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Mathematical and Theoretical Physics

Research Article Volume 1 Issue 4

Some characteristics of the function methodology to describe the reinjection process in chaotic intermittency

Ezequiel del R o,2 Luis F Guti rrez Marcantoni,1 Sergio Elaskar1

1Departamento de Aeronautica, Universidad Nacional de Cordoba and CONICET, Argentina
2E.T.S.I. Aeronautica y Espacio, Universidad Politecnica de Madrid, Spain

Correspondence: Sergio Elaskar, Departamento de Aeronáutica, FCEFyN, Instituto de Estudios Avanzados en Ingenierııía y Tecnologıía, IDIT, Universidad Nacional de Córdoba and CONICET, Córdoba, Argentina, Tel +5435 1535 3800

Received: July 29, 2018 | Published: August 27, 2018

Citation: Elaskari S, del Río E, Gutiérrez Marcantoni L. Some characteristics of the M function methodology to describe the reinjection process in chaotic intermittency. Open Acc J Math Theor Phy. 2018;1(4):168-173 DOI: 10.15406/oajmtp.2018.01.00029

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Abstract

The M function methodology allows calculating the statistical properties of intermittency in a broad class of one-dimensional maps. It has shown to be very accurate in type I, II, III and V intermittencies; and it also includes the uniform reinjection as a particular case. This paper studies some properties of theM function methodology. We establish the conditions that a reinjection probability density function must verify to obtain m=1/2,m=1/2,and we describe new pathological cases where the reinjection is not uniform, but the characteristic relation is the same that for uniform reinjection.

Keywords: m function methodology, reinjection process, chaotic intermittency, one-dimensional maps

Introduction

Intermittency is a traditional route to chaos, where a system evolves from regular behavior to chaotic behavior and trajectories alternate chaotic bursts and regular phases. The regular or laminar phases correspond to regions of pseudo-equilibrium or pseudo-periodic solutions, while the bursts are regions with chaotic evolution. In the early eighties, intermittency was classified, according to the system Floquet multipliers or the local Poincaré map eigenvalues, into three different types known as I, II and III.1‒4 Nevertheless, more recent advances have included other types such as V, X, on-off, in-out, ring and eyelet.5‒10 In engineering, biology, physics and chemistry there are several systems in which chaotic intermittency has been observed.11‒21 Furthermore, intermittency has been found in economics and medicine systems.22‒24 Consequently, a more accurate intermittency description might help to increase the knowledge about all these phenomena.

Intermittency behavior is defined by both the local map around the unstable or vanished fixed point and the reinjection mechanism.1,2,4 The reinjection mechanism maps trajectories from the chaotic zone to the laminar one, which is described by the reinjection probability density function (RPD). Accordingly, the accurate evaluation of the RPD function has a strong influence to describe the intermittency phenomenon correctly. Note that, the calculation of the RPD function from data series (experimental or numerical), is not a simple activity owing in no small amount of data needed and the statistical fluctuations involved in the numerical computations or experimental measurements. In consequence, many approaches have been utilized to describe the RPD function, where the most common one was a constant RPD (uniform reinjection).

A more general methodology to achieve the reinjection probability density function has been elaborated in the last decade, which is named the M function methodology. It includes the uniform reinjection as only a particular case. This methodology has shown to be very accurate for a wide class of maps showing type I, II, III and V intermittencies.4,25‒38 In this paper, we analyze some characteristics of this methodology. We have shown the M function methodology works very accurately for several maps, but there are pathological cases where it can only partially describe the intermittency reinjection process.

Evaluation of the RPD function

Let us consider a general one-dimensional map:x n+1 = F(x n),x n+1 = F(x n),which shows chaotic intermittency. To describe the reinjection process, we should evaluate the RPD function,φ(x),φ(x),which determines the probability density that trajectories are reinjected into a point x inside the laminar interval.1,2,4

A new theoretical scheme to evaluate the intermittency reinjection process is the M function methodology, in which the RPD is indirectly obtained. First, the M(x)M(x)function must be evaluated:4,25‒39

M(x)={xˆxτϕ(τ)dτxˆxϕ(τ)dτ,  if xˆxϕ(τ)dτ0,0,                otherwise,M(x)=⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪ ⎪ ⎪xˆxτϕ(τ)dτxˆxϕ(τ)dτ,  if xˆxϕ(τ)dτ0,0,                otherwise, (1)

Where ˆxˆxis the lower boundary of reinjection. From the data series the ˆxˆxcalculation is straightforward:

M(x)1NNJ=1xj M(x)1NNJ=1xj  (2)

Where the reinjection points {x,j}Nj=1{x,j}Nj=1must be sorted from the lowest to the highest,25‒33 i.e. xjxj+1.xjxj+1.

Previous studies have shown a linear M(x)M(x)for a wide class of maps with type I, II, III and V intermittencies:4,25‒39

M(x)={m(xˆx)+x,       if     ˆxxc,0,                 otherwhise, M(x)={m(xˆx)+x,       if     ˆxxc,0,                 otherwhise, (3)

c is the limit of the laminar interval.

In the previous equation the main parameter is m(0,1)m(0,1) (the M(x)M(x)function slope) which is determined by the nonlinear map. When M(x)M(x)is a linear function, it has been shown the RPD function is a power law:4,25-26,28,31,38,39

ϕ(x)=b(α)(xˆx)α,     with    α=2m11mϕ(x)=b(α)(xˆx)α,     with    α=2m11m (4)

Where b(α)=α+1(cˆx)(α+1)b(α)=α+1(cˆx)(α+1)is the normalization parameter.

From Eq. (4) two parameters are only needed to describe the RPD function: m and ˆx,ˆx,which are calculated from M(x).M(x). m is the slope of M(x),M(x),and it verifies M(ˆx) =ˆx,M(ˆx) =ˆx,i.e. it allows to calculate ˆx.ˆx.Then, the M(x)M(x)function stores the map nonlinear information.4,35,36

Finally, note that uniform reinjection (α = 0)(α = 0)is only a particular case of the new theoretical formulation when m=1/2.m=1/2.

Some characteristic of the M(x) function

In this section some properties of the M(x)M(x)function are studied. We have special interest to describe the behavior of ϕ(x)ϕ(x) and M(x)M(x)functions, mainly when the M(x)M(x)function slope is m=1/2.m=1/2.

In the following Theorem, will expose two conditions that the RPD must verify to obtain m=1/2m=1/2in Eq. (3).

Theorem:

If the reinjection probability density function, φ(x),φ(x),satisfies the following conditions:

  1. ϕ(ˆx)0ϕ(ˆx)0
  2. dϕ(x)dx|x=ˆx  is boundeddϕ(x)dxx=ˆx  is bounded

Then the M(x)M(x)function can be approximated as M(x)=x/2M(x)=x/2for points close enough to x =ˆx.x =ˆx.

Proof:

To prove this hypothesis, the slope of the M(x)M(x)function for x close to ˆxˆxcan be derived from Eq. (1):

limxˆxdM(x)dx=limxˆxxϕ(x)xˆxϕ(τ)dτϕ(x)xˆxτϕ(τ)dτ(xˆxϕ(τ)dτ)2limxˆxdM(x)dx=limxˆxxϕ(x)xˆxϕ(τ)dτϕ(x)xˆxτϕ(τ)dτxˆxϕ(τ)dτ2 (5)

Which by the L’Hôpital rule, gives

limxˆxdM(x)dx=limxˆxϕ(x)xˆxϕ(τ)dτ+xϕ(x)xˆxϕ(τ)dτϕ(x)xˆxτϕ(τ)dτ2ϕ(x)xˆxϕ(τ)dτ (6)

The last expression results:

limxˆxdM(x)dx=limxˆx[12+12xϕ(x)ϕ(x)12ϕ(x)ϕ(x)xˆxτϕ(τ)dτxˆxϕ(τ)dτ] (7)

Applying the L’Hôpital rule only on the last term, we can obtain:

limxˆxdM(x)dx=limxˆx[12+12xϕ(x)ϕ(x)12xϕ(x)ϕ(x)] (8)

Because  ϕ(x)0and dϕ(x)dx|x=ˆxis bounded, we obtain 1/2 for the limit given in Eq. 6.

This Theorem generalizes previous developments in whichˆx=x0=0,where x0=0is the vanished or unstable fixed point.4,26 For the Theorem, the LBR (ˆx)can be any point inside the laminar interval, being the vanished or unstable fixed point x0only a particular case.

If the conditions (1) and (2) are verified, then for a small laminar interval (c 0),the RPD can be uniform, and the classical theory is valid. However, there could be RPDs satisfying the two conditions and not having uniform reinjection; we call these RPDs as pathological cases.

Pathological case for type II intermittency

Type II intermittency starts from a subcritical Hopf bifurcation.2,4 Then, two complex-conjugate eigenvalues of the system leave the unit circle. When a control parameter (ε)is less than a critical value, the map has a stable fixed point. On the other hand, if the control parameter rises above this threshold, a bifurcation leads to a new chaotic attractor. The old attractor remains as a subset of the new attractor.

In several previous studies the Theorem of Section 3 has been verified for type I, II, III and V intermittencies with and without lower boundary of reinjection.4,25‒38

In this section, we study the intermittency process for a map showing type II intermittency. The map is an extension of those studied25,40 and it is given by:

F(x)={F1(x)=(1ε)xax3,  x<xrF2(x)=w(xxr1xr)γ2,  x>xr (9)

xris obtained from F1(xr)=1,and 0  w  1. The map has a fixed point x0=0 which is stable for 2 < ε < 0.It becomes unstable for ε > 0and type II intermittency can happen. The laminar interval is [0, c]. In Eq. (9), F1(x)function is the typical local map for type II intermittency and the nonlinear function, F2(x),generates the reinjection process. Therefore, the exponents γ1 and γ2will drive the reinjection mechanism.

To numerically analyze type II intermittency phenomenon for this map, we consider the following parameters: γ1=2,γ1=0.87,w=1/2,ε=0.001andN=100000,and we apply the M function methodology to calculate the RPD function. Figure 1 displays the results. Red points represent the numerical data and blue lines the theoretical results calculated using Eqs. (2) and (4).

Figure 1 Type II intermittency. M(x) and ϕ(x) functions for γ1=2, γ2=0.87,w=0.5, ε=0.001 and N=100000. Red points represent the numerical data, and the blue line the theoretical results calculated using Eqs. 2 and 4.

From Figure 1(A) we can observe the theoretical RPD cannot approach accurately the numerical RPD. Beside this, the numerical RPD verifies the conditions ϕ(ˆx)0 and bounded dϕdx|x=ˆx; therefore following the previous theorem the M(x)function slope must be m=1/2.Using the date shown in Figure 1(B) we can calculate m0.5060.5in agreement with the theory.

Following Section 2, if m0.5, the RPD must be constant (uniform reinjection). However, the numerical data in Figure 1(B) (red points) do not correspond to uniform reinjection. Therefore, it is a pathological case for the M function methodology; then the theoretical RPD (calculated using the M function methodology) does not show high accuracy with the numerical data.

To analyze the intermittency behavior for this pathological test, we study the map at points previous to reinjection (pre-reinjection points).39 If the reinjection points are xn+1=F(xn), we study the map derivative at xn points; because they have influence on the RPD function.35 The derivative of map (9) can be written as:

dF(x)dx={dF1(x)dx=(1+ε)+3ax2,                                                                       xxrdF2(x)dx=w(1xr)γ1γ1(xxr)γ11 +1w(1xr)γ2γ2(xxr)γ21,                  x>xr   (10)

Figure 3 shows the Eq. (10) (red line) inside the interval [xr0.682,F12(c)0.726473] which contains all the pre-reinjection points, i.e xn points previous to reinjection. Moreover, this figure shows the derivative of the map used in25,40 with γ=0.4(blue line) and γ=1(green line):

F1(x)={F1(x)=(1+ε)x+ax3,                      xxrdF2(x)dx=(xxr1xr)γ,                        x>xr (11)

xr verifies F1(xr)=1, and x0=0is a fixed point of the map, which is stable for 2<ε<0.

Note some differences between the three curves. The derivative of Eq. (9) (red line) has lower values regarding the derivative of Eq. (11) (blue line) –which is important for points close to xr– and the second derivative of map (9) goes from negative to positive values inside the interval [0.682, 0.726473]. This derivative behavior generates a different reinjection process between maps (9) and (11).

The RPD for γ=0.4is given in Figure 2(B), note as xˆx, the derivative dF(x)/dx,which implies ϕ(x)0.Also, for γ=0.4,the derivative dF(x)/dxdecreases inside the laminar interval.41‒47 On the other hand, for Eq. (9), the derivative F(x)/dx|xˆx does not tend to infinite, but it takes higher values than in other points in the laminar interval. This result might seem contradictory with the analytical limit of Eq. (10) which tends to infinite when xˆx.This contradiction can be explained because Figure 3 is obtained by a discrete process where there are a large but finite number of reinjected points, and not by a continuous process with infinite number of reinjected points (like as of Eq. (10)).

Figure 2 Type II intermittency. M(x) and ϕ(x) functions for γ=0.4,ε=0.0001, c = 0.1 and N = 20000. Red points represent the numerical data, and the blue line the theoretical results calculated using Eqs. (2) and (4).

As the derivative F(x)/dx|xˆxreaches higher values than in other points inside the laminar interval for map (9), then the RPD acquires lower and non-zero values close to ˆx(Figure 1(B)). As x increases, dF(x)/dx decreases and ϕ(x)increases until dF(x)/dx=0(inlection point), from this point dF(x)/dxgrows very softly and ϕ(x)decreases very softly too. The inflection point is xin0.707,it can be calculated by:

xin = xr+(1xr) (1wwγ1(1γ2)γ2(γ11))1γ1γ2(12)

Finally, we highlight that for uniform reinjection dF(x)/dxis constant (green line in Figure 3).

Figure 3 Type II intermittency. The derivative of maps (9) and (11). The parameters for map 9 are γ1=2, γ2=0.87, w=0.5, ε=0.001 and N=100000. The parameters for map 11 are γ=0.4 (blue) and γ=1 (green), ε=0.001. Red points correspond to map 9. Blue and green points correspond to map (11) γ=0.4with and γ=1respectively. Figure (B) is a zoom of Figure (A).

The characteristic relation is an important expression to determine the intermittency behavior. Using the M function methodology, this relation for type II intermittency is:4

ˉlαεβ(13)

Where the critical exponent βis given:25

β= 3α231=3(1m)1(31)(1m)  the same thing (14)

and  ˆlis the average laminar length, which can be calculated using:

ˉI=x0+cx0cϕ(x)l(x,c)dx (15)

l(x, c)is the laminar length for each point in the laminar interval, which measures the number of iterations inside the laminar interval.2,4

To study the characteristic relation, we have carried out several numerical tests. Table 1 shows m and αfor different ε. From the table, we can observe that αdoes not depend on the control parameter, ε.

From Table 1, we have found α0 and m1/2;then the theoretical evaluation for the characteristic relation exponent, given by Eq. (14), is β1/2.To verify this exponent, we calculate ˉlfor the tests included in Table 1. The results are exposed in Figure 4, which shows the numerical characteristic relation for the map (9). Red points are the numerical data, and the blue line is its linear interpolation. The slope of this straight line is approximately -0.495, which is very close to the theoretical exponent β=1/2predicted using the M function methodology.4,25 Another consequence of the previous Theorem is that the traditional values of the characteristic relation exponent,β=1/2,can be obtained not only for uniform reinjection but also for any RPD holding the conditions (1) and (2). Figure 4 shows this behavior. Therefore, the test studied in this subsection does not have a uniform RPD, but its characteristic relation is equal to those obtained with uniform reinjection.

ε

m

α

0.05

0.508

0.035

0.01

0.507

0.032

0.005

0.504

0.0189

0.001

0.506

0.024

0.0005

0.504

0.0188

0.0001

0.506

0.024

0.00001

0.505

0.0229

0.000001

0.505

0.0229

Table 1 Values of m, for different Parameters: w=0.5 and c=0.1

Figure 4 Type II intermittency. Characteristic relation for γ1=2,γ1=0.87, and w=0.5. Red points are the numerical average laminar length, and the blue line the linear interpolation.

Analysis and conclusion

In this paper, we have analyzed a theorem describing some properties of the M(x)and ϕ(x)functions. We have been able to identify numerically two reinjection mechanisms satisfying this theorem. One of them corresponds to uniform reinjection process, and the M function methodology captures and describes it very well. Pathological cases like Eq. (9) give the other one. For these cases, the M(x)function slope verifies m=1/2, but there is not uniform reinjection, and the M unction methodology would seem to introduce errors in the RPD evaluation, but it calculates the characteristic relation correctly.

The map derivative for pathological cases at points previous to reinjection, xnxn+1=F(xn)where xn+1are the reinjected points–shows a different behavior regarding to the non-pathological cases (Figure 3). For pathological behavior, the map derivative does not tend to infinity close to ˆxwhich is produced by the discrete process and the finite number of reinjected points. Another aspect to highlight is that inside the laminar interval the second derivative goes from negative to positive values.

TheM function methodology has theoretically established that the characteristic relation exponent,β, depends on αusing Eqs. (13) and (14). Here, we have found numerically that the exponent,β,depends on αin the same way for both pathological cases and uniform reinjection processes. The characteristic relation for all cases with α0.5results:

ˉlαεβ (16)

It is a valuable property because the characteristic relation is used to describe the intermittency process. Therefore, when we obtain from experimental or numerical data a characteristic relation like Eq. (16), we have to study other parameters to determine the intermittency behavior, because it can be a process with uniform reinjection or a pathological case.

In some previous papers, we have verified that the M function methodology works accurately in several maps with different intermittency types. Besides, here we have shown there are pathological cases where this methodology partially describes the intermittency reinjection process. For these cases, theM function methodology captures very well de characteristic relation, but it could introduce errors in the RPD function evaluation. This phenomenon occurs because a discrete reinjection process around x0=0 is studied; this process has a large but finite number of reinjected points. In this case none of the two exponents γin Eq. (9)(γ1=2and γ2=0.87) dominate the reinjection mechanism. However, if we study a process with an infinite number of reinjected points, the RPD will tend to zero for x0=0as shown Eq. (10), the Theorem of Section 4 would not be applied, and the M function methodology would work correctly.

Acknowledgements

This work was supported by CONICET, Universidad Nacional de Córdoba, and Universidad Politécnica de Madrid.

Conflict of interest

The author declares that there is no conflict of interest.

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