Luo A.C.J. (Ed.) Dynamical Systems: Discontinuity, Stochasticity and Time-Delay

Подождите немного. Документ загружается.

16 J.H. Peng and J.S. Tang

On the other hand, differentiating (2.5) one more time yields

«x

.t/ D

@f . x

.t/: (c)

The forms of (b) are the same with (c), hence the trial solution x

.t/ DPx

.t/ is a

solution of (2.9).

Now, we prove that the trial solution x

.t/ is the real solution of (2.9). The ﬁrst

derivative and second derivative can be easily obtained as follows

.t/ DRx

.t/

Œ Px

.t/

2

dt C Œ Px

.t/

1

;

.t/ D«x

.t/

Œ Px

.t/

2

dt CRx

.t/Œ Px

.t/

2

Rx

.t/Œ Px

.t/

2

D«x

.t/

Œ Px

.t/

2

Substituting representation (b) into this equation yields

.t/ D«x

.t/

Œ Px

.t/

2

dt D

@f . x

.t/

Œ Px

.t/

2

dt: (d)

Meanwhile, we substitute x

.t/ into (2.9)

.t/ D

@f . x

.t/

Œ Px

.t/

2

dt:

It is exactly the same with representation (d), and it shows that the trial solution

.t/ is the solution of (2.9). So far, the fundamental solutions of homogeneous

equation (2.9) have been determined.

It is obvious that x

.t/ and x

.t/ are linearly independent, thereby the general

solution of (2.9) is the linear combination of x

.t/ and x

.t/,thatis

.t/ D c

.t/ C c

.t/:

According to the method of variation of constants, the general solution of the

ith-order perturbation equation

@f . x

C h

.t; x

;:::;x

i1

IPx

; Px

;:::; Px

i1

/; i D 1;2;::: (2.11)

can be expressed as

.t/ D c

.t/x

.t/ C c

.t/x

.t/: (2.12)

Differentiating it with respect to t yields

.t/ DPc

.t/x

.t/ CPc

.t/x

.t/ C c

.t/ Px

.t/ C c

.t/ Px

.t/: (2.13)

2 Sensitivity of Chaos to Initial Condition and Perturbation 17

Because there is only one equation (2.11) that the two unknown functions c

.t/ and

.t/ must fulﬁll, we can set another expression that the two unknown functions

should hold. Representation (2.13) implying that the second derivatives of c

.t/

and c

.t/ is undesired in Rx

.t/ we can set

.t/x

.t/ CPc

.t/x

.t/ D 0; (2.14)

so that

.t/ D c

.t/ Px

.t/ C c

.t/ Px

.t/: (2.15)

differentiating (2.15) with respect to t one more time results in

.t/ DPc

.t/ Px

.t/ CPc

.t/ Px

.t/ C c

.t/ Rx

.t/ C c

.t/ Rx

.t/: (2.16)

Substituting (2.12)and(2.16)into(2.11), one has

.t/ Px

.t/ CPc

.t/ Px

.t/ C c

.t/



.t/ 

@f . x

.t/



.t/



.t/ 

@f . x

.t/



D h

: (2.17)

Employing representations (2.9)and(2.10), this equation becomes

.t/ Px

.t/ CPc

.t/ Px

.t/ D h

: (2.18)

Combining equations of (2.14)and(2.18), the condition is that the solutions of

unknown variables c

.t/ and c

.t/ are nontrivial is

W D





D x

 x

¤ 0: (2.19)

That is

W D x

 x

DPx

.t/



.t/

Œ Px

.t/

2

dt CŒ Px

.t/

1







.t/

Œ Px

.t/

2



.t/ DPx

.t/Œ Px

.t/

1

D 1: (2.20)

Condition (2.19) is fulﬁlled, and then the nontrivial solutions of c

.t/ and c

.t/ are

(

.t/ Dx

.t/h

.t/ D x

.t/h

: (2.21)

18 J.H. Peng and J.S. Tang

Integrating these equations yields

(

.t/ D C

x

.t/h

.t/ D C

.t/h

; (2.22)

where C

are the integral constants that are related to the initial conditions.

With the initial conditions (2.4) and expressions (2.12), (2.15), we obtain C

and

as follows

(

D x

.0/ Px.0/ Px

.0/x.0/

DPx

.0/x.0/  x

.0/ Px.0/

: (2.23)

Substituting them into representations (2.22) yields

(

.t/ D x

.0/ Px.0/ Px

.0/x.0/ C

x

.t/h

.t/ DPx

.0/x.0/  x

.0/ Px.0/ C

.t/h

: (2.24)

To simplify (2.24), we choose a special time t

D A

and it fulﬁlls

.0/ Px.0/ Px

.0/x.0/ 

.t/h

dt D 0: (2.25)

With the same method, we select t

D A

and it holds

.0/x.0/  x

.0/ Px.0/ 

.t/h

dt D 0: (2.26)

Therefore, representations (2.24) reduce to

(

.t/ D

.t/h

.t/ D

.t/h

; (2.27)

where A

and A

are the constants determined by the initial conditions. Substitut-

ing them into (2.12), we get the general solution of ith-order perturbation equation

.t/ D x

.t/

.t/h

dt x

.t/

.t/h

dt; i D 1;2;::: (2.28)

Differentiating it with respect to t results in

.t/ DPx

.t/

.t/h

dt Px

.t/

.t/h

dt; i D 1;2;::: (2.29)

2 Sensitivity of Chaos to Initial Condition and Perturbation 19

2.4 Sensitivity to the Initial Conditions

According to Ref. [16], the homoclinic point and heteroclinic point both are the

core of the chaotic structure. Therefore, what the chaotic motion is sensitivity to

the initial conditions is transferred to that of the homoclinic orbit or the heteroclinic

orbit is sensitivity to the initial conditions. Setting that the solution fx

.t/; Px

.t/g

of unperturbed equation (2.5) of nonlinear second differentiating dynamical system

(2.1) is a homoclinic orbit. In the light of the deﬁnition of homoclinic orbit, the orbit

will tent to the saddle as t !1[17]. It implies Px

.t/ ! 0; Rx

.t/ ! 0 as t !1.

Therefore, the fundamental solution x

.t/ (see (2.10)) is a divergent function. That

is, x

.t/ ! 0 as t !˙1.

Setting that there is a small change of the initial condition in (2.1) and it gives

rise to the integral lower limits of (2.28) transforming from A

and A

into A

and A

.i D 1;2;:::/, Consequently, the general solution of i th order equation

and its derivative transit from x

.t/, Px

.t/ to x

.t/, Px

.t/ .i D 1;2;:::/. The differ-

ences, which is caused by the small change of initial condition, between x

.t/ and

.t/ are

.t/  x

.t/ D x

.t/

.t/h

dt x

.t/

.t/h





.t/

.t/h

dt x

.t/

.t/h



D x

.t/

.t/h

dt 

.t/h

D Ax

.t/  Bx

.t/; i D 1;2;::: (2.30)

and the differences between Px

.t/ and Px

.t/ are

.t/ Px

.t/ DPx

.t/

.t/h

dt Px

.t/

.t/h





.t/

.t/h

dt Px

.t/

.t/h



D A Px

.t/  B Px

.t/; i D 1;2;::: (2.31)

The two equations given above show that only if A

D A

and A

D A

the

function in all square brackets equal zero, so that x

.t/ D x

.t/, Px

.t/ DPx

.t/.

Otherwise, x

.t/ ¤ x

.t/, Px

.t/ ¤Px

.t/. With (2.25)and(2.26) we know that any

small perturbation of initial condition will give rise to A

¤ A

, A

¤ A

(2.30)and(2.31). Therefore, we have x

.t/ ¤ x

.t/, Px

.t/ ¤Px

.t/. Inserting them

in (2.2), we obtain x.t/ ¤ x

.t/, Px.t/ ¤Px

.t/, that is to say the two orbits corre-

sponding to two sets of little different initial conditions respectively are different.

Equations (2.30)and(2.31) imply that x

.t/  x

.t/ !1and Px

.t/ Px

.t/ !1

20 J.H. Peng and J.S. Tang

as t !1because of the inﬁnity of x

.1/ and Px

.1/, namely, the difference of

the two orbits tends inﬁnity. The above mentioned analysis clearly shows that the

chaotic motion is sensitively dependent on the initial conditions.

2.5 Sensitivity to Perturbations

In this section, we prove the sensitivity of chaotic motion to the perturbations

with the same approach we used in the last section. To this end, we sup-

pose that the perturbation acting on the system (2.1) changes from g.t; Px/ into

.t; Px/. It makes h

.t; x

;:::;x

i1

IPx

; Px

;:::; Px

i1

/ of ith order perturba-

tion equation change into h

.t; x

;:::;x

i1

IPx

; Px

;:::; Px

i1

/ Px

; Px

;:::; Px

i1

Hence, the solution of ith order perturbation equation .i D 1;2;:::/ tran-

sits from x

.t/ to x

.t/, Px

.t/ to Px

.t/ .i D 1;2;:::/. The chaotic orbit,

.t/; x

.t/; : : : ; x

.t/; : : : IPx

.t/; Px

.t/; : : : ; Px

.t/; : : :g of the system turns into

the new orbit of fx

.t/; x

.t/; : : : ; x

.t/; : : : IPx

.t/; Px

.t/; : : : ; Px

.t/; : : :g. The dif-

ference between the two orbits can be determined by calculating the differences

.t/  x

.t/ and Px

.t/ Px

.t/. Because the initial conditions of the two orbits

are identical so that A

and A

remain the same, and the unperturbed solution

is independent of perturbation as well as, we can easily obtain the following results

.t/  x

.t/ D x

.t/



.t/.h

 h

/dt



x

.t/



.t/.h

 h

/dt



; (2.32)

.t/ Px

.t/ DPx

.t/



.t/.h

 h

/dt



Px

.t/



.t/.h

 h

/dt



; (2.33)

where i D 1;2;:::. These expressions state that as long as h

 h

does not equal

to zero, x

.t/  x

.t/ and Px

.t/ Px

.t/ will not be vanished. Since h

¤ h

and

lim

t!1

.t/ !1, lim

t!1

.t/ !1, above two equations become

lim

t!1

Œx

.t/  x

.t/ D lim

t!1



.t/

.t/.h

 h

/dt

x

.t/

.t/.h

 h

/dt



 lim

t!1



.t/

.t/.h

 h

/dt



!1;

2 Sensitivity of Chaos to Initial Condition and Perturbation 21

lim

t!1

Œ Px

.t/ Px

.t/ D lim

t!1



.t/

.t/.h

 h

/dt

Px

.t/

.t/.h

 h

/dt



 lim

t!1



.t/

.t/.h

 h

/dt



!1;iD 1;2;:::

These results clearly shows that any small change of the perturbation acting on the

system will cause the orbit of the system to go far away from the original one. That

is to say that the chaos possesses the property of sensitivity to the perturbation.

References

1. Lorentz EN (1963) Deterministic non-periodic ﬂow. J Atmos Sci 20:130–141

2. Li T, Yorke JA (1975) Period three implies chaos. Am Math Monthly 82:985–992

3. Peng JH, Tang JS, YU DJ, et al (2002) Solutions bifurcations and chaos of the nonlinear

Schrodinger equation with weak damping. Chin Phys 11:213–217

4. Lee CH, Hai WH, Lei S, et al (2001) Chaotic and frequency-locked atomic population oscilla-

tions between two coupled Bose–Einstein condensates. Phys Rev A64:053604

5. Hai WH, Zhang XL, Huang WL, et al (2001) Chaotic solution of the rf-driven Josephson

system with quadratic damping. . Int J BifurcatChaos 11:2263–2269

6. Hai WH, Duan YW, Zhu XW, et al (1998) Stable orbits embedded in a chaotic attractor for a

trapped ion interacting with a laser ﬁeld. JPhysAMathGen31:2991–2996

7. Peng JH, Tang JS, Yu DJ, et al (2003) Suppressing Chaos by parametric perturbation at doubled

frequency of periodic perturbation. Chin phys 12:17–21

8. Pecora IM, Carrol TL (1998) Synchronization in chaotic systems. Phys Rev Lett 64:821

9. Badola P, Tambe SS, Kulkami BD (1992) Driving systems with chaotic signals. Phys Rev

A46:6735

10. Van Wiggeren GD, Roy R (1998) Communication with chaotic lasers. Science 279:1198–1200

11. Gauthier DJ (1998) Chaos has come again. Science 279:1156–1157

12. Lin WW, Wang YQ (2002) Chaotic synchronization in coupled map lattices with periodic

boundary conditions. SIAN J Appl Dyn Syst 1:175–189

13. Eckman JP, Ruelle D (1985) Ergodic theory of chaos and strange attractors. Rev Mod Phys

57:617–656

14. Ruelle D (1980) Measures describing a turbulent ﬂow. AnnNYAcadSci357:1–9

15. He D-H, Xu J-X, Chen Y-H (2000) Study on Lyapunov characteristic exponents of a nonlinear

differential equation system. Acta Phys Sin 49:833–837

16. Liu SD, Liu SS (1994) Solitons and turbulences. Shanghai Scientiﬁc and Technological

Education Publishing House, Shanghai, p 102

17. Liu Z (1994) Perturbation criteria for chaos. Shanghai Scientiﬁc and Technological Education

Publishing House, Shanghai, p 12

Chapter 3

Study on the Multifractal Spectrum of Local

Area Networks Trafﬁc and Their Correlations

YanLiuandJia-ZhongZhang

Abstract Due to the singularity in the Local Area Network (LAN) trafﬁc, the

multifractal spectrums are used to study the characteristics of network trafﬁc from

the viewpoint of nonlinear dynamic system. First, the multifractal spectrums of the

LAN trafﬁc are introduced and established to investigate the complex features of

the systems. Then, the distributions of spectrum parameter vs. the network trafﬁc

are studied in detail, and some important phenomena, which are related with the

complicated networks trafﬁc, are captured. Furthermore, the correlations between

multifractal spectrum and logarithm of mean trafﬁc are presented, and they can be

feasibly applied to the prediction for the networks trafﬁc. Some conclusions can

be drawn that the variation of width of multifractal spectrum is similar to that of

network trafﬁc in a sense. To some degree, the difference between maximum and

minimum probability of multifractal spectrum is ahead to the oscillation of network

trafﬁc, and it is a fundamental route for the network trafﬁc prediction.

3.1 Introduction

There exists a rich variety of nonlinear phenomena in Network trafﬁc. As the study

of the characteristics of network trafﬁc is in-depth, especially on the study of TCP

data streams as well as the behaviors in scale of the global area network, it is found

that the network trafﬁc in larger time scales shows single fractal characteristics,

while in smaller time scales shows the multifractal characteristics of network traf-

ﬁc obviously, that is, the network trafﬁc behaves as a complex unsteady ﬂow, and

in particular, there exists a local breaking and bursting in the trafﬁc [1–4]. Tra-

ditional methods of network trafﬁc analysis and the models are focused on the

self-similarity of the network trafﬁc, in the relatively large time scale [5–8]. How-

ever, these methods and models could not be used to describe and investigate the

J.-Z. Zhang (



)

School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi 710049,

People’s Republic of China

e-mail: jzzhang@mail.xjtu.edu.cn

A.C.J. Luo (ed.), Dynamical Systems: Discontinuity, Stochasticity and Time-Delay,

DOI 10.1007/978-1-4419-5754-2

 Springer Science+Business Media, LLC 2010

24 Y. Liu and J.-Z. Zhang

local characteristic of the time sequence of network trafﬁc in small or ﬁne scale in

detail, and the breaking and bursting behaviors could not be captured. Hence, the

multifractal is introduced to the analysis of the time-sequence of networks trafﬁc in

this study, especially for the rich variety of nonlinear information.

Further, it is found that there exists the complex singularity in the time-sequence

of network trafﬁc in the global area network trafﬁc. As for the local network, there

are a few studies on such topics. With such background, the current work focuses

on the essential features of the fractal structure in the time-sequence of local area

networks trafﬁc, especially the multilevel feature. First, the essential features of the

local area network trafﬁc are studied by the multifractal spectrum, and the rela-

tionship between the multifractal spectrum and the ﬂux of the system is analyzed

further. Some important results are obtained, and can be applied to the prediction of

the network trafﬁc.

3.2 Multifractal and the Spectral Parameters

for Networks Trafﬁc

In a sense, multifractal is the decomposition of the fractal object, and the component

will have its own fractal dimension. Assume that there exists a unit interval with a

unit mass, and it is further divided into several subintervals with the length ı of each

interval for. Then, a nonnegative mass  is allocated in the kth sub-interval, and

the sequence f

;k  1g denotes a stochastic process. Hence, the exponent for

singularity of 

at time t

can be deﬁned as follows:

˛.t

/ D lim

ı!0

ln.

ln ı

; (3.1)

where 

denotes the mass of the interval containing t

in time. Normally, singu-

larity exponent ˛.t/ is also called as H˘older exponent, which describes the fractal

dimension in the fractal geometric theory, namely, local fractal dimension, and can

describe the probability of growing of the small interval.

If there exists no limit in (3.1), then the singularity exponent could not be deﬁned

at time t

;If˛.t/ is a constant, then the singularity of the sequence at the overall

scale can be described by only one global exponent, which is the feature of single

fractal; If ˛.t/ is the function of time t, that is, the feature of its scale is related

to time, then the sequence has a multifractal characteristics. In comparison with the

single fractal, the concept of multifractal extends the understanding of the scale, and

the scale relevant to time could describe the nonregular behaviors in the local time

interval.

The multifractal is used to describe fractal dimension of the domain with a

large number of small area, and ˛.t/ would extend the single fractal exponent

(Hurst parameter) to the one with multivalues. To this end, it needs to know the

probability of ˛.t/ with different values, in order to analyze the characteristics of

network trafﬁc. As the results, a spectrum f.˛/composed by inﬁnite sequence con-

taining various ˛.t/,andf.˛/ fractal dimension of subsets with same ˛, namely,

the probability of the occurrence of ˛ in the system, and it can be called multifractal

spectrum. The value of f.˛/ values should be in the interval [0,1], and usually be

the convex shape, which can be considered the probability density for the exponent

˛.t/.

Multifractal spectrum f.˛/ could represent the nonuniform property of network

trafﬁc in the fractal structure, so that much more information on the structure can

be gained, when compared with the simple single fractal dimension. If the multi-

fractal spectrum ˛.t/ < 1 for a sequence, then the sequence in the small interval

around a certain point will behave as bursting and breaking in all of the scales.

If ˛.t/ > 1, it when demonstrate that the network trafﬁc varies slightly without

obvious bursting and breaking. Hence, it can use the range of ˛.t/ to determine

bursting and breaking properties of network trafﬁc. Multifractal spectrum width

˛ D ˛

max

 ˛

min

could describe the property of nonuniform of the probability

for the entire fractal structure, so that the features, related with the different area,

different level, and different local area, can be described in detail. In addition, the

multifractal spectrum parameters ˛

max

and f.˛

max

/ represent the property of the

minimum subset of the probability, and ˛

min

and f.˛

min

/ represent the property of

the maximum subset of the probability. Hence, multifractal spectrum is a measure

for the complexity, non-regularity and nonuniform property of the fractal structure

of the sequence.

3.3 Relationship Between Variables of Multifractal Spectrum

A function 

."/, namely, distribution function, can be deﬁned by the follows,



."/ D

."/

D "

.q/

; (3.2)

where .q/ is the mass function or structure function. If (3.2) can be satisﬁed, that

then the mass function can be deﬁned as

.q/ D

ln 

."/

ln "

: (3.3)

According to the deﬁnition of multifractal, the generalized multifractal dimension

by .q/ can be expressed as

.q/

q  1

ln 

."/

.q  1/ ln "

: (3.4)

3 Multifractal Spectrum of LAN Trafﬁc

26 Y. Liu and J.-Z. Zhang

Following Legendre transform, there exist some relationships between ˛; f .˛/;

.q/ and D

,theyare

q  1

Œq˛  f.˛/; (3.5)

f.˛/ D q˛  .q/; (3.6)

˛ D

d.q/

: (3.7)

By (3.5), D

can be obtained as ˛ and f.˛/ are given, and ˛ can be obtained

from (3.7), in combination with the derivation of .q/. Multifractal spectrum can

be further obtained by (3.6)and(3.7),

f.˛/ D q

d.q/

 .q/: (3.8)

Furthermore, following (3.4)and(3.7), yields,

˛ D

d.q/

Œ.q  1/D

: (3.9)

It implies that ˛ can be also obtained if D

is prescribed. In the above mentioned

equation, q is the weighting factor, and the multifractal spectrum can be used to

analyze the different level of fractal structure decomposed by q.

3.4 Numerical Simulation and Analysis

LAN trafﬁc sequence used in this study comes from the internal network of a central

group of network monitoring system during March 16, 2005 to March 28, 2005.

This type of LAN is the Ethernet host whose components come from hundreds

of medium-sized local area networks, and the network trafﬁc data is obtained by

snifﬁng tools, recording the number of network packets and the amount of data with

a time interval of 1 s, and resulting in a nonnegative time series. The transmission

of information includes web browsing, ﬁle transfer, network operating systems, etc.

3.4.1 Multifractal Spectrum and Parameters

Figure 3.1 shows the analysis of the multifractal spectrum of the time sequence ob-

tained during March 16, 2005, and the multifractal spectrum parameters are shown

in Table 3.1. It can be seen from the numerical simulation results that there ex-

ists a signiﬁcant difference in the width of the multifractal spectrum and its shape.