzenko A, Ryabov Ya. Non-Debye dielectric relaxation in complex materials. J Chem Phys 2002;284(1–2):139–68. [10] Klafter J, Shlesinger MF. On the relationship among three theories of relaxation in disordered systems. Proc Natl Acad Sci USA 1986;83(4):848–51. [11] Blumen A, Klafter J, Zumoven G. Models for reaction dynamics. In: Zschokke I, editor. Optical spectroscopy of glasses. Dordrecht: Reidel Publishing Co.; 1986. [12] Metzler R, Klafter J. The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys Rep 2000;339(1): 1–77. [13] Sornette D. Discrete scale invariance and complex dimensions. Phys Rep 1998;297:239–70.

R.R. Nigmatullin et al. / Chaos, Solitons & Fractals 76 (2015) 166–181 [14] Sornette D, Johansen A, Arneodo A, Muzy JF, Saleur H. Complex fractal dimensions describe the hierarchical structure of diffusion-limited-aggregate clusters. Phys Rev Lett 1996;76(2): 251–4. [15] Johansen A, Sornette D, Hansen AE. Punctuated vortex coalescence and discrete scale invariance in two-dimensional turbulence. Physica D 2000;138(3–4):302–15. [16] Stauffer D, Sornette D. Log-periodic oscillations for biased diffusion on random lattice. Physica A 1998;252:271–7. [17] Pietronero L, Tosatti E, editors. Fractals in physics: the proceedings of the sixth Trieste international symposium on fractals in physics, Trieste, Italy: July 9–12, 1985. New York: Elsevier Science Publ. Co.; 1986. [18] Nigmatullin RR, Tenreiro Machado JA, Menezes R. Self-similarity principle: the reduced description of randomness. Cent Eur J Phys 2013;11(6):724–39. [19] Nigmatullin RR. Is it possible to replace the probability distribution function by its Prony’s spectrum? (I). J Appl Nonlinear Dyn 2012;1(2): 173–94. [20] Nigmatullin RR. The fluctuation metrology based on Prony’s spectroscopy (II). J Appl Nonlinear Dyn 2012;1(3):207–25. [21] Nigmatullin RR, Ionescu C, Baleanu D. NIMRAD: novel technique for respiratory data treatment. J Signal Image Video Process 2012;8(8): 1517–32.

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