Volume. 4., 2013 ISSN 2253-0371 - arctbds.com · Aims and scope Annual Review of Chaos Theory, Bifurcations and Dynamical Systems (ARCTBDS) () is a multidisciplinary international

Volume. 4., 2013 ISSN 2253-0371

Copyright (c) 2013 Annual Review of Chaos Theory, Bifurcations and Dynamical Systems (ARCTBDS). ISSN 2253-0371. All Rights Reserved.

www.arctbds.com.

Aims and scope

Annual Review of Chaos Theory, Bifurcations and Dynamical Systems (ARCTBDS)(www.arctbds.com) is a multidisciplinary international peer reviewed journal of chaos theory, bifurcations and dynamical systems publishing high-quality articles quarterly. The primary objective of this review journal is to provide a forum for this multidisciplinary discipline of chaos theory, bifurcations and dynamical systems as well as general nonlinear dynamics. This review journal will be a periodical or series that is devoted to the publication of review (and original research) articles that summarize the progress in particular areas of chaos theory, bifurcations and dynamical systems during a preceding period. Also, the journal publishes original articles and contributions on the above topics by leading researchers and developers.

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Annual Review of Chaos Theory, Bifurcations and Dynamical Systems (ARCTBDS)(www.arctbds.com) covers all aspects of chaos theory, bifurcations and dynamical systems. Topics of interest include, but are not limited to:

Mathematical modeling, computational methods, principles and numerical simulations Chaos, bifurcation, nonlinear dynamical systems, complexity in nonlinear science and engineering and numerical methods for nonlinear differential equations. Fractals, pattern formation, solitons, coherent phenomena and nonlinear fluid dynamics. Control theory and stability and singularity on fundamental or applied studies. Real applications in all areas of science.

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Volume. 4., 2013

Table of Contents

Evidence of Noisy Chaotic Dynamics in the Returns of Four Dow Jones Stock Indices

John Francis T. Diaz 01-15

Dynamic Behaviour of a Unified Two-Point Fourth Order Family of Iterative Methods

D. K. R. Babajee and S. K. Khratti 16-29

A Common Fixed Point Theorem of Presic Type for Three Maps in Fuzzy Metric Space

P. P. Murthy, Rashmi 30-36

Analysis of Dual Functions

Farid Messelmi 37-54

Chaotic Dynamical Behavior of Recurrent Neural Network

A. Zerroug, L. Terrissa, A. Faure 55-66

Annual Review of Chaos Theory, Bifurcations and Dynamical SystemsVol. 4, (2013) 1-15, www.arctbds.com.Copyright (c) 2013 (ARCTBDS). ISSN 2253–0371. All Rights Reserved.

Evidence of Noisy Chaotic Dynamics in theReturns of Four Dow Jones Stock Indices

John Francis T. DiazChung Yuan Christian University, College of Business, Chung-li City, Taiwan

e-mail: [email protected]

Abstract

This research finds evidence of noisy chaotic properties in the returns of fourDow Jones indices, based on three tests of non-linearity and chaos. The study usesan average of 24,815 data points to correctly simulate chaos in financial time-series.The data consists of the Dow Jones Industrial Average (29,229 observations); DowJones Transportation Average (29,121 observations); Dow Jones Utility Average(21,150 observations) and the Dow Jones Composite Average (19,906 observations).The a) Brock, Dechert, and Scheinkman (BDS) test indicates that most of the DowJones indices are not iid series, except for the filtered residuals from the GARCHof the Dow Jones Utility Average. The b) rescaled range analysis shows that afterscrambling the data, all Hurst exponents are above 0.5, and a trend-reinforcingproperty, which helps in the conclusion of having a chaotic process. Lastly, the c)correlation dimension analysis complements the initial findings and concludes thepresence of a high dimensional noisy chaotic structure in the four Dow Jones indices.

Keywords: Stock returns and volatility, Dow Jones indices, Statistical Physics, Noisychaotic process

Manuscript accepted: May 13, 2013.

1 Introduction

The straightforward solutions given by linear models are becoming inadequate with thegrowing complexities of financial time-series. Investors’ treatment of risk and expectedreturns, strategic interactions between financial market participants and the way by whichinformation is integrated into security prices, all behave in a nonlinear manner [1]. The

2 John Francis T. Diaz

authors conclude that the natural tendency in modeling financial time-series is to con-sider nonlinear dynamics. The importance of investigating nonlinearities in time-seriesdata provides a hint on the hidden structure of the data, which makes it easier to dis-tinguish between their random and chaotic properties [2]. Econometric models related tothese two tendencies can be both stochastic and deterministic in nature. The study of [3]provided an initial evidence on the existence of chaos in financial markets. The authorexplained that a chaotic structure is a nonlinear system that appears to be random innature making it easily misinterpreted as a random process by linear econometric method.

Chaos as an area of Econophysics has been starting to be recognized in the literaturebecause of the availability of longer time-series data, which according to [4], enhancesthe accuracy of results and recommended a volume of at least 5,000 observations to suf-ficiently detect deterministic dynamics. The authors studied the S&P Composite PriceIndex using 16,127 observations and concluded that the S&P 500 showed strong evidenceof chaos based on Grassberger-Procaccia’s correlation dimension measurement with non-linear noise filtering and a surrogate technique. A year ago, [5] used daily observations of7,917 data points and observed nonlinear properties on the returns of the Swedish StockIndex. In [6], the author used a much higher frequency data of 27,523 continuous obser-vations of the Dow Jones Industrial Average and found that long-range correlations asshown by fractal fluctuations are related to quantum-like chaos. Although the minimumnumber set by [4] was not strictly followed, a recent study of [7] still found conclusiveevidences of nonlinearities and chaos in the returns of the main stock market indices ofCzech Republic with 4,369 returns, Hungary with 4,577 observations and Poland using4,575 data points.

Following the recommendation of having a longer time-series to identify deterministicprocesses, this paper was motivated to study the oldest group of stock indices still in usetoday, the four Dow Jones stock indices. Although the oldest index in the family is theDow Jones Transportation Index (DJTA), which was established in 1884, the Dow JonesIndustrial Average (DJIA) is the best known index, which was created in 1896. With theDJTA together with the DJIA, also experiencing record highs during the first quarter of2013, economists contend that the US economy might be experiencing growth even in asubtle manner. This research is also interested in studying the relatively younger indices,the Dow Jones Utility Average (DJUA) and Dow Jones Composite Average (DJCA),which was created in 1929 and 1939, respectively.

This research examines evidences on the possibility of finding nonlinearities, particularlychaotic tendencies of the four Dow Jones stock indices. The persistence and nonlinearproperties of the DJIA using different time-series have been established in the literature(e.g., [8-12]), but none of these have carefully studied the nonlinear tendencies of the otherDow Jones indices, specifically its chaotic properties. Another unique contribution of thispaper is the sole consideration of a very long time-series without structural breaks in thedata. The minimum requirement of atleast 5,000 daily data limited the paper on havingstructural breaks, because the interval of some crises and possible structural breaks occurin less than 5,000 daily observations [4].

Evidence of Noisy Chaotic Dynamics in the Returns of Four Dow Jones Stock Indices 3

Three different approaches in testing nonlinear and chaotic properties have been alreadyestablished in the literature, namely, Brock, Dechert, and Scheinkman (BDS) test of [13],Rescaled Range (R/S) analysis originated by [3], and the Correlation Dimension (CD)analysis of [14]. This study contributes to the literature of applying Statistical Physicsmethodologies to Economics or what is now known as the field of Econophysics, and aimsto 1) provide additional evidence on the nonlinearities in financial time-series through ex-amining chaotic properties of the DJIA, DJTA, DJUA and DJCA returns. This researchalso wants to 2) challenge the validity of the efficient market hypothesis (EMH) of [15],which has been the explanation on the stochastic attributes of financial time-series. Tothe best of the author’s knowledge, no research yet has been done to determine chaos inthe returns of these four Dow Jones indices. The strength of studying a relatively longertime-series can benefit the investing community in understanding the Dow Jones indicesmarket behavior and provides a considerable amount of knowledge for both academiciansand researchers in providing potential avenues for future research.

The paper is structured as follows. Section 2 presents the data and methodology of thethree tests of Chaos, namely, BDS test, R/S analysis and the CD analysis; Section 3interprets empirical findings; and Section 4 provides the conclusion.

2 Data and Methodology

This research analyzes daily closing prices of the four Dow Jones indices from the FederalReserve Bank of St. Louis database until February 26, 2013. Although the oldest stockindex, the DJTA was created in 1884, the oldest available data provided started fromMay 27, 1896 and has a total of 29,229 observations for DJIA; DJTA began from October27, 1896 and has 29,121 data points; DJUA started from January 3, 1929 and has 21,150observations; and DJCA began from January 3, 1934 with 19,906 data points. The seriesof returns were computed as, yt = 100 (log pt − log pt−1) where pt represents the DowJones index price at time t.

2.1 Chaos methodologies

According to Peters (1994), the existence of a fractal dimension and sensitivity dependenceon initial conditions are the two necessary conditions for a process to become chaotic. Theresearch utilizes three different approaches in testing the chaotic dynamics of the four DowJones indices. The detailed three methodologies presented in this research are as follows:

2.1.1 Brock, Dechert, and Scheinkman test

The BDS test [13] is a way to detect dependency in financial time series. The test givesdelineation between a random series from deterministic chaos or from nonlinear stochasticseries. However, the BDS test has a low power against the autoregressive (AR) andautore-gressive conditional heteroscedasticity (ARCH) models [3]. To compensate this shortcom-ing, this study pre-filters the financial time series with linear filter like the autoregressive


moving average (ARMA) and a nonlinear filter like generalized autoregressive conditionalheteroscedasticity (GARCH) before proceeding with the BDS test. These processes elim-inate linearity from the returns, and any dependence found in the residuals.

The sample correlation integral is the basis in calculating the BDS test statistic, whichcan be defined as:

CN (l, T ) =2

TN (TN − 1)

∑t<s

Il

(xN

t , xNs

), (1)

where TN = T − N + 1.

The correlation integral of the time-series is dependent on a sequence xt := 1, ....T ofobservations which are independent and identically distributed (iid), and N-dimensionalvectors

[xN

t = (xt, xx+1

]called the ”N-histories”.

The null hypothesis xt under the test is that the increments of the data time-seriesis iid with a non-degenerative density F , CN(l, T ) → C1(l)

N with probability of one,as T → ∞, for any fixed N and l; and that

√T[CN(l, T ) − C1(l, T )N

]has a normal

distribution with zero mean and variance [13].

σ2

N (l) = 4

[KN + 2

N−1∑j=1

KN−1C2j + (N − 1)2C2N − N2KC2N−2

], (2)

where C = C(l) =∫

[F (z+1)−F (z−1)]dF (z), K = K(l) =∫ ∫

[F (z + 1) − F (z − 1]2dF (z).

The term C1(l, T ) is a consistent estimate of C(l), and

K (l, T ) =6

TN (TN−1)(TN−2)

∑t<s<r

Il(xt, xs)Il(xs, xr). (3)

Furthermore, σN (l) can be also estimated by σN (l, T ), which C1(l, T ) and K1(l, T ) can beput in place of C(l) and K(l) in the equation, because Eq.(3) is also a consistent estimateof K(l). Following a normal distribution, the BDS test structure can be completed asfollows:

wN (l, T ) =√

T[CN(l, T ) − C1(l, T )N

]/σN (l, T ), (4)

where σN (l, T ) denotes the standard deviation of the correlation integrals.

2.1.2 Rescaled Range analysis: Hurst exponent

The R/S statistic or the so-called rescaled range defines the R/S analysis, which wasdeveloped by [16]. The initial rescaled range procedure was improved by [17] and [18],which has the limitation of determining range dependencies without discriminating be-tween short and long dependencies in the series of data [19]. The improvements madethe modified R/S analysis to remove short-term dependencies and also able to identify


long term dependencies. This research initially transformed the financial time-series intologarithmic returns and is given by:

S1 = ln(Pt/Pt−1), (5)

where St = logarithmic returns at time t, and Pt = Dow Jones price index at time t.The St series undergoes a process called pre-whitening to minimize the effect of lineardependency and non-stationarity which is calculated as follows:

St = α + βSt−1 + εt, (6)

where St−1 denotes the logarithmic return at time period t − 1 and ε and β exhibit theparameters to be estimated and εt represents the residual.

The data is separated into A adjacent sub-periods of length n, such thatA×n = N , whereN represents the extent of the series Nt , which is similar to the processes of [20] and [21].Each sub-period is defined as Ia, a = 1, 2, 3, , A. The time series in Ia is marked Nk,a,k = 1, 2, 3, , n and the average value ea for each Ia of length n is

ea =

⎧⎪⎩1

n

⎫⎪⎭×n∑

k=1

Nk,a. (7)

RIa = max(Xk,a) − min(Xk,a), where1 ≤ k ≤ n, 1 ≤ a ≤ A, (8)

The range RIa denotes the difference between the maximum and minimum value Xk,a

within each sub-period Ia, and expressed as:Xk,a =

∑k

i=1(Ni,a − εa), k = 1, 2, 3, ..., n exhibits the elements for each sub-period of

departures from the mean value. The R/S analysis requires that RIa should be normalizedby dividing the sample by the standard deviation SIa corresponding to it and is computedbelow:

SIa =

[1

n×

n∑k=1

(Nk,a − εa)2

]0.50

(9)

The average R/S value for the length n is computed as:⎧⎪⎩R

S

⎫⎪⎭n

=

⎧⎪⎩ 1

A

⎫⎪⎭×A∑

a=1

(RIa/SIa). (10)

The final procedure in the analysis is the application of an ordinary least squares (OLS)regression with log(n) as the independent variable and log(R/S) as the dependent vari-able. The H exponent can have the following values: H = 0.5, which denotes the DowJones series is a random walk; 0 ≤ H < 0.5, which represents an anti-persistent series, ora medium memory; and 0.5 < H < 1 , which exhibits a persistent series, or a series withlong-memory. The R/S analysis computation can be derived from the expected values ofthe R/S statistics:

E (R/S) =

[(n − 0.5

n

)×(

n × π

2

)]−0.50

×n−1∑r=1

√(n − r)

r. (11)


The Hurst exponent,H is computed from the slope of the regression of E(log(R/S)n onlog(n). The variance of the Hurst exponent can be shown as:

V ar(H)n =1

T(12)

where T denotes the total number of observations in the series.

2.1.3 Correlation Dimension Analysis

The CD method as introduced by [14] differentiates deterministic and stochastic timeseries. The methodology examines the amount of complexity of a time-series data, whichhelps in determining possible signs of chaos in the four Dow Jones indices. Based onthe recommendation of [14] and [3], the CD analysis requires the initial filtering of theobservations through the ARMA and GARCH processes to eliminate possible problemsof autocorrelation and conditional heteroscedasticity, respectively. The filtering processis followed by creating n-histories of the filtered data, the process is shown below:

1 − history : x1

t = xt, (13)

2 − history : x2

t = (xt−1, xt), (14)

n − history : xnt = (xt−n+1,...,, xt). (15)

where n-history represents a particular point in the n-dimensional space.

This is followed by the calculation of correlation integral to define the correlation dimen-sion which can be shown as:

Cn(ε) = limT→∞ (t, s), 0 < t, s, < T : ‖xn

t − xns‖ < ε /T 2, (16)

where correspond to the number of points in the set, and ‖ ‖ represents the sup- ormax- norm making the correlation integral Cn(ε) the fraction of pairs (xn

s , xnt ), which are

close to each other, based on the limit:

maxi=0,...,n−1 |xs−i − xt−i| < ε. (17)

The last step calculates for the slope of logCn(ε) on log(ε) for small values of ε with thefollowing equation:

vn = limε→0logCn(ε)/logε. (18)

Deterministic chaos behavior is present if the embedding dimension increases while thevalue of correlation dimension (vn) does not converge to a stable value. A stochastic chaosis determined if the correlation dimension increases without any bound.

3 Empirical Results

Table 1 shows that the four Dow Jones indices have positive returns with the DJCA post-ing the highest average returns of 1% for the whole data sample, followed by the DJIA


(0.9%), DJTA (0.7%) and the DJUA (0.3%) has the lowest average returns. Although theDJCA index posted the highest returns, it has the lowest volatility with 0.428 standarddeviation. The DJTA which posted the highest dispersion of 0.562 is only third in theranking of average returns. This paper concludes that the Modern Portfolio Theory [22],stating that a higher risk is compensated with higher returns, does not conform withthe chosen data sets. All of the four Dow Jones indices are negatively skewed and haveleptokurtic distributions. The Jarque-Bera statistic for residual normality shows that theindices returns are under a non-normal distribution assumption.

Table 2 shows the filtering done by this study by initially establishing the stationarity ofthe data through the Augmented Dickey-Fuller (ADF) test. The orders of the ARMA,ARMA residual and GARCH residual models use the minimum values of the Akaike Infor-mation Criterion. Most of the Dow Jones indices data period passed the serial correlationexamination based on the results of the Lagrange Multiplier test. This study utilizesthe ARCH-LM process to test heteroscedasticity problem and shows that we can applyGARCH filtering models for each of the designed periods, because the null hypothesis wasrejected. The final test for ARCH effect showed that all data sets have already constantvariance from the GARCH residual models.

Table 3 illustrates that the BDS statistics are significant for most values of ε/σ from 0.5to 2.0, and m ranging from 2 to 6 for the filtered index returns, ARMA and GARCHresiduals for all the identified data sets, except for the DJUA. The residuals of the utilityindex from the GARCH model failed to reject the null hypothesis of iid observations. Therejection of the null hypothesis can have three major possibilities of having the form ofnon-stationarity, linear serial dependence, or non-linear dependence, which can be eitherchaotic or stochastic. Earlier we have already established the stationarity of the datathrough the ADF test, and its nonlinearity through the ARMA and GARCH filtrationprocess. Therefore, this study can conclude that the Dow Jones Indices are not iid ornot pure random series, and conventional linear methodologies are not appropriate fortheir analysis. In earlier studies, [23] and [21] have the same findings of non-stochasticprocesses in the returns of S&P 500 cash index and FTSE index, respectively; and [24]and [25] regarding the chaotic properties of the Istanbul Stock Exchange Index, and thereal estate investment trusts (REITs) and the Russell 2000 stock index, respectively. Thisstudy does not conform to the EMH [15], because the weak-form efficiency of the fourDow Jones indices were not validated.

The BDS test cannot conclude the iid properties for all the GARCH residuals of DJUA,wherein the presence of significant result cannot be discounted and may hint a possibilityof having a stochastic process. Since BDS test is just the beginning in testing for chaosand cannot exactly determine chaotic properties in the Dow Jones indices, this literatureconducts further tests and utilizes the R/S and correlation dimension analyses to supple-ment this initial examination.


Tab

le1:

The

Sam

ple

Siz

ean

dPer

iods

ofth

efo

ur

Dow

Jon

esin

dic

es

Dow

Jones

indic

es

retu

rns

Sta

rt

ofD

ata

Obs.

Mean

Std

.Dev.

Skew

Kurt.

J-B

era

Dow

Jon

esIn

dust

rial

Ave

rage

(DJIA

)M

ay27

,18

9629

,229

0.00

90.

503

-0.8

3427

.572

7387

45.4

***

Dow

Jon

esTra

nsp

orta

tion

Ave

rage

(DJTA

)O

ctob

er27

,189

629

,121

0.00

70.

562

-0.1

9316

.707

2281

42.5

***

Dow

Jon

esU

tility

Ave

rage

(DJU

A)

Jan

uar

y3,

1929

21,1

500.

003

0.50

6-0

.270

27.4

5852

7420

.2**

*D

owJon

esC

ompos

ite

Ave

rage

(DJC

A)

Jan

uar

y3,

1934

19,9

060.

010

0.42

8-0

.912

32.9

0874

4676

.9**

*Source:F

ederalR

eserve

Bank

ofSt.

Louis

database

Note:

*,*

*and

***

are

sig

nifi

cant

at

10,5,and

1%

levels

,respectiv

ely

;p-v

alu

es

are

inparentheses.

Tab

le2:

Sum

mar

ySta

tist

ics

ofU

nit

Root

test

,an

dA

RM

A-G

AR

CH

filt

erin

g

DJ

AD

FA

RM

AA

ICLM

-test

AR

MA

AIC

LM

-A

RC

H-

GA

RC

HA

ICA

RC

H-

Indic

es

Res.

test

LM

Res.

LM

DJIA

-82.

435*

**(2

,2)

1.46

19.

009*

*(1

,1)

1.46

00.

062

1576

.038

***

(2,2

)1.

013

53.6

40D

JTA

-81.

610*

**(1

,2)

1.68

00.

004*

*(2

,1)

1.67

90.

267

2035

.287

***

(2,2

)1.

229

1.78

1D

JU

A-6

1.75

2***

(2,2

)1.

473

0.00

3**

(2,2

)1.

468

1.00

837

91.5

41**

*(2

,2)

0.62

73.

714

DJC

A-8

0.37

2***

(2,2

)1.

136

18.5

30**

(2,2

)1.

133

1.70

511

15.9

15**

*(2

,2)

0.78

85.

367

Note:

*,*

*and

***

are

sig

nifi

cant

at

10,5,and

1%

levels

,respectiv

ely

;p-v

alu

es

are

inparentheses.


Tab

le3:

BD

Ste

stfo

rth

eD

owJon

esIn

dic

esre

turn

san

dre

sidual

s

DJIA

DJIA

retu

rns

AR

MA

resid

uals

GA

RC

Hresid

uals

ε/σ

0.5

1.0

1.5

2.0

0.5

1.0

1.5

2.0

0.5

1.0

1.5

2.0

20.0

12***

0.0

22***

0.0

21***

0.0

15***

0.0

12***

0.0

22***

0.0

21***

0.0

14***

-0.0

01***

-0.0

01***

-0.0

01***

-0.0

01*

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

06)

(0.0

02)

(0.0

05)

(0.0

68)

30.0

13***

0.0

42***

0.0

47***

0.0

37***

0.0

13***

0.0

42***

0.0

47***

0.0

37***

-0.0

00***

-0.0

02***

-0.0

02***

-0.0

01**

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

07)

(0.0

02)

(0.0

02)

(0.0

45)

40.0

10***

0.0

53***

0.0

72***

0.0

61***

0.0

10***

0.0

53***

0.0

72***

0.0

61***

-0.0

00***

-0.0

02***

-0.0

03***

-0.0

02**

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

08)

(0.0

01)

(0.0

01)

(0.0

26)

50.0

07***

0.0

56***

0.0

93***

0.0

85***

0.0

07***

0.0

56***

0.0

93***

0.0

85***

-0.0

00**

-0.0

01***

-0.0

03***

-0.0

02**

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

32)

(0.0

04)

(0.0

04)

(0.0

46)

60.0

04***

0.0

56***

0.1

09***

0.1

08***

0.0

04***

0.0

56***

0.1

09***

0.1

08***

-0.0

00

-0.0

01**

-0.0

02**

-0.0

02

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.2

15)

(0.0

45)

(0.0

27)

(0.1

43)

Note:

*,*

*and

***

are

sig

nifi

cant

at

10,5,and

1%

levels

,respectiv

ely

;p-v

alu

es

are

inparentheses.

DJTA

DJTA

retu

rns

AR

MA

resid

uals

GA

RC

Hresid

uals

ε/σ

0.5

1.0

1.5

2.0

0.5

1.0

1.5

2.0

0.5

1.0

1.5

2.0

20.0

13***

0.0

25***

0.0

22***

0.0

16***

0.0

13***

0.0

25***

0.0

23***

0.0

16***

-0.0

01***

-0.0

02***

-0.0

01***

-0.0

01***

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

01)

(0.0

00)

(0.0

01)

(0.0

04)

30.0

15***

0.0

46***

0.0

50***

0.0

39***

0.0

15***

0.0

46***

0.0

50***

0.0

39***

0.0

00***

-0.0

02***

-0.0

03***

-0.0

02***

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

40.0

11***

0.0

57***

0.0

76***

0.0

63***

0.0

11***

0.0

57***

0.0

76***

0.0

64***

0.0

00***

-0.0

02***

-0.0

04***

-0.0

04***

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

50.0

08***

0.0

61***

0.0

96***

0.0

88***

0.0

07***

0.0

61***

0.0

97***

0.0

89***

0.0

00***

-0.0

02***

-0.0

04***

-0.0

04***

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

01)

(0.0

00)

(0.0

00)

(0.0

00)

60.0

05***

0.0

60***

0.1

12***

0.1

11***

0.0

05***

0.0

60***

0.1

13***

0.1

12***

0.0

00**

-0.0

01***

-0.0

03***

-0.0

04***

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

18)

(0.0

05)

(0.0

01)

(0.0

01)

Note:

*,*

*and

***

are

sig

nifi

cant

at

10,5,and

1%

levels

,respectiv

ely

;p-v

alu

es

are

inparentheses.


(con

tinued

)

DJU

AD

JU

Aretu

rns

AR

MA

resid

uals

GA

RC

Hresid

uals

ε/σ

0.5

1.0

1.5

2.0

0.5

1.0

1.5

2.0

0.5

1.0

1.5

2.0

20.0

31***

0.0

43***

0.0

32***

0.0

21***

0.0

31***

0.0

43***

0.0

32***

0.0

21***

0.0

00

0.0

01

0.0

01

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.1

40)

(0.1

48)

(0.1

48)

(0.1

31)

30.0

41***

0.0

84***

0.0

72***

0.0

51***

0.0

41***

0.0

84***

0.0

73***

0.0

52***

0.0

00

0.0

00

0.0

00

0.0

00

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.5

13)

(0.5

77)

(0.6

05)

(0.4

60)

40.0

37***

0.1

12***

0.1

11***

0.0

84***

0.0

37***

0.1

12***

0.1

12***

0.0

85***

0.0

00

0.0

00

0.0

00

0.0

00

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.9

20)

(0.8

07)

(0.7

15)

(0.9

19)

50.0

30***

0.1

29***

0.1

46***

0.1

16***

0.0

30***

0.1

30***

0.1

47***

0.1

18***

0.0

00

0.0

00

-0.0

01

-0.0

01

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.5

12)

(0.4

07)

(0.3

24)

(0.5

02)

60.0

23***

0.1

37***

0.1

75***

0.1

48***

0.0

23***

0.1

38***

0.1

76***

0.1

49***

0.0

00

0.0

00

-0.0

01

-0.0

01

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.6

77)

(0.4

41)

(0.3

28)

(0.4

78)

Note:

*,*

*and

***

are

sig

nifi

cant

at

10,5,and

1%

levels

,respectiv

ely

;p-v

alu

es

are

inparentheses.

DJC

AD

JC

Aretu

rns

AR

MA

resid

uals

GA

RC

Hresid

uals

ε/σ

0.5

1.0

1.5

2.0

0.5

1.0

1.5

2.0

0.5

1.0

1.5

2.0

20.0

09***

0.0

19***

0.0

17***

0.0

12***

0.0

09***

0.0

18***

0.0

17***

0.0

12***

-0.0

01**

-0.0

01**

-0.0

01*

0.0

00

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

35)

(0.0

26)

(0.0

77)

(0.5

81)

30.0

10***

0.0

35***

0.0

40***

0.0

31***

0.0

10***

0.0

34***

0.0

40***

0.0

31***

-0.0

01**

-0.0

02***

-0.0

02**

-0.0

01

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

18)

(0.0

09)

(0.0

17)

(0.1

87)

40.0

08***

0.0

43***

0.0

61***

0.0

52***

0.0

08***

0.0

42***

0.0

60***

0.0

52***

0.0

00**

-0.0

02***

-0.0

03***

-0.0

02

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

22)

(0.0

08)

(0.0

10)

(0.1

13)

50.0

05***

0.0

46***

0.0

78***

0.0

73***

0.0

05***

0.0

45***

0.0

78***

0.0

72***

0.0

00*

-0.0

01**

-0.0

03**

-0.0

02

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

84)

(0.0

30)

(0.0

28)

(0.1

89)

60.0

03***

0.0

45***

0.0

92***

0.0

93***

0.0

03***

0.0

44***

0.0

91***

0.0

92***

0.0

00

-0.0

01

-0.0

02

-0.0

01

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.0

00)

(0.4

51)

(0.2

22)

(0.1

34)

(0.3

83)

Note:

*,*

*and

***

are

sig

nifi

cant

at

10,5,and

1%

levels

,respectiv

ely

;p-v

alu

es

are

inparentheses.


Table 4: Hurst exponents of the four Dow Jones indices

Stock returns DJIA DJTA DJUA DJCA

Original Series 0.000141 0.000846 0.000353 0.000403Scrambled Series 0.523801 0.538644 0.521252 0.520887

ARMA residuals DJIA DJTA DJUA DJCA


GARCH residuals DJIA DJTA DJUA DJCA


Table 5: Correlation Dimension Analysis estimates of the four Dow Jones indices

Correlation Embedding Dimensions

Dimensions 1 2 3 4 5 6 7 8 9 101. DJIA returns 1.017 2.066 3.025 3.839 4.479 5.035 5.418 5.725 6.101 6.181ARMA residuals 1.028 2.067 3.025 3.839 4.477 5.039 5.421 5.729 6.104 6.181GARCH residuals 1.030 2.072 3.037 3.883 4.580 5.191 5.547 6.014 6.241 6.434

2. DJTA returns 1.011 2.071 3.037 3.864 4.521 4.950 5.363 5.828 5.992 6.126ARMA residuals 1.029 2.068 3.035 3.865 4.523 4.955 5.367 5.835 5.997 6.133GARCH residuals 1.028 2.072 3.038 3.884 4.581 5.197 5.554 6.020 6.244 6.438

3. DJUA returns 0.892 2.064 3.026 3.822 4.431 4.966 5.323 5.608 5.993 6.022ARMA residuals 1.029 2.069 3.024 3.829 4.435 4.970 5.328 5.614 6.005 6.029GARCH residuals 1.029 2.073 3.037 3.885 4.579 5.184 5.547 6.009 6.248 6.448

4. DJCA returns 1.017 2.071 3.035 3.805 4.515 5.060 5.495 5.769 5.986 6.306ARMA residuals 1.029 2.066 3.039 3.806 4.519 5.064 5.501 5.776 5.994 6.316GARCH residuals 1.027 2.071 3.033 3.867 4.567 5.181 5.625 5.991 6.345 6.388


Table 4 shows the results of the R/S analysis through the Hurst exponents. A time-seriesis determined by a chaotic process, the Hurst exponent would be much closer to 0.5 afterscrambling the data than the one before the procedure [20]. Most of the Hurst exponentsof the four Dow Jones indices returns, ARMA and GARCH residuals are way below 0.5;however, after scrambling the data, all Hurst exponents are above 0.5, which concludesthat the long time-series data of the four Dow Jones indices have a persistent and a trend-reinforcing series, wherein having an upward (downward) trend in the last period, willcontinue to be positive (negative) in the next period. These findings support the initialfindings that the four indices do not follow a random walk and an anti-persistent series.These results are again consistent with [21] when they studied the FTSE index using theHurst test.

Table 5 presents the correlation dimension estimates for the four Dow Jones indices dataseries. The correlation dimension analysis is a procedure necessary for confirming chaotic

behavior [26]. This research observed that as the embedding dimensions gradually increasefrom 1 to 10, the correlation dimension increases less quickly. This can be related to thepresence of a high dimensional underlying noisy chaotic structure [27]. The tendencyshows that the Dow Jones index returns, ARMA and GARCH residuals can be consistentwith chaos and these findings also conforms to the study of [28] and [27] on the chaoticproperties of the French CAC40 index and metal futures prices in the London MetalExchange, respectively. These results warn technical investing analysts to be cautious inutilizing linear processes in modeling the Dow Jones indices; and provide an understandingto academicians and researchers that the long time-series of the Dow Jones indices signifiesnoisy chaotic tendencies and shows evidences that the EMH [15] may not be true on theseset of time-series.

4 Conclusions

The study employs three tests of non-linearity and chaotic behavior, namely the BDStest, R/S analysis and correlation dimension analysis on the daily stock returns of thefour Dow Jones indices - DJIA, DJUA, DJTA and DJCA. This research uses an averageof 24,815 data points to correctly simulate chaos in financial time-series, following therecommendation of higher accuracy in having longer time-series data. The BDS test re-sults rejected the presence of linearity and indicates that most of the Dow Jones indicesare not iid or not pure random series. BDS statistics are significant for most values ofε/σ from 0.5 to 2.0, and m ranging from 2 to 6 for the filtered index returns, ARMA andGARCH residuals, except for the DJUA, wherein the filtered residuals from the GARCHmodel failed to reject the null hypothesis of iid observations. The R/S analysis demon-strates that the initial Hurst exponents of the four Dow Jones indices returns, ARMAand GARCH residuals are way below 0.5. However, after scrambling the data, all Hurstexponents are above 0.5, which concludes that the four time-series data have a persistentand a trend-reinforcing dynamics. Furthermore, the correlation dimension analysis sup-plements the first two initial tests, and finds that as the embedding dimensions increasefrom 1 to 10, the correlation dimension increases less quickly. This can be a sign of the


presence of a high dimensional underlying noisy chaotic structure in the four Dow Jonesindices.

These findings challenge the validity of the random walk hypothesis and the long time-series of the Dow Jones stock indices may be characterized by nonlinearity and chaotictendencies. These results also caution the investing community in the risk of using linearprocesses in modeling the four Dow Jones stock indices; and give further research avenueto academicians and researchers in the chaotic tendencies of longer time-series returns.Differences in findings with regards to the non-linear characteristics can be affected bythe volume of the data. The accuracy of test results improves with the increase in thelength of the time series [4]. For future studies, it is suggested that researchers use largerdata sets or around 5,000 or more in determining chaotic behavior of financial time-series.This minimum requirement of data may also provide some limitations on having structuralbreaks in the data, because some crises and possible structural breaks occur in less than5,000 daily observations. This occurrence also limited this study in the possibility ofputting structural breaks in the long time-series of the Dow Jones indices.

Acknowledgement: The author expresses deep gratitude to Jocelyn Flores Villaverdeof the Mapua Institute of Technology, and Mark Nolan Confesor of the Iligan Institute ofTechnology for the technical assistance.

References

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[4] R. Harrison, Y. Dejin, L. Oxley, W. Lu, D. George, Non-linear noise reduction anddetecting chaos: Some evidence from the S&P Composite Price Index, Math. Com-put. Simul., 48 (1999) 497-502.

[5] H. Amilon, H. Bystrom, The search for chaos and nonlinearities in Swedish stockindex returns, Department of Economics, Lund University (1998).

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[8] R. Genceay, T. Stengos, Moving average rules, volume and the predictability ofsecurity returns and feedforward networks, J. Forecasting, 17(1998) 401-414.


[9] R. Engle, A. Patton, What good is a volatility model?, Quant. Financ., 1(2001)237-245.

[10] L.G. Moyano, J. Souza, S.M. Duarte, Multi-fractal structure of traded volume infinancial markets, Physica A, 371 (2006) 118-121.

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[12] M. Chikhi, A. Feisolle, M. Terraza, SEMIFARMA-HYGARCH modeling of DowJones return persistence, Aix Marseille School of Economics Working Papers (2012).

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[20] E. Peters, Fractal market analysis: Applying chaos theory to investment and Eco-nomics, Wiley: New York (1994).

[21] K. Opong, G. Mulholland, A. Fox, K. Farahmand, The behaviour of some UK equityindices: An application of Hurst and BDS tests, J. Empir. Financ., 6 (1999) 267-82.

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[26] A. Wei, R. Leuthold, Long agricultural prices: ARCH, long memory or chaos pro-cesses? OFOR Paper (1998) 98-03.

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Dynamic Behaviour of a Unified Two-PointFourth Order Family of Iterative Methods

D. K. R. BabajeeAfrican Network for Policy Research & Actions for Sustainability (ANPRAS), Mauritius


S. K. KhrattiDepartment of Engineering, Stord Haugesund University College, Norway


Abstract

Many variants of existing multipoint methods have been developed. Recently,Khratti et al. (2011) developed a unifying family of two-point fourth order methodswhich contains the well-known Ostrowski method. The authors also obtained somenew methods which are variants of Ostrowski’s method. However, it is difficult tocompare the methods with the same of the order of convergence. The dynamicbehaviour of the methods can be used as a tool for comparison.In this work, we study the dynamic of six members of the unifying family forsome quadratic and cubic polynomials. By means of computer generated plots,we draw their polynomiographs for the polynomials f(z) = z2−1 and f(z) = z3−1and explain their respective dynamic behaviour by analyzing the free critical andadditional fixed points. Our results show that the methods exhibit different fractalbehaviour and the most efficient method based on the size of its basins of attractionswas found to the well-known Ostrowski method. This shows that these fourth ordervariants of Ostrowski’s method are inefficient.

Keywords: Fourth order methods, Dynamic behaviour, fractal, Polynomiography,Scaling Theorem, free critical points, additional fixed points, Julia set, Basins of attractions,Efficient method

Manuscript accepted May 28, 2013.

Dynamic Behaviour of a Unified Two-Point Fourth Order Family of Iterative Methods 17

1 Introduction

Iterative methods are used to find approximate solutions of nonlinear equations, f(z) = 0which arise from various problems in mathematical and engineering sciences.

The Newton-Raphson method (2ndNR) is one of the best known and probably themost used method for solving such nonlinear equations and is given by

zk+1 = ψ2ndNR(zk), (1)

where

ψ2ndNR(zk) = zk − u(zk)

and

u(zk) =f(zk)

f ′(zk).

Several higher order variants of the Newton’s method free from second derivatives havebeen proposed in the literature (see [2] and the references therein). Khratti et al. [7]developed a unifying family of multipoint optimal fourth order methods (4thUF ) whichis given by:

zk+1 = ψ4thUF (zk), (2)

where

ψ4thUF (zk) = zk − u(zk)(1 + v(zk) + 2v(zk)

2 + αv(zk)3F(v(zk))

),

v(zk) =f(ψ2ndNR(zk))

f(zk)

and

F(v(zk)) =∞∑

j=0

aj v(zk)j ,

aj ∈ R is a converging power series. This family is termed as two-point methods becauseits member requires the evaluations of the function at two different points. Khratti et al.[7] proved that the methods are fourth order convergent in the real plane. Their resultextends to the complex plane in the following theorem:

Theorem 1. [7] Let the function f : D ⊂ C → C has a simple root z∗∈D in the openinterval D. Furthermore the first, second and the third derivatives of the function f(z)belongs in the open interval D. Then the methods of the iterative family (2) are at leastfourth order convergent for any choice of the real parameter α and the real power seriesF . The methods of the family (2) satisfies the error equation

ek+1 = −c2 ((−5 + α a0) c22 + c3c1) ek

4

c13

+ O(ek

5), (3)

where the error after k iterations ek = zk − z∗, the constants ck = f (j)(z∗)/j! with j ≥ 1and a0 is the coefficient of the power series.

18 D. K. R. Babajee and S. K. Khratti

Table 1: Some members of the 4thUF family for different selections of α and F(v(z)).

Method α F(v(z)) ψ(z)

4thOst [10] 1 4 + 18v(z) + 16v(z)2 + ... z − u(z)

(1 − v(z)

1 − 2v(z))

)4thAmit[9] 1 1 + v(z) + v(z)2 + ... z − u(z)

(v(z)2 +

1

1 − v(z)

)4thKLW [8] 2 1 + v(z) + v(z)2 + ... z − u(z)

(1 + v(z)2

1 − v(z)

)4thKNS1 [7] 0 - z − u(z)

(1 + v(z) + 2v(z)2

)4thKNS2 [7] 5 1 z − u(z)

(1 + v(z) + 2v(z)2 + 5v(z)3

)4thKNS3 [7] 1

∞∑m=0

5m+1

2mv(z)m z − u(z)

(2 − 3v(z) − v(z)2

2 − 5v(z)

)

He also rediscovered some methods including the well-known Ostrowski method (4thOst)and obtained new fourth order methods for different selections of α and F(v(z)). We makea summary of the some of the methods in Table 1. The 4thKNS2 and 4thKNS3 havethe following error equation

ek+1 = −c2c3

c12

ek4 + O

(ek

5)

and are fifth order methods for quadratic functions since c3 = 0. For this case we termthese methods as 5thKNS2q and 5thKNS3q methods, respectively.It is difficult to compare these methods because they have same order of convergence.They enjoy their higher order convergence only if the starting point is chosen close to theroot. So it is important to find the basins of attractions of the methods by studying theirdynamic behaviour in the complex plane.In this work, we introduce the basic notations and definitions. We prove the ScalingTheorem for the unifying family. We study the dynamic behaviour of its six membersfor the polynomials, f(z) = z2 − 1 and f(z) = z3 − 1 by analyzing their free critical andadditional fixed points. Bahman Kalantari [6] coined the term ”polynomiography” to bethe art and science of visualization in the approximation of roots of polynomial usingiteration methods. We draw the polynomiographs of the methods and use them to findthe most efficient method based on the size of the basins of attractions.


2 Basic Notations and Preliminaries

Definition 1. [6, p. 89] A fixed point of the rational function R is a point z∗ such thatR(z∗) = z∗.

Definition 2. [6, p. 90] Given a fixed point z∗ ∈ C the quantity ϕ = R′(z∗) is awell-defined point in C and is called its multiplier. There are four different basic types offixed points:

z∗ :

super-attractive, if |ϕ| = 0attractive, if 0 < |ϕ| < 1repelling, if |ϕ| > 1indifferent, if |ϕ| = 1.

An indifferent fixed point is said to be rationally different or parabolic if ϕ is a root ofunity, i.e there exists a natural number n1 such that ϕn1 = 1, otherwise, irrationallyindifferent.

Let z∗ be an attracting fixed point of R(z). Its basin of attraction is the set

B(z) = z ∈ C : Ra(z) → z∗ as a → ∞. (4)

Definition 3. [6, p. 123] The Julia set of a given rational function R, denoted by J (R)

is the set of points z ∈ C where Ra is not normal at the point. The complement J (R) iscalled the Fatou set and is denoted by F(R).

We list the properties of the Julia set [5]:

1. JR

is the closure of the repelling periodic points.

2. JR

is non-empty.

3. JR

is completely invariant under R; i.e. R(JR) = J

R= R−1(J

R).

4. JR

is the boundary of the basin of attraction of each fixed point or attractive cycle.

5. If z∗ ∈ JR, then the closure of

z|Ra(z) = z∗, for some non-negative integer a,

the backward iterates of z∗, is the whole of JR.

Remark 2. [5] Property 4 guarantees that, if there are more than two roots, JR

willbe a fractal set. Property 1 guarantees that the Julia set is an unstable set. Iterates ofpoints close to the Julia set will move away from that set. Hence, higher order methodsare very sensitive to initial conditions when the initial point is near the Julia set. Nearbypoints could converge to different roots or might not converge at all. Ideally, if you startwith a point actually on the Julia set, Property 3 implies that the iterates will also beon the Julia set. However, in practice, because the Julia set is unstable, the iterates willmost likely be thrown off the set because of rounding errors.


Remark 3. [4] The zeros which are not zeros of f are referred as additional (or extraneous)fixed points. Their appearance is a striking example of the caution necessary for theselection of starting values of these higher order iteration sequences, which generally leadto smaller basins of attraction. These points are either attracting or repelling but, in anycase, they affect the roots’ basins of attraction. If the extra fixed points are repelling, theybelong to the Julia set, so they cannot trap an iteration sequence.

To detect the existence of attracting cycles which could interfere with the search for theroots, we observe the orbits of the free critical points of the iteration function.

Definition 4. [4] Critical values of a function are defined as those values q ∈ C for whichf(z) = q has a multiple root. The multiple root z = z∗ is called the critical point of f .This is equivalent to the condition f ′(z∗) = 0.

The solutions of the equation ψ′

,q(z) = 0 that are not solutions of the equation ofq(z) = 0 are called free critical points. The free critical points of a higher order methodsatisfy the equation

ψ′

IF,q(z)

q(z)p−1= 0,

where p is the order of the method

Theorem 4 (Fatou-Julia). Let ψ(z) be a rational map. If z0 is an attracting periodicpoint, then the immediate basin of attraction B∗(z0) contains at least one critical point.

Remark 5. [1] As a consequence of Theorem 4, to detect the existence of an attractingperiodic point that interferes with our search of a root of the equation q(z) = 0, the orbit ofeach free critical point must be computed and its set of limit point determined. If the set oflimit points of the orbit of some free critical points is not a root, that is, a super-attractingfixed point of any iterative method ψ,q(z) under consideration, then it must be an attractingperiodic orbit.

Definition 5. [1] Let f and g be two maps in the Riemann sphere into itself. An analyticconjugacy between f and g is an analytical diffeomorphism h from the Riemann sphereonto itself such that h f = g h (conjugacy equation).

Conjugacy plays a central role in understanding the behaviour of classes of maps fromdynamical system point of view in the sense that it preserves fixed and periodic pointsand their type as well as basin of attraction.

Theorem 6 (Scaling Theorem). [1] Let f(z) be an analytic function on the Riemannsphere, and let T (z) = Y1z + Y2, Y1 = 0, be an affine map. If g(z) = f T (z), thenT ψ,g T−1 = ψ,f . That is ψ,f and ψ,g are analytically conjugated by T .

Remark 7. [1] If q(z) = ς1q(z), where ς1 is a constant, a straightforward calculationshows that ψ(, q) = ψ(, q), that is, the identity map is a conjugacy between the mapsψ(, q) and ψ(, q), therefore their dynamics are equivalent. The Scaling Theorem allows usto, modulo suitable changes of coordinates, reduce the study of the dynamics of iterations


ψ(, q), to the study of specific families of iterations of simpler maps. For instance, everyquadratic polynomial q(z) = b1z

2+b2z+b3 reduces, via an affine change of coordinates, toa polynomial belonging to the one-parameter family pc(z) = z2 − c, where c = b2

2−4b1b3.This is nothing but an appropriate re-scaling that puts ψ(, q) inside the conjugacy class ofψ(, pc) for some c.

Theorem 8. [3, p. 8] Suppose that q is a polynomial of degree d ≥ 2. The unit circleS1 = z ∈ C : |z| = 1 is completely invariant if and only if q(z) = lzd, where |l| = 1.

From Theorem 8, J (q) = S1.We prove the Scaling Theorem for the 4thUF family in the next section.

(a) (b)

Figure 1: Polynomiographs of 4thOM and 4thAmit methods for f(z) = z2 − 1

3 Scaling Theorem for the 4thUF family

Theorem 9. 4thUF’s family satisfies the Scaling Theorem, that is,T ψ4thUF,g T−1(z) = ψ4thUF,f(z).

Proof. It is enough to show that T ψ4thUF,g(z) = ψ4thUF,f T (z).Since g(z) = f(T (z)), we have by induction

g(a)(z) = Y a

1 f (a)(T (z)), for a ∈ N. (5)

We have

ug(z) =g(z)

g′(z)

=f(T (z))

Y1f ′(T (z)), using eq. (5) with a = 1,

=1

Y1

uf(T (z)). (6)


Then,

g(ψ2ndNR,g)(z) = g

(z − 1

Y1

uf(T (z))

)= f

(T

(z − 1

Y1

uf(T (z))

)). (7)

Now, from the definition of T (z), we have

T

(z − 1

Y1

uf(T (z))

)= Y1

(z − 1

Y1

uf(T (z))

)+ Y2

= Y1z + Y2 − uf(T (z))

= T (z) − uf(T (z)). (8)

Using eqs. (7) and (8), we have

vg(z) =g(ψ2ndNR,g(z))

g(z)

=f (T (z) − uf(T (z)))

f(T (z))

= vf (T (z)). (9)

Using eqs. (6) and (9), we finally obtain

T ψ4thUF,g(z)

= Y1ψ4thUF,g(z) + Y2

= Y1

[z − ug(z)

(1 + vg(z) + 2vg(z)2 + αvg(z)3

∞∑j=0

aj vg(z)j

)]+ Y2

= T (z) − uf(T (z))

(1 + vf(T (z)) + 2vf (T (z))2 + αvf(T (z))3

∞∑j=0

aj vf(T (z))j

)= ψ4thUF,f T (z).

4 Study of the members of the 4thUF family for the

Generic Quadratic Polynomial

In this section, we analyze the dynamic behaviour of the family for the generic quadraticpolynomial.

Proposition 10. For qc = z2 − c, where c ∈ C, the Julia set of the Ostrowski method,J (ψ4thOst,qc

) is a straight line.


(a) (b)

Figure 2: Polynomiographs of 4thKLW and 4thKNS1 methods for f(z) = z2 − 1

Proof. Following [4], we apply the Mobius transformation

M(z) =z +

√c

z −√c,

the inverse of which is

M−1(z) =

√c

z + 1

z − 1,

so that, using a computer algebra software such as Maple, we have

Mψ4thOstM−1(z) = z4. (10)

From Theorem 8, we have J (Mψ4thOstM−1(z)) = S1 with interior B(0) and exterior

B(∞).

However, since

Mψ4thAmitM−1(z) =

z8 + 5z7 + 10z6 + 9z5 + 4z4

4z4 + 9z3 + 10z2 + 5z + 1,

Mψ4thKLW M−1(z) =

z6 + 3z5 + 3z4

3z2 + 3z + 1,

Mψ4thKNS1M−1(z) =

z6 + 4z5 + 5z4

5 ∗ z2 + 4 ∗ z + 1,

Mψ5thKNS2qM−1(z) =

z8 + 6z7 + 14z6 + 14z5

14z3 + 14z2 + 6z + 1,

and

Mψ5thKNS3qM−1(z) =

2z6 + 3z5

3z + 2,

the Julia sets of these methods are not straight lines. This implies that only the Ostrowskimethod will generate an ”ideal” fractal. This will be confirmed by a numerical study inthe following section.


4.1 Numerical Study for the Quadratic Polynomial f(z) = qc=1 =z2 − 1.

We now draw the polynomiographs of f(z) = qc=1 = z2−1 with roots z∗1 = −1 and z∗2 = 1.Let z0 = x + i y be the initial point. A square grid of 80000 points, composed of 400columns and 200 rows corresponding to the pixels of a computer display would representa region of the complex plane [11]. We consider the square R×R = [−2, 2]× [−2, 2]. Eachgrid point is used as a starting value z0 of the sequence zk+1 = ψmethod(zk) and the numberof iterations until convergence is counted for each gridpoint. We assign pale blue colourif the iterates zk of each grid point converge to the root z∗1 = −1 and green colour if theyconverge to the root z∗2 = 1 in at most 100 iterations and if |z∗j − zk| < 0.0001, j = 1, 2.In this way, the basin of attraction B(z∗j ) for each root would be assigned a characteristiccolour. The common boundaries of these basins of attraction constitute the Julia set ofthe methods. If the iterates do not satisfy the above criterion for convergence we assignthe dark blue colour. The polynomiographs are generated in MATLAB R2010a. Fig. 1(a) shows the polynomiograph of the 4thOM method and we see that its Julia set is theimaginary axis because the 4thOM method satisfies Theorem 8. The polynomiographs ofthe other five methods can be shown in Figs. 1 (a), 2 and 3. It can be observed that theirJulia sets are not straight lines because these methods do not satisfy Theorem 8. In figs.1 to 3, we denote ∗ as the roots, o as the free critical points and + as the additional fixedpoints.

(a) (b)

Figure 3: Polynomiographs of 5thKNS2q and 5thKNS3q methods for f(z) = z2 − 1

We denote FCP o and AFP+ as the free critical point and additional fixed pointof the methods, respectively. We also denote No, N+ and ND as the number of freecritical points, number of additional fixed points and number of diverging starting points,respectively. Table 2 shows a comparison of these numbers. It also gives the values of thefree critical and additional fixed points of the methods. All these points are repelling andare found in the Julia set. All starting points converge for the methods considered. The5thKNS3q method is found to be the best of the 5 methods which do not satisfy Theorem


8 as it has no free critical points and 2 additional fixed points which lie on the imaginaryaxis. Its Julia set is the least complex as shown in 3 (b). For the other methods, thepresence of the free critical and additional fixed points may interfere with the root search,thus resulting in complex fractal shapes like petals and hearts as shown in Figs. 1 (b),2 and 3 (a). The 4thOM method is found as the most efficient since it has the largestbasins of attractions for the quadratic polynomial.

Table 2: Comparison of number of free critical points, additional fixed points and divergingpoints of the members of UF family for the quadratic polynomial f(z) = z2 − 1.

method No FCP o N+ AFP+ ND

4thOM 0 - 2 ±0.5744i 0

4thAmit 4 ±0.3175 ± 0.3175i 6 ±0.4535i, ±0.4535 ± 0.2542i 0

4thKLW 2 ±0.3780i 4 ±0.3882 ± 0.3031i 0

4thKNS1 0 - 4 ±0.4916 ± 0.2446i 0

5thKNS2q 0 - 6 ±0.5392, ±0.5183 ± 0.4017i 0

5thKNS3q 0 - 4 ±1.0248i, ±0.2239 0

5 Numerical Study for the Cubic Polynomial f(z) =

qe=1 = z3 − 1.

The dynamic of the 4thUF Family for the Generic Cubic Polynomial is rather complex.Therefore we limit ourselves to the numerical study of the Cubic Polynomialf(z) = qe=1 = z3 − 1. The roots are z∗1 = 1, z∗2 = −0.5000 + 0.8660i and z∗3 = −0.5000 +0.8660i. Each grid point over the region [−2, 2]× [−2, 2] is coloured accordingly, brownishyellow for convergence to z∗1 , blue for convergence to z∗2 and pale green for convergence toz∗3 . We use the same conditions for convergence as in the quadratic polynomial. Using theconjugate map S(z) = ψ

(1

z

), we can verify that ψ4thUF,qe=1

(∞) = ∞ and ∞ is a repellingfixed point of the six members of the 4thUF family since |S ′(0)| > 1.Fig 4 (a) shows the polynomiograph of the 4thOM method. There are 6 repelling free

critical and 6 repelling additional fixed points. The free critical points are usually onthe perpendicular bisector of any two roots. Fig 4 (b) shows the polynomiograph of the4thAmit method. There are 18 free critical and 18 additional fixed points which are allrepelling. They are located at the ends of the petals centered at the origin where wecan observe some diverging starting points. It is the presence of these repelling points


(a) (b)

Figure 4: Polynomiographs of 4thOM and 4thAmit methods for f(z) = z3 − 1

which cause the 4thAmit iterates to diverge. Fig 5 (a) shows the polynomiograph of the

(a) (b)

Figure 5: Polynomiographs of 4thKLW and 4thKNS1 methods for f(z) = z3 − 1

4thKLW method and Julia set appears to be butterfly-shaped. There are 12 repellingfree critical and 12 repelling additional fixed points for this method. Fig 5 (b) shows thepolynomiograph of the 4thKNS1 method. There are 6 free critical and 12 additional fixedpoints. These points are repelling and we observe some diverging points at the origin.However, the number of diverging points is less than of the 4thAmit method. Fig 6 (a)shows the polynomiograph of the 4thKSN2 method. There are 12 free critical points and18 additional fixed points. 3 free critical points ( 1.0223, ±0.8853i−0.5111) are attractingsince ϕ = 0.0016 < 1 while the rest are repelling. Points near the attracting free criticalpoints converge to the super-attracting fixed points of f(z). All additional fixed points arerepelling. It is also observed that this method has the highest number of diverging pointsbecause the repelling points surround the origin. Fig 6 (b) shows the polynomiograph of


(a) (b)

Figure 6: Polynomiographs of 4thKNS2 and 4thKNS3 methods for f(z) = z3 − 1

the 4thKNS3 method. There are 12 free critical and 12 additional fixed points. 3 freecritical points ( 1.3411, ±1.1614i−0.6705) are attracting since ϕ = 0.04 < 1 while the restare repelling. All additional fixed points are repelling. However, all starting points areconvergent. This is because the repelling free critical points and additional fixed pointsare mainly located on the perpendicular bisector of any two roots. They do not affectthe iterates of nearby points. The 4thOM method is again found as the most efficientsince it has the largest basins of attractions for this cubic polynomial. Finally, we include

Table 3: Comparison of number of free critical points, additional fixed points and divergingpoints of the members of UF family for the cubic polynomial f(z) = z3 − 1.

Methods No N+ ND

4thOM 6 6 0

4thAmit 18 18 110

4thKLW 12 12 0

4thKNS1 6 12 36

4thKNS2 12 18 692

4thKNS3 12 12 0

the polynomiographs of the six methods for generic cubic polynomial f(z) = qe=0 whoseroots are −1, 0, 1. They are shown in Figs. 7 and 8. All starting points are convergent


for the six methods. We can find bulb and petal shapes in the Julia set. The 4thOMmethod is the most efficient method as it has the smallest Julia set. The 4thAmit and4thKNS2 methods are the most chaotic methods because the shape of their Julia set aremost complex. These observations were also made with the first two polynomials.

(a) (b) (c)

Figure 7: Polynomiographs of 4thOM, 4thAmit and 4thKLW methods for f(z) = z3 − z

(a) (b) (c)

Figure 8: Polynomiographs of 4thKNS1, 4thKNS2 and 4thKNS3 methods for f(z) = z3−z

6 Conclusion

In this work, we prove the Scaling Theorem for the unifying family. We explain thedynamic behaviour of its six members for the polynomials, f(z) = z2−1 and f(z) = z3−1by considering their free critical and additional fixed points. We found that these pointscan interfere with the root search and cause the method to behave chaotically and thusreducing their basins of attractions. We found that the Ostrowski method is the bestefficient of the six methods as it behaves the least chaotically and has the largest basins ofattractions. We conclude that our analysis on the dynamic behaviour of iterative methodscan be used as a tool for comparing methods of same of convergence order using computergenerated plots. This enable us to choose the best efficient method from a family.


References

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[3] A F Beardon. Iteration of Rational Functions. Springer-Verlag, New York, 1991.

[4] V Drakopoulos. How is the dynamics of Konig iteration functions affected by theiradditional fixed points. Fractals, 7(3):327–334, 1999.

[5] W Gilbert. Generilizations of Newton’s method. Fractals, 9(3):251–262, 2001.

[6] B Kalantari. Polynomial root-finding and polynomiography. World ScientificPublishing Co. Pte. Ltd, Singapore, 2009.

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A Common Fixed Point Theorem of Presic Typefor Three Maps in Fuzzy Metric Space

P. P. MurthyDepartment of Pure and Applied Mathematics

Guru Ghasidas Vishwavidyalaya (A Central University)Koni, Bilaspur (Chhattisgarh), 495 009, India


RashmiDepartment of Pure and Applied Mathematics

Guru Ghasidas Vishwavidyalaya (A Central University)Koni, Bilaspur (Chhattisgarh), 495 009, India


Abstract

The present paper deals with a common fixed point theorem in Fuzzy metricspace by implementing the concept of Presic fixed point theorem [16]. In this paperwe have proved a unique common fixed point theorem of Presic type for three mapsin a Fuzzy metric space. Also we have obtained the main theorem of R. George[Some fixed point results in dislocated fuzzy metric spaces, Journal of AdvancedStudies in Topology, (2012), 3(4), 41-52] as a corollary by employing the conditionsof our theorem for dislocated spaces.

Keywords: Fixed point theorem of Presic type, Three maps in Fuzzy metric space

Manuscript accepted September 5, 2013.

1 Introduction and Preliminaries

The pioneer concept of fuzzy set theory introduced by Lofti Zadeh [1] of Univ. ofCalifornia, Berkeley in 1965. There are many view points of the notion of the metricspace in fuzzy topology. We can divide them into two groups: First group involved to

A Fixed Point Theorem of Presic Type 31

those results in which a fuzzy metric on a set X is treated as a map d : X × X → R+,where X represents the totality of all fuzzy points of a set and satisfies some axioms whichare analogous to the ordinary metric axioms. For such approach numerical distances areset up between the fuzzy objects. Where as the second group studied such results inwhich the distances between objects are fuzzy and the objects themselves may or maynot be fuzzy. Erceg [2], Kaleva and Seikkala [4] and Kramosil and Michalek [3] discussedfuzzy metric spaces in detail. Grabiec’s [6] proved a fixed point theorem in fuzzy metricspace by generalizing the contraction mapping principle due to Banach. Subramanyam [5]generalized Grabiec’s result for a pair of commuting maps in the lines of Jungck [8]. Recentdefinition of Fuzzy Metric spaces credit goes to George and Veermani [7] who modifiedthe concept of fuzzy metric spaces and defined a Hausdorff topology on this space andit has numerous applications in quantum particle physics particularly in connection withboth string and E - infinity theory. They have also shown that every metric induces afuzzy metric in Hausdroff topology.In this paper we have proved common fixed point theorems of Presic type in Fuzzy metricspaces which extends the results of R. George [14].We recall the following definitions:

Definition (1) : A fuzzy set A in X is a function with domain X and values in [0, 1].Definition (2): A binary operation ∗ : [0, 1] × [0, 1] → [0, 1] is a continuous t-norm if

it satisfies the following conditions:

(a) ∗ is associative and commutative;

(b) ∗ is continuous;

(c) a ∗ 1 = a for every a ∈ [0, 1];

(d) a ∗ b ≤ c ∗ d if a ≤ c and b ≤ d for all a, b, c, d ∈ [0, 1].

Definition (3): A triplet (X, M, ∗) is said to be a fuzzy metric space, if X is an arbitraryset, ∗ is a continuous t-norm and M is a fuzzy set on X × X × (0,∞) satisfying thefollowing conditions for all x, y, z ∈ X and t, s > 0;

(1) M(x, y, t) > 0,

(2) M(x, y, t) = 1 if and only if x = y,

(3) M(x, y, t) = M(y, x, t),

(4) M(x, y, t) ∗ M(y, z, s) ≤ M(x, z, t + s),

(5) M(x, y, ·) : (0,∞) → [0, 1] is continuous.

In view of (1) and (2), it is worth pointing out that 0 < M(x, y, t) < 1 for all t > 0,provided x = y. In view of Definition (1), George and Veermani [7] introduced the conceptof Hausdorff topology on fuzzy metric spaces and showed that every metric space inducesa fuzzy metric space.In fact, we can fuzzify metric spaces into fuzzy metric spaces in a natural way as is shown

32 P. P. Murthy and Rashmi

by the following example. In other words, every metric induces a fuzzy metric.Example (1):Let (X, d) be a metric space and define a ∗ b = ab for all a, b ∈ [0, 1]. Alsodefine M(x, y, t) = t

t+d(x,y)for all x, y ∈ X and t > 0. Then (X, M, ∗) is a fuzzy metric

space called standard fuzzy metric space induced by (X, d).Definition (4): Let (X, M, ∗) be a fuzzy metric space, then

i ) a sequence xn in X is said to be convergent to x if

limn→∞M(xn, x, t) = 1,

for all t > 0;

ii ) a sequence xn in X is said to be a Cauchy sequence if for any ε > 0, there existsn0 ∈ N, such that

M(xn, xm, t) > 1 − ε,

for all t > 0 and n, m ≥ n0;

iii ) a fuzzy metric space (X, M, ∗) is said to be complete if and only if every Cauchysequence in X is convergent.

Here, we recall the definition of weakly compatible pair of maps for 2k tuples given byRao, Kishore and Ali [15].

Definition (5): Let X be a non empty set and T : X2k → X, f : X → X.(f, T ) issaid to be 2k - weakly compatible pair, if f(T (p, p, · · · , p)) = T (fp, fp, · · · , fp) wheneverp ∈ X such that fp = T (p, p, · · · , p).

2 Main Rresults

Before proving the main theorem we define the following: Let a function φ : [0, 1]2k → [0, 1]such that:

(i) φ is an increasing function, i.e., x1 ≤ y1, x2 ≤ y2, · · · , x2k ≤ y2k implies φ(x1, x2, · · · , x2k) ≤φ(y1, y2, · · · , y2k).

(ii) φ(t, t, t, · · · ) ≥ t, for all t ∈ X.

(iii) φ is continuous in each coordinate variables.

Now we are ready to prove our main:Theorem Let (X, M, ∗) be a Fuzzy Metric space and S, T : X2k → X, f : X → X bemappings satisfying for each positive integer k :

M(S(x1, x2, · · · , x2k−1, x2k), T (x2, x3, · · · , x2k, x2k+1), qt) ≥ φM(fxi, fxi+1, t) : 1 ≤ i ≤ 2k(1)

for all x1, x2, · · · , x2k+1 ∈ X, 0 < q < 1

2and t ∈ [0,∞);

M(T (y1, y2, · · · , y2k−1, y2k), S(y2, y3, · · · , y2k, y2k+1), qt) ≥ φM(fyi, fyi+1, t) : 1 ≤ i ≤ 2k(2)


for all y1, y2, · · · , y2k+1 ∈ X; and

M(S(u, u, · · · , u), T (v, v, · · · , v), qt) > M(fu, fv, t). (3)

for all u, v ∈ X, with u = v. Suppose that f(X) is complete and either (f, S) or(f, T ) is 2k-weakly compatible pair. Then there exists a unique point p in X such thatfp = p = S(p, p, · · · , p) = T (p, p, · · · , p).Proof : Suppose x1, x2, · · · , x2k are arbitrary points in X and for n ∈ N.Define fx2k+2n−1 = S(x2n−1, x2n, x2n+1, · · · , x2n+2k−2)andfx2k+2n = T (x2n, x2n+1, x2n+2, · · · , x2n+2k−1).Let αn = M(fxn, fxn+1, qt).

Claim: αn ≥ (K−θn

K+θn )2 for all n ∈ N, where θ = 1

2qand K = min θ(1+

√

α1)

1−√

α1,

θ2(1+√

α2)

1−√

α2, · · · ,

θ2k(1+√

α2k)

1−√

α2k

.By selection of k we have αn ≥ (K−θn

K+θn )2 for n = 1, 2, · · · , 2k.Now

α2k+1 = M(fx2k+1, fx2k+2, qt)

= M(S(x1, x2, · · · , x2k−1, x2k), T (x2, x3, · · · , x2k, x2k+1), qt)

≥ φM(fxi, fxi+1, t) : i = 1, 2, · · · , 2k;by (1), we have

α2k+1 ≥ φα1, α2, · · · , α2k−1, α2k

≥ φ(K − θ1

K + θ1)2, (

K − θ2

K + θ2)2, · · · , (

K − θ2k−1

K + θ2k−1)2, (

K − θ2k

K + θ2k)2

≥ φ(K − θ2k

K + θ2k)2, (

K − θ2k

K + θ2k)2, · · · , (

K − θ2k

K + θ2k)2, (

K − θ2k

K + θ2k)2

≥ (K − θ2k

K + θ2k)2

≥ (K − θ2k+1

K + θ2k+1)2

Thus α2k+1 ≥ (K−θ2k+1

K+θ2k+1 )2.

Similarly,we have

α2k+2 = M(fx2k+2, fx2k+3, qt)

= M(T (x2, x3, · · · , x2k, x2k+1), S(x3, x4, · · · , x2k+2), qt)

≥ φM(fxi, fxi+1, t) : i = 2, 3 · · · , 2k + 1 by (1)

= φαi : i = 2, 3, · · ·2k + 1

≥ φ(K − θ2

K + θ2)2, (

K − θ3

K + θ3)2, · · · , (

K − θ2k

K + θ2k)2, (

K − θ2k+1

K + θ2k+1)2

≥ φ(K − θ2k+1

K + θ2k+1)2, (

K − θ2k+1

K + θ2k+1)2, · · · , (

K − θ2k+1

K + θ2k+1)2, (

K − θ2k+1

K + θ2k+1)2

≥ (K − θ2k+1

K + θ2k+1)2

≥ (K − θ2k+2

K + θ2k+2)2


Thus α2k+2 ≥ (K−θ2k+2

K+θ2k+2 )2. Then our claim is true.

Now we show that the sequence xn is a Cauchy sequence in X. For any n, p ∈ N, wehave

M(fxn, fxn+p, t) ≥ M(fxn, fxn+1,t

2) ∗ M(fxn+1, fxn+2,

t

22) ∗ · · · ∗ M(fxn+p−1, fxn+p,

t

2p)

≥ αn ∗ αn+1 ∗ · · · ∗ αn+p−1

≥ (K − 2n

K + 2n)2 ∗ (

K − 22n

K + 22n)2 ∗ · · · ∗ (

K − 2np

K + 2np)2

→ 1 ∗ 1 ∗ · · · ∗ as n → ∞.

Hence fxn is Cauchy sequence. Since f(X) is a complete sub space of X, then thereexists z in f(X) such that limn→∞

fxn = z. For z in f(X), there exists p ∈ X such thatz = fp. Then for any integer n, using (1) and (2), we have

M(S(p, p, p · · · , p), fp, t) = limn→∞M(S(p, p, · · · , p), fx2n+2k−1, t)

= limn→∞M(S(p, p, · · · , p), S(x2n−1, x2n, · · · , x2n+2k−2), t)

≥ limn→∞M(S(p, p, · · · , p), T (p, p, · · · , x2n−1),

t2)

∗M(T (p, p, · · · , x2n−1), S(p, p, · · · , x2n−1, x2n), t23 ) ∗ · · ·

∗M(T (p, x2n−1, · · · , x2n+2k−3), S(x2n−1, x2n, · · · , x2n+2k−2),t

2k−1 )

≥ limn→∞φM(fp, fp, t), M(fp, fp, t), · · · , M(fp, fx2n−1, t)

∗φM(fp, fp, t), M(fp, fp, t), · · · , M(fx2n−1, fx2n, t) ∗ · · ·

∗φM(fp, fx2n−1, t), M(fx2n−1, fx2n, t), · · · , M(fx2n+2k−3, fx2n+2k−2, t)

→ 1,i.e., M(S(p, p, · · · , p), fp, t) = 1 and so c(f, T ) = Φ where c(f, T ) denote the set of allcoincidence points of the mappings f and T.So that

S(p, p, · · ·p) = fp (4)

Consider

M(fp, T (p, p, · · · , p), t) = M(S(p, p, · · · , p), T (p, p, · · · , p), t)

≥ φM(fp, fp, t), M(fp, fp, t), · · · , M(fp, fp, t)≥ M(fp, fp, t)

= 1

ThusT (p, p, · · · , p) = fp. (5)


Now suppose that (f, S) is 2k - weakly compatible pair. Then we havef(S(p, p, · · · , p)) = S(fp, fp, · · · , fp).f 2p = f(fp) = f(S(p, p, · · · , p)) = S(fp, fp, · · · , fp).Suppose that fp = p. Then from (3), we have

M(f 2p, fp, t) = M(S(fp, fp, · · · , fp), T (p, p, · · · , p), t)

≥ M(f 2p, fp, t), M(f 2p, fp, t), · · · , M(f 2p, fp, t)≥ M(f 2p, fp, t).

It is a contradiction.Therefore fp = p. Now from (4) and (5), we havefp = p = S(p, p, · · · , p) = T (p, p, · · · , p).Uniqueness of p: Suppose there exists a point q = p in X such thatfq = q = S(q, q, · · · , q) = T (q, q, · · · , q)Consider

M(fp, fq, t) = M(S(p, p, · · · , p), T (q, q, · · · , q), t)

≥ φM(fp, fq, t), M(fp, fq, t), · · · , M(fp, fq, t) ≥ M(fp, fq, t).

It is a contradiction. Therefore p = q.When S = T and 2k is replaced by k in main theorem, we get the following corollary.Corollary(1): Let (X, M, ∗) be a FM - space, k is a positive integer T : Xk → X, f :X → X be mappings satisfying

M(T (x1, x2, · · · , xk), T (x2, x3, · · · , xk+1, qt) ≥ φM(fxi, fxi+1, t) : 1 ≤ i ≤ 2kfor all x1, x2, · · · , xk+1 in X, 0 < q < 1

2and t ∈ [0,∞)

M(T (u, u, · · · , u), T (v, v, · · · , v), qt) > M(fu, fv, t), ∀u, v ∈ X with u = v

Suppose that f(X) is complete and (f, T ) is k - weakly compatible pair. Then there existsa unique point p ∈ X such that fp = p = S(p, p, · · · , p) = T (p, p, · · · , p).

Acknowledgement: First author (P.P. Murthy) thankful to University Grants Commission,New Delhi, India for financial assistance through Major Research Project File number42-32/2013 (SR).

References

[1] L.A. Zadeh, Fuzzy sets, Information & Control, 8 (1965), 338-353.

[2] M.A. Erceg, Metric space in fuzzy set theory, Journal of Mathematical Analysis andApplications 69 (1979), 205-230.

[3] I.Kramosil and J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetica15 (1975), 326-334.


[4] O. Kaleva and S. Seikkala, On fuzzy metric spaces, Fuzzy Sets and Systems 12 (1984),215-229.

[5] P.V. Subramanyam, A common fixed point theorem in fuzzy metric spaces,Information Sciences 83(3-4) (1995), 109-112.

[6] M. Grabiec, Fixed points in fuzzy metric space, Fuzzy Sets and Systems 27(1988),385-389.

[7] A.George and P.Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets andSystems 64 (1994), 395-399.

[8] G. Jungck, Commuting mappings and fixed points, American Mathematical Monthly83(4) (1976), 261-263.

[9] B. Schweizer and A. Sklar, Statistical metric spaces, Pacific Journal of Mathematics10 (1960), 314-334.

[10] O. Kaleva and S. Seikkala, On fuzzy metric spaces, Fuzzy Sets and Systems 12 (1984),215-229.

[11] M.S. Khan, M. Swaleh and S. Sessa, Fixed point theorems by altering distancesbetween the points, Bulletin of the Australian Mathematical Society 30 (1984), 1-9.

[12] S.N. Mishra, N. Sharma and S.L. Singh, Common fixed points of maps on fuzzymetric spaces, International Journal of Mathematics and Mathematical Sciences 17(1994), 253-258.

[13] H.K. Pathak, S.S. Chang and Y.J. Cho, Fixed point theorems for compatiblemappings of type (P), Indian Journal of Mathematics 36(2) (1994), 151-156.

[14] R. George, Some fixed point results in dislocated fuzzy metric spaces, Journal ofAdvanced Studies in Topology,(2012), Vol. 3, No. 4, 41-52, eISSN: 2090 - 388X.

[15] K. P. R. Rao, G. N. V. Kishore and Md. Mustaq Ali, A generalization of the Banachcontraction principle of presic type for three maps, Mathematical Sciences, Vol. 3,No. 3(2009) 273 - 280.

[16] S.B. Presic, Sur une classe dinequations aux differences finies et sur la convergencede certaines suites, Publ. Inst. Math. (Beograd), 5(19)(1965), 75-78.


Analysis of Dual Functions

F. MesselmiDepartment of Mathematics and LDMM Laboratory

University Ziane Achour of Djelfa, Algeriae-mail: [email protected]

Abstract

The purpose of this paper is to develop a theory, inspired from complex analysis,of dual functions. In detail, we introduce the notion of holomorphic dual functionsand we establish a general representation of holomorphic dual functions. As anapplication, we generalize some usual real functions to the dual plane. Finally,we will define the integral trough curves of any dual functions as well as the dualprimitive.

Keywords: Dual number; dual function; holomorphicity; integration; primitive.

Manuscript accepted October 2, 2013.

1 Introduction

A dual number z is an ordered pair of real numbers (x, y) associated with the real unit 1and the dual unit ε, where ε is an nilpotent number i.e. ε2 = 0 and ε = 0. A dual numberis usually denoted in the form

z = x + yε. (1)

Thus, the dual numbers are elements of the 2−dimensional real algebra

D = R [ε] =z = x + yε | (x, y) ∈ R

2, ε2 = 0 and ε = 0

, (2)

generated by 1 and ε.There are many manners to choose the dual unit number ε, see for more details and

examples the references [1, 5].

38 Farid Messelmi

Addition and multiplication of the dual numbers are defined by

(x1 + y1ε) + (x2 + y2ε) = (x1 + x2) + (y1 + y2) ε, (3)

(x1 + y1ε) . (x2 + y2ε) = (x1x2) + (x1y2 + x2y1) ε. (4)

This multiplication is commutative, associative and distributes over addition.The algebra of dual numbers D has the numbers εy, y ∈ R, as divisors of zero. No

number εy has an inverse in the algebra D.One can verify, using (4), that

(x + yε)n = xn + nxn−1yε. (5)

The conjugate z of the dual number z = x + εy is defined by

z = x − yε. (6)

Sozz = x2. (7)

The division of two dual numbers can be computed as

z1

z2

=z1z2

z2z2

=x1x2 + (x1y2 − x2y1) ε

x22

, (8)

Thus, the division z1

z2is possible and unambiguous if x2 = 0.

Dual numbers can be represented as follows:

• Gaussian representation: z = x + yε.

• Polar representation: z = x (1 + ε arg z) , where arg z = y

x, x = 0, is the argument

of z.

It follows from this representation that

arg (z1 + z2) = arg z1 + arg z1. (9)

If z = x+ εy is a dual number, we denote by real and imaginary (dual) parts of z, thereal numbers

real (z) = x and Im (z) = y. (10)

The theory of algebra of dual numbers has been originally introduced by W. K. Clif-ford [1] in 1873, and he showed that they form an algebra but not a field because onlydual numbers with real part not zero possess an inverse element. In 1891 E. Study [11]realized that this associative algebra was ideal for describing the group of motions ofthree-dimensional space. At the turn of the 20th century, A. Kotelnikov [6] developeddual vectors and dual quaternions.

Algebraic study of dual numbers are the topic of numerous papers, e.g. [1, 5].This nice concept has lots of applications in many fields of fundamental sciences;

such, algebraic geometry, Riemannian geometry, quantum mechanics and astronomy. It

Analysis of Dual Functions 39

also arises in various contexts of engineering: aerospace, robotic and computer science.For more details about the applications of dual numbers, we refer the reader to [2, 3, 4,6, 8, 9, 11, 12, 13]

However, up to now there are only a few attempts in the mathematical study of dualfunctions (functions of dual variable). An early attempt may be due to E. E. Kramer [7]in 1930. Later, in 2011, Z. Ercan and S. Yuce [3] obtained generalized Euler’s and DeMoivre’s formulas for functions with dual Quaternion variable.

In the study of dual functions, some natural questions raise:

• When and under what conditions a dual function is differentiable ?.

• It is possible to generalize some elementary results of complex analysis to dualfunctions ?.

• How can one extend regularly real functions to dual variable ?.

The purpose of this work is to contribute to the development of dual functions andwe will try to answer some questions.

We start by generalizing the notion of holomorphicity to dual functions. To this aim,as in complex analysis, we study the Differentiability of dual functions. The notion ofholomorphicity has been introduced and a general representation of holomorphic func-tions was shown. Moreover, we provide the basic assumptions that allow us to extendholomorphically real functions to the wider dual plane and we ensure that such an ex-tension is meaningful. As an application, we generalize some usual real functions to dualplane.

Further, we also outline the concept of integral along curves of dual functions as wellas the primitives of holomorphic dual functions.

2 Holomorphicity of dual functions

We start by giving some topological definitions and properties of the dual plan D.Let us introduce the mapping P : D −→ R+,

P (z) = |real (z)| . (11)

It is easy to check that⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩zz = P (z)2 ∀z ∈ D.

P (z1 + z2) ≤ P (z1) + P (z2) ∀z1, z2 ∈ D,P (z1z2) = P (z1)P (z2) ∀z1, z2 ∈ D,P (λz) = |λ| P (z) ∀z ∈ D, ∀λ ∈ R,

P (0) = 0.

(12)

Particularly, P defines a semi-modulus in D.

40 Farid Messelmi

Thus, we can define the dual disk and dual circle of centre z0 = x0 + y0ε ∈ D andradius r > 0, respectively, by

D (z0, r) = z = x + yε ∈ D | p (z − z0) < r= z = x + yε ∈ D | |x − x0| < r, y ∈ R , (13)

S (z0, r) = z = x + yε ∈ D | p (z − z0) = r= z = x + yε ∈ D | |x − x0| = r, y ∈ R . (14)

S (z0, r) is also called Galilean circle.Definition 1 1. We say that Ω is a dual subset of the dual plan D if there exists a

subset O ⊂ R such that

Ω = O × R. (15)

O is called the generator of Ω.We say that Ω is an open dual subset of the dual plan D if the generator of Ω is an

open subset of R.2. Ω is said to be a closed dual subset of D if its complementary is an open subset of

D.3. Ω is said to be a connected dual subset of D if its generator is a connected subset

of R (real interval).We discuss now some properties of dual functions. We investigate the continuity of dual

functions and the derivability in the dual sense, which can be also called holomorphicity,as in complex case.

To this end, we will need these.Definition 2 A dual function is a mapping from a subset Ω ⊂ D to D.Definition 3 A dual function f defined on subset Ω ⊂ D is called homogeneous dual

function if

f (real (z)) ∈ R.

In following definitions, we admit that D is provided by the usual topology of R2.Let Ω be an open subset of D, z0 = x0 + y0ε ∈ Ω and f : Ω −→ D a dual function.Definition 4 We say that the function f is continuous at z0 if

limz→z0

f (z) = f (z0) . (16)

Definition 5 The function is continuous in Ω ⊂ D if it is continuous at every pointof Ω.

Definition 6 The dual function f is said to be differentiable at z0 = x0 + y0ε (in thedual sense), if the limit below exists

df

dz(z0) = lim

z→z0

f (z) − f (z0)

z − z0

, (17)

df

dz(z0) is called the derivative of f at the point z0.

If f is differentiable for all points in a neighborhood of the point z0 then f is calledholomorphic at z0.


Definition 7 The function f is holomorphic in Ω ⊂ D if it is holomorphic at everypoint of Ω.

The definition of derivative in the dual sense has to be treated with a little more carethan its real companion; this is illustrated by the following example.

Example 1 The function f (z) = z is nowhere differentiable. To this aim, a simplecalculation gives

limz→z0

f (z) − f (z0)

z − z0

= limz→z0

z − z0

z − z0

= limz→z0

z − z0

2

(x − x0)2

= 1 − 2ε limx→x0,y→y0

y − y0

x − x0

.

But this limit does not exist.The basic properties for derivatives are similar to those we know from real calculus.

In fact, one should convince oneself that the following rules follow mostly from propertiesof the limit.

Lemma 1 Suppose f and g are differentiable at z ∈ D, and that c ∈ D, n ∈ Z, andh is differentiable at g (z) .

1. d(f+cg)

dz= df

dz+ cdg

dz.

2. d(f.g)

dz= df

dzg + f dg

dz.

3.d( f

g)

dz=

df

dzg−f

dg

dz

g2 (we have to be aware of division by zero).

4. dhg

dz= dh

dz(g) dg

dz.

In the following results we generalize the Cauchy-Riemann formulas to dual functions.Theorem 2 Let f be a dual function in Ω ⊂ D, which can be written in terms of its

real and dual parts asf = ϕ + εψ.

f is holomorphic in Ω ⊂ D if and only if the derivative of f satisfies

df

dz=

∂f

∂x=

∂ϕ

∂x+

∂ψ

∂xε. (18)

Proof. Suppose that f is holomorphic in Ω and let z0 = x0 + εy0 be an arbitraryelement of Ω.

By definition of derivative, we have

df

dz(z0) = lim

x→x0

f (x + yε) − f (x0 + y0ε)

(x + yε) − (x0 + y0ε)

= limx→x0,y→y0

ϕ (x, y) − ϕ (x0, y0)

x − x0

+ limx→x0,y→y0

ψ (x, y) − ψ (x0, y0)

x − x0

ε +

limx→x0,y→y0

(ϕ (x, y) − ϕ (x0, y0)) (y − y0)

(x − x0)2

ε.

=∂ϕ

∂x(x0, y0) +

∂ψ

∂x(x0, y0) ε +

limx→x0,y→y0

(ϕ (x, y) − ϕ (x0, y0)) (y − y0)

(x − x0)2

ε.

42 Farid Messelmi

Then, the limit exists if and only if it does not depend on limit of the bounded ratioy−y0

x−x0. Hence, we have two cases to deal with

First case. Let us remark that

limx→x0,y→y0

(ϕ (x, y) − ϕ (x0, y0)) (y − y0)

(x − x0)2

= limx→x0,y→y0

ϕ (x, y) − ϕ (x0, y0)

x − x0

y − y0

x − x0

.

Therefore, the above limit exists if

limx→x0,y→y0

ϕ (x, y) − ϕ (x0, y0)

x − x0

=∂ϕ

∂x(x0, y0) = 0. (19)

This implies thatdf

dz(z0) =

∂ψ

∂x(x0, y0) ε.

Which gives (18).Second case. We can also write

limx→x0,y→y0

(ϕ (x, y) − ϕ (x0, y0)) (y − y0)

(x − x0)2

= limx→x0,y→y0

ϕ (x, y) − ϕ (x0, y0)

y − y0

(y − y0

x − x0

)2

.

Hence, the limit exists if

limx→x0,y→y0

ϕ (x, y) − ϕ (x0, y0)

x − x0

=∂ϕ

∂y(x0, y0) = 0. (20)

Clearly, (18) follows. This permits us to conclude the proof.Corollary 3 Let f be a dual function in Ω ⊂ D, which can be written in terms of

its real and dual parts as f = ϕ + εψ and suppose that the partial derivatives of f exist.Then,

1. f is holomorphic in Ω ⊂ D if and only if its partial derivatives satisfy

ε∂f

∂x=

∂f

∂y. (21)

2. f is holomorphic in Ω ⊂ D if and only if the following formula holds∂ϕ

∂x= ∂ψ

∂y,

∂ϕ

∂y= 0.

. (22)

Proof. 1. In view of (18), we can assert that the total differential of f can be written

df =

(∂ϕ

∂x+

∂ψ

∂xε

)d (x + εy)

=

(∂ϕ

∂x+

∂ψ

∂xε

)dx +

∂ϕ

∂xεdy. (23)

This yields, ∂f

∂x= ∂ϕ

∂x+ ∂ψ

∂xε,

∂f

∂y= ∂ϕ

∂xε,

(24)


and so

ε∂f

∂x= ε

∂ϕ

∂x.

Which eventually, combined with the second equation in (24), gives (21).2. The formula (21) leads to(

∂ϕ

∂x+

∂ψ

∂xε

)ε =

∂ϕ

∂y+

∂ψ

∂yε.

Hence, (22) results.Theorem 4 The function f is holomorphic in the open subset Ω ⊂ D, (with respect

to the topology of R2), if and only if there exists a pair of real functions ϕ and k, such thatϕ ∈ C1 (Px (Ω)) , dϕ

dxis differentiable in Px (Ω) and k is differentiable in Px (Ω) , where Px

is the first projection (parallel to the vertical axis), so that the function f can be writtenexplicitly

f (z) = ϕ (x) +

(dϕ

dxy + k (x)

)ε ∀z ∈ Ω. (25)

Proof. Since f is holomorphic in Ω, we find, employing (22)∂ϕ

∂x= ∂ψ

∂y,

∂ϕ

∂y= 0.

It follows thatϕ (x, y) = ϕ (x) .

Hence∂ψ

∂y=

dϕ

dx.

So, we find

ψ (x, y) =dϕ

dxy + k (x) .

This achieves the proof.Remark 1 1. The formula (25) gives, taking into account the fact that df

dz= ∂f

∂x,

df

dz=

dϕ

dx+

(d2ϕ

dx2y +

dk

dx

)ε. (26)

In addition, if the functions ϕ and k are enough regular, we can generalize this relationby recurrence.

2. If, in particular, f is an homogeneous function, formula (2.15) gives k ≡ 0.3. We remark that f is linear with respect to the variable y. Then, f can be holomor-

phically extended to the dual subset Px (Ω) × R.The lemma below can be easily deduced.Lemma 5 Let f be an homogeneous holomorphic function in the open dual subset

O × R ⊂ D. Thenf (z) = f (z) ∀z ∈ O × R. (27)

44 Farid Messelmi

We are interested now to some properties regarding constant dual functions. To thisaim, we need the following definition.

Definition 8 Let f be a dual function in the subset Ω ⊂ D.1. f is said to be bounded in Ω if there exists a positive constant c such that

‖f (z)‖R2 ≤ c ∀z ∈ Ω, (28)

which means that

∃c > 0 | (realf (z))2 + (Imf (z))2 ≤ c ∀z ∈ Ω. (29)

2. f is said to be bounded in the dual sense in Ω if there exists a positive constant csuch that

P (z) ≤ c ∀z ∈ Ω. (30)

Proposition 6 1. Let f be an holomorphic function in the open dual subset O×R ⊂D. If there exists a connected and compact subset K, (bounded interval), contained inO such that f is bounded in K. Then, there exists a differentiable real function k and aconstant C (K) , depending on K, such that the following formula holds

f (z) = C (K) + k (x) ε ∀z ∈ K × R. (31)

2. Let f be an homogeneous holomorphic function in the open connected dual subsetO × R ⊂ D. If f is bounded in O × R, then it is necessary constant in O × R.

Proof. 1. Making use (25), we have

f (z) = ϕ (x) +

(dϕ

dxy + k (x)

)ε ∀z ∈ O × R,

whence ϕ ∈ C1 (O) , dϕ

dxis differentiable in O and k is differentiable in O.

Suppose that there exists a compact K contained in O such that f is bounded in K,then there exists c > 0 verifying, for all x ∈ K and y ∈ R,

|ϕ (x)| ≤ c,∣∣∣∣(dϕ

dxy + k (x)

)∣∣∣∣ ≤ c.

Thus, we can infer that ϕ is bounded in O and∣∣∣∣dϕ

dx

∣∣∣∣ |y| ≤ c + |k (x)| ∀x ∈ K, ∀y ∈ R.

Which asserts us, keeping in mind that k is continuous in K, that∣∣∣∣dϕ

dx

∣∣∣∣ |y| ≤ c + maxK

|k (x)| ∀x ∈ K, ∀y ∈ R.

Consequently, since O is connected, we deduce that ϕ is constant.


2. Let f be an homogeneous holomorphic function in an open connected dual subsetO × R ⊂ D. Then, by definition, we have k ≡ 0.

Hence, since O × R is supposed to be connected, the result follows, by proceeding asin the proof of the first assertion.

Proposition 7 If an holomorphic function f is defined in an open connected dualsubset O × R ⊂ D and df

dz= 0 for all z in O × R, then f is constant.

Proof. In view of Theorem 4, there exists a pair of real functions ϕ and k, whereϕ ∈ C1 (O) , dϕ

dxis differentiable in O and k is differentiable in O, such that

f (z) = ϕ (x) +

(dϕ

dxy + k (x)

)ε ∀x ∈ O.

Under this formula the derivative of f can be written

df

dz=

dϕ

dx+

(d2ϕ

dx2y +

dk

dx

)ε.

From which, we can infer

dϕ

dx= 0 ∀x ∈ O,

d2ϕ

dx2y +

dk

dx= 0 ∀x ∈ O, ∀y ∈ R.

Then, since O is connected, we conclude that ϕ and k are constant.The following Proposition shows that we can extend any regular real function to the

dual plane D.Proposition 8 Let O an open subset of R, f ∈ C1 (O) and df

dxis differentiable in O.

Then, there exists a unique homogeneous holomorphic function F : O × R ⊂ D −→ D

such thatF (x) = f (x) ∀x ∈ O. (32)

Moreover, if f ∈ Ck (O) then F ∈ Ck−1 (O × R) .Proof. Considering the dual function F given by

F (z) = F (x + εy) = f (x) +df

dxyε ∀z ∈ O × R.

It is clear that F is an homogeneous holomorphic function in O × R and verifies

F (x) = f (x) ∀x ∈ O.

Suppose that there exists another holomorphic function g in O × R such that

g (x) = f (x) ∀x ∈ O.

It is well known that there exists a real function ϕ, where ϕ ∈ C1 (O) and dϕ

dxis

differentiable in O, satisfying

g (z) = ϕ (x) +dϕ

dxyε.

46 Farid Messelmi

Hence,g (x) = ϕ (x) ∀x ∈ O,

and soϕ (x) = f (x) ∀x ∈ O.

The second assertion is an immediate consequence.

3 Usual dual functions

We can think of applying the statement of proposition 8, which asserts that any reg-ular real function on its domain can be holomorphically extended to dual numbers, tobuild homogeneous dual functions similar to the usual real functions, obtained as theirextensions.

3.1 The dual exponential function

The real exponential function ex defined for all x ∈ R can be extended to the dual planeD as follows

exp (z) = ez = ex + exyε = ex (1 + yε) . (33)

The derivative of ez is

dez

dz=

dex

dx+

dex

dxyε = ez ∀z ∈ D. (34)

By recurrence, we find

dnez

dzn= ez ∀z ∈ D, ∀n ∈ N. (35)

Thus, any dual number z = x + yε ∈ D, x = 0 has the exponential representation

z = xe(arg z)ε. (36)

Some properties of the dual exponential function are collected in the following.Proposition 9 1. ez1+z2 = ez1ez2 .2. e−z = 1

ez .3. ez = 0 ∀z ∈ D.4. exp ∈ C∞ (D) .

3.2 The dual trigonometric functions

The trigonometric functions: sine, cos, tangent, etc, have their dual analogues. In fact,we can define them by the formulas

sin z = sin x + (cosx) yε ∀z ∈ D, (37)

cos z = cos x − (sin x) yε ∀z ∈ D, (38)


tan z = tan x − y

cos2 xε =

sin z

cos z∀z ∈ D − (2k + 1)π, k ∈ Z × R. (39)

The next properties follow mostly from the previous definition.Proposition 10 1. sin, cos and tan are 2π−periodic functions.2. sin (−z) = − sin z, cos (−z) = cos z.3. sin (z1 + z2) = sin z1 cos z2 + cos z1 sin z2.4. cos (z1 + z2) = cos z1 cos z2 − sin z1 sin z2.5. sin2 z + cos2 z = 1.6. cos (εx) = 1, sin (εx) = εx.7. d sin z

dz= cos z.

8. d cos zdz

= − sin z.9. sin, cos ∈ C∞ (D) and tan ∈ C∞ (D − (2k + 1)π, k ∈ Z × R) .

3.3 The dual hyperbolic functions

The dual hyperbolic functions are defined by

sinh z = sinh x + (cosh x) yε ∀z ∈ D, (40)

cosh z = cosh x + (sinh x) yε ∀z ∈ D, (41)

tan z =sinh z

cosh x∀z ∈ D. (42)

These are equivalent, as in the real case, to

sinh z =ez − e−z

2∀z ∈ D, (43)

cosh z =ez + e−z

2∀z ∈ D, (44)

tanh z =ez − e−z

ez + e−z∀z ∈ D. (45)

The following collects some basic properties.Proposition 11 1. sinh (−z) = − sinh z, cosh (−z) = cosh z.2. cosh2 z − sinh2 z = 1.3. cosh (εx) = 1, sinh (εx) = εx.4. d sinh z

dz= cosh z.

5. d cosh zdz

= sinh z.6. sinh, cosh, tanh ∈ C∞ (D) .

3.4 The dual logarithmic function

We define the dual Logarithmic function by the formula

log z = log x +y

xε = log x + (arg z) ε ∀z ∈ R

∗

+ × R ⊂ D. (46)

The dual Logarithmic function, satisfies some properties, given by

48 Farid Messelmi

Proposition 12 1. log(

1

z

)= − log z.

2. log (z1z2) = log z1 + log (z2) .3. elog z = log (ez) = z.4. log (zα) = α log z ∀z ∈ R∗

+ × R, ∀α ∈ R.

5. d log z

dz= 1

z.

6. log ∈ C∞

(R∗

+ × R).

Example 2 We can evaluate the quantity zw for all z = x + yε ∈ R∗

+ × R andw = a + bε ∈ D as follows

zw = e(a+bε)(log x+y

xε)

= ea log x+(ay

x+b log x)ε

= ea log x(1 +

(ay

x+ b log x

)ε)

.

4 Integration of dual functions

We begin integration by focusing on ”1−dimensional” integrals over lines.At first sight, dual integration is not really anything different from real integration.

For a continuous dual-valued function f : [a, b] ⊂ R −→ D, we define

b∫a

f (t) dt =

b∫a

real (f (t)) dt + ε

b∫a

Im (f (t)) dt. (47)

For a function which takes dual numbers as arguments, we integrate along a curve γ(instead of a real interval). Suppose this curve is parameterized by γ (t) , a ≤ t ≤ b. Thefollowing definition is used.

Definition 9 Suppose γ is a smooth curve parameterized by γ (t) , a ≤ t ≤ b, and fis a continuous dual function on γ. Then, we define the integral of f on γ as∫

γ

f (z) dz =

b∫a

f (γ (t))dγ

dtdt. (48)

This definition can be easily and naturally extended to piecewise smooth curves. Inwhat follows, we will usually state our results for smooth curves, bearing in mind thatpractically all can be extended to piecewise smooth curves.

The dual integral has some standard properties, most of which follow from real case.The first property to observe is that the actual choice of parameterization of γ does notmatter.

Proposition 13 Let γ be a smooth curve and let f a continuous dual function on γ.Then, the integral

∫γf (z) dz is independent of the parameterization of γ chosen.

We give now the following Proposition, which is a direct consequence of the definitionof dual integral.

Proposition 14 1. Suppose γ is a smooth curve, f and g are dual functions whichare continuous on γ and let c ∈ D. Then∫

γ

(cf (z) + g (z)) dz = c

∫γ

f (z) dz +

∫γ

g (z) dz. (49)


2. If γ is parameterized by γ (t) , a ≤ t ≤ b, define the curve −γ by −γ (t) =γ (a + b − t) . Then ∫

−γ

f (z) dz = −∫

γ

f (z) dz. (50)

3. If γ1 and γ2 are smooth curves so that γ2 starts where γ1 ends then define the curveγ1 γ2 by following γ1 to its end, and then continuing on γ2 to its end. Then∫

γ1γ2

f (z) dz =

∫γ1

f (z) dz +

∫γ2

f (z) dz. (51)

4. Suppose that γ is a smooth curve parallel to the vertical axis, i.e. γ is parameterizedby γ (t) = (α, y (t)) , a ≤ t ≤ b and α ∈ R, then

real

(∫γ

f (z) dz

)= 0, (pure dual number). (52)

Now, we claim and prove the following Theorem, which generalizes that of Cauchy-Goursat to dual functions.

Theorem 15 Let O be an open connected subset of R and let f be an holomorphicfunction in the dual subset O×R ⊂ D. Then, the integral of f vanishes along any rectanglecontained in O × R.

Proof. To simplify the proof we restrict ourselves to the rectangle R = z1, z2, z3, z4 ,whence z1 = x1+εy1, z2 = x2+εy1, z3 = x2+εy2, z4 = x1+εy2, such that a ≤ x1 ≤ x2 ≤ band y1 ≤ y2.

As already known, there exists a pair of real functions ϕ and k, such that ϕ ∈ C1 (O) ,dϕ

dxis differentiable in O and k is differentiable in O, satisfying

f (z) = ϕ (x) +

(dϕ

dxy + k (x)

)ε ∀z ∈ O × R.

Then, the integral of f becomes∫∂R

f (z) dz =

∫∂R

[ϕ (x) +

(dϕ

dxy + k (x)

)ε

](dx + εdy)

=

∫∂R

[ϕ (x) +

(dϕ

dxy + k (x)

)ε

]dx + εϕ (x) dy

=

x2∫x1

[ϕ (x) +

(dϕ

dxy1 + k (x)

)ε

]dx + ε

y2∫y1

ϕ (x2) dy

−x2∫

x1

[ϕ (x) +

(dϕ

dxy2 + k (x)

)ε

]dx − ε

y2∫y1

ϕ (x1) dy

= ε (ϕ (x2) − ϕ (x1)) y1 + ε (y2 − y1)ϕ (x2) − ε (ϕ (x2) − ϕ (x1)) y2

−ε (y2 − y1)ϕ (x1) .

Thus,∫

∂Rf (z) dz = 0.

50 Farid Messelmi

Theorem 16 Let γ1 and γ2 be two homotopic smooth curves parameterized, respec-tively, by γ1 (t) and γ2 (t) , 0 ≤ t ≤ 1 such that γ1 (0) = γ2 (0) and γ1 (1) = γ2 (1) and Hdenotes a regular homotopy between γ1 and γ2 defined by⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

H : [0, 1]2 −→ R,H (t, 0) = γ1 (t) ∀t ∈ [0, 1] ,H (t, 0) = γ2 (t) ∀t ∈ [0, 1] ,

H (0, s) = γ1 (0) = γ2 (0) ∀s ∈ [0, 1] ,H (1, s) = γ1 (1) = γ2 (1) ∀s ∈ [0, 1] .

(53)

Let f be an holomorphic function in a dual subset O×R containing the curves γ1 andγ2. If H has continuous second partial derivatives, then∫

γ1

f (z) dz =

∫γ2

f (z) dz. (54)

Proof. Let γs be the curve parameterized by H (t, s) , 0 ≤ t ≤ 1. Consider the functionI (s) =

∫γs

f (z) dz.We will show that I is constant with respect to s. To do this, considering the derivative

of I. By some calculations, we obtain

dI

ds=

d

ds

1∫0

f (H (t, s))∂H

∂tdt

=

1∫0

(f ′ (H (t, s))

∂H

∂s

∂H

∂t+ f (H (t, s))

∂2H

∂s∂t

)dt

If we assume assumption that H has continuous second partial derivatives, the follow-ing equality holds

dI

ds=

1∫0

(f ′ (H (t, s))

∂H

∂t

∂H

∂s+ f (H (t, s))

∂2H

∂t∂s

)dt

=

1∫0

∂

∂s

(f (H (t, s))

∂H

∂s

)dt.

ThendI

ds= f (H (1, s))

∂H

∂s(1, s) − f (H (0, s))

∂H

∂s(0, s) = 0.

This obviously completes the proof of the Theorem.Corollary 17 Let f be an holomorphic function in a connected dual subset O × R

and γ ⊂ O × R be a closed smooth curve. Then∫

γf (z) dz = 0.

Proof. It is enough to remark that O × R is simply connected (with respect to thetopology of R2). Hence, the curve γ is contractible, and so Theorem 16 affirms us that∫

γf (z) dz = 0.


We introduce now the concept of primitive of dual functions.Definition 10 Let O×R be an open subset of D. For any functions f, F : O×R −→ D,

if F is holomorphic in O ×R and dFdz

= f (z) for all z ∈ O ×R, then F is a primitive of fin O × R.

Theorem 18 Every holomorphic function in an open connected subset O × R of D

has a primitive.Proof. Let f : O × R −→ D be an holomorphic function in O × R.Fix z0 = x0 + y0ε ∈ Int (O × R) and consider the dual function F defined by

F (z) =

1∫0

(z − z0) f (z0 + t (z − z0)) dt ∀z ∈ O × R.

Thus, F possesses partial derivatives with respect to the variables x and y, given by

∂F

∂x=

1∫0

(f (z0 + t (z − z0)) + (z − z0) t

df

dz(z0 + t (z − z0))

)dt,

∂F

∂y= ε

1∫0

(f (z0 + t (z − z0)) + (z − z0) t

df

dz(z0 + t (z − z0))

)dt.

We can infer∂F

∂y= ε

∂F

∂x.

Which implies that F is holomorphic in O × R and its derivative is

dF

dz=

∂F

∂x=

1∫0

(f (z0 + t (z − z0)) + (z − z0) t

df

dz(z0 + t (z − z0))

)dt

=

1∫0

d

dt(tf (z0 + t (z − z0))) dt = f (z) .

We conclude that F is a primitive of f.Theorem 19 Suppose f is continuous in an open connected dual subset of O×R ⊂ D

and ∫γ

f (z) dz = 0, (55)

for all smooth closed paths γ ⊂ O × R. Then f has a primitive in O × R.Proof. Fixing a point z0 = x0 + y0ε ∈ Int (O × R) . For each point z ∈ O × R, let γz

be a smooth curve in O × R from z0 to z.Introducing the function

F (z) =

∫γz

f (w) dw.

52 Farid Messelmi

We obtain the following both equalities (56) and (57), by integrating along, respec-tively, the two closed paths below and using the hypothesis (55),

[x0, x] × y0ε ∪ x × [y0ε, yε]∪ −γz ,

x0 × [y0ε, yε] ∪ [x0, x] × yε ∪ −γz ,

x∫x0

f (s + y0ε) ds +

y∫y0

f (x + tε) εdt − F (z) = 0, (56)

y∫y0

f (x0 + tε) εdt +

x∫x0

f (s + yε)ds − F (z) = 0. (57)

Owing (56) and (57), we deduce that F possesses partial derivatives with respect thevariables x and y, given by

∂F

∂x= f (z) and

∂F

∂y= εf (z) .

This yields

ε∂F

∂x=

∂F

∂y.

Which implies, making use (21), that F is holomorphic in O × R and its derivative is

dF

dz=

∂F

∂x= f (z) .

We establish here a general representation of primitives of any holomorphic function.Proposition 20 Let O × R be an open connected subset of D. Let f : O × R −→ D

be an holomorphic function in O × R given by the representation

f (z) = ϕ (x) +

(dϕ

dxy + k (x)

)ε ∀z ∈ O × R. (58)

Then, the primitive of f can be calculated via the following formula

F (z) =

∫f (z) dz =

∫ϕ (x) dx +

(ϕ (x) y +

∫k (x) dx

)ε + c ∀z ∈ O × R. (59)

where c is an arbitrary dual constant.Proof. Let F be primitive of f, i.e. dF

dz= f.

It is well known that there exists a pair of real functions ϕ and k, such that ϕ ∈ C1 (O) ,dϕ

dxis differentiable in O and k is differentiable in O, satisfying

F (z) = ψ (x) +

(dψ

dxy + r (x)

)ε ∀z ∈ O × R.

So

f (z) =dF

dz=

∂F

∂x=

dψ

dx+

(d2ψ

dx2y +

dr

dx

)ε ∀z ∈ O × R.


Which implies, employing (58)

dψ

dx= ϕ (x) ∀x ∈ O,

d2ψ

dx2y +

dr

dx=

dϕ

dxy + k (x) ∀x ∈ O, ∀y ∈ R.

Hence, for every x ∈ O ψ (x) =

∫ϕ (x) dx + c1,

r (x) =∫

k (x) dx + c2,

where c1 and c2 are two real constants.Thus, we obtain

F (z) =

(∫ϕ (x) dx + c1

)+

(ϕy +

∫k (x) dx + c2

)ε

=

∫ϕ (x) dx +

(ϕ (x) y +

∫k (x) dx

)ε + (c1 + c2ε) ∀z ∈ O × R.

This completes the proof of the Theorem.We are now ready to state the following Theorem, showing that the integral of holo-

morphic functions along any path dependent only on the extremities.Theorem 21 Suppose O×R is an open connected dual subset. Let f : O×R −→ D

be an holomorphic function in O×R, γ ⊂ O×R be a smooth curve with parameterizationγ (t) , a ≤ t ≤ b. If F is any primitive of f in O × R then∫

γ

f (z) dz = F (γ (b)) − F (γ (a)) . (60)

Proof. By definition of the integral along the curve γ, one can check that∫γ

f (z) dz =

∫γ

dF

dzdz

=

b∫a

dF

dz(γ (t))

dγ

dtdt

=

b∫a

d

dtF (γ (t)) dt = F (γ (b)) − F (γ (a)) .

References

[1] W. K. Clifford. Preliminary sketch of bi-quaternions, Proc. London Math. Soc. 4,381-395, 1873.

[2] A. C. Coken, A. Gorgulu, On the dual Darboux rotation axis of the dual space curve,Demonstratio Math Vol. 35, No. 2, 385–390, 2002.

54 Farid Messelmi

[3] Z. Ercan, S. Yuce, On properties of the dual quaternions, Eur. J. Pure Appl. Math,Vol. 4, No. 2, 142-146, 2011.

[4] J. A. Fike, J. J. Alonso, The development of hyper-dual numbers for exact second-derivative calculations, 49th AIAA Aerospace Sciences Meeting including the NewHorizons Forum and Aerospace Exposition, Orlando, Florida, 4-7 January 2011.

[5] W. B. V. Kandasamy, F. Smarandache, Dual numbers, ZIP Publishing, Ohio, 2012.

[6] A. P. Kotelnikov, Screw calculus and some applications to geometry and mechanics,Annal. Imp. Univ. Kazan, 1895. (Russian).

[7] E. E. Kramer, Polygenic functions of the dual variable w = u + jv, Amer. J. Math.Vol. 52, No. 2, pp. 370-376, Apr 1930.

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[9] C. Soule, Rational K−theory of the dual numbers of a ring of algebraic integers,Springer Lec. Notes 854, 402-408, 1981.

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[11] E. Study, Geometrie der dynamen, Teubner, Leipzig, 1901.

[12] G. R. Veldkamp, On the use of dual numbers, vectors and matrices in instantaneousspatial kinematics, Mechanisms and Machine Theory, vol. 11. pp. 141-156, 1976.

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Chaotic Dynamical Behaviorof Recurrent Neural Network

A. ZerrougDepartment of Mathematics,

University of Biskra, BP 145 Biskra, 07000, Algeriae-mail: [email protected]

L. TerrissaDepartment of informatique,

University of Biskra, BP 145 Biskra, 07000, Algeriae-mail: [email protected]

A. FaureLaboratoire GREAH, Universite du Havre. Place Robert Schuman, 76610


Abstract

On account of their role played in the fundamental biological rhythms and byconsidering their potential use in information processing, the dynamical propertiesof an artificial neural network are particularly interesting to investigate. In orderto reduce the degree of complexity of this work, we have considered in this papera fully connected neural network of two discrete neurons. We have proceeded to aqualitative and quantitative study of their state evolution by means of numericalsimulation. The first aim was to find the possible equilibrium states. Other authorshave already shown that some oscillating state can occur. So, the second aim wasto analyze the dynamical properties of each of them. We have computed the valueof the Lyapunov’s exponents and the fractal dimension. The sensitivity of thedynamical characteristics to parameters such as the weights of the connections andthe shape of the activation function has been studied.

Keywords: Bifurcation, Chaotic behaviour, Dynamical Systems, Neural Oscillator,Recurrent Neural Networks, Strange attractors.

Manuscript accepted October 20, 2013.

56 A. Zeroug, et al.

Figure 1: Neural network.

1 Introduction

Chaotic dynamical behavior in the brain has recently been observed and discussed. Whetherchaotic behavior of neural networks has any application in biological modeling such uslearning and information processing is a very controversial issue [1]. An important ques-tion is the biological implications of chaos in neural networks [11, 16]. Neverthles, a lotof effort has gone into modeling and analyzing the chaotic behavior of biological systems,especially for the Hodgkin -Huxley axon model [22]. In this paper, we have tried to studythis phenomenon by using a recurrent neural network, which we find to be conceptu-ally simpler, and more appropriate for dealing with discrete or symbolic data [21]. Thisrecurrent neural network is constituted of only two neurons with the sigmoıdal neuronactivation function and has no external inputs and no time delay. Our practical objectiveto studied this kind of network is to design network dynamics for performing a given taskby adjusting network parameters. This involves two major issues in dynamical systemtheory, namely, dynamics and bifurcation. Dynamics is concerned with the asymptoticbehavior of the networks, which includes limit sets (e.g., fixed points, periodic orbits andchaotic invariant sets) and their asymptotic stability (e.g., stable, unstable and saddle),while bifurcation is concerned with how the dynamics changes as parameters are varied[24]. In this context, we have performed numerical simulations for determinate the equi-librium states and their asymptotic stability in the first time, and measure the chaos byevaluating the Lyaounov’s exponents, as well as the fractal dimension in the second time.The influence of the parameters of the system on the dynamic behavior has been alsostudied.

2 Model and equations

The study proposed network is a variant of the Mac-Culloch and Pitt’s model, where theactive parts are two discrete neurons. The first is an excitatory neuron, and the secondis an inhibitory one. The network is fully connected and we assign a weight wij to eachconnection (Fig 1). The representative state evolution of this model is given by the systemof equations (1).

x(t + 1) = f(w11x(t) + w12y(t) + e1)y(t + 1) = f(w21x(t) + w22y(t) + e2)

(1)

Chaotic Dynamical Behavior of Recurrent Neural Network 57

Where x and y are the state variables of the two neurons, their values lie in the rangeI = [0 1]. The synaptic weights are wij. Some external inputs e1 and e2 have their valueassigned to zero in a first step.

The activation function f has a sigmoıdal form, and is called Fermi’s function. It isgiven by the following equation:

f (z) =1

1 + e−4σz(2)

Where σ is the constant of Fermi or neuron gain . In this paper, we have speciallyconsidered this constant on account of its contribution to the activation function and thedynamical sensitivity of the system [22]. In fact:

• It controls the maximal slope of the function f : the larger the value of σ is, thecloser the function f approximates the step function.

• It serves a purpose (at least mathematically) that a change of σ causes a changeof all connectivity weights wij and therefore affects the dynamic behavior of thenetwork. The effect of an increasing or decreasing value of σ is similar to the effectof learning modifying the weights.

For this network it has been proven mathematically that with an increasing fermiconstant σ the period doubling route is leading to chaos [22]. In particular, it showschaotic behavior for the parameter setting used here, σ = 1 [12, 22]. See also the diagramsof bifurcation (figure 5).

3 Equilibrium states

To determinate the equilibrium states, we assume that a such state exists, so:

xi(k + 1) = xi(k) (3)

This condition applied to the components of the state variables of the studied system(1) gives:

x(k + 1) = x(k)y(k + 1) = y(k)

(4)

It follows that the two algebraic equations shows in (5) must be simultaneously solved:f(w11x(k) + w12y(k)) = x(k)f(w21x(k) + w22y(k)) = y(k)

(5)

As the function f is a non-linear function, the analytical solving of this system ofequations is not considered for the time being [7, 6]. A graphical method of solving isapplied. This method consists to draw respectively on the same reference the followinggraphs: x(k) vs y(k) and y(k) vs. x(k). The possible points of intersection correspond tothe sought equilibrium states (Fig 2).


Figure 2: Equilibrium state.

To carry out this task, the determination of the elements of the synaptic matrix Wis necessary, because they are as relevant as the neuron gain σ and that affects directlythe dynamic behavior of the system. The choice of this matrix is arbitrary and doesnot follow a general law. As usual, we consider that the mutual connection between theneurons are symmetrical, and wij = wji. This implies that the matrix is symmetrical.This assumption is funded in the case where the learning law of Hebb is taking account.During the process of learning, the mutual connections lead to reach the same valuefor the weights. More, the symmetrical form of W leads to some important properties.But if we consider the biological aspect, the same value for the mutual connection is notimperative itself. Some biological measurements have shown that the excitatrice (wij > 0),or inhibitrice (wij < 0) synaptic property depends only of the neuron j, transmitter ofsignal [10]. Unfortunately, the asymmetrical networks are much more difficult to analyze,what explains the restricted number of works which were dedicated to them [17].

In this work, we are interested in the study of the behavior and the properties ofthe biologic nervous system from an already established model, therefore the considerednetwork is not a processing system whose objective is to carry out a precise task. So, thephase of learning is not necessary to establish the weight matrix.

In the previous works, one finds several approaches to establish this matrix. Som-polinsky [15,19], for example, demonstrated that continuous-time networks with randomasymmetrical connection will be chaotic asymptotically as number of neurons n tends toinfinity, provided that the origin is not a stable fixed point. But it is not clear how ”spon-taneous” chaos occurs an autonomous neural networks of finite neurons with no time delayand no external inputs. On the other hand, X.Wang [22, 4, 8] proved, that for a certainclass of connection weight matrices, the simple neural network is dynamically equivalentto a one-parameter full family of S-unimodal maps on the interval [0 1] ,which is well-known to become chaotic through the periodic-doubling route as the parameter varies.In this paper we use this mathematical property, which seems to us to be convenient forthis study.

We consider now the following matrix:

W =

(w11 w12

w21 w22

)(6)

During the stationary state of the network, the values of the weights stabilize, and we


can show that: w11 = −w12

w21 = −w22

, (7)

letting w11 = a and w21 = b, one obtains the following new matrix:

W =

(a −ab −b

)(8)

According to [22, 18], the dynamic evolution of the network becomes chaotic if one

chooses b < a < 0 withb

a> 2 (see figure 5).

4 Study of the stability

The figure 2 shows a single intersection point in the interval [0 1], corresponding to theequilibrium state. Its coordinates in the space phase are (xe, ye) = (0.5, 0.5). To analysethe condition of stability in the vicinity of this equilibrium state, we consider the effectof a perturbation and we study the resulting evolution [5]. Let:

−→X (k + 1) =

−→X (k) at the equilibrium state with

−→X (k) =

(x(k)

y(k)

)(9)

Or:−→X (k + 1) =

−→F (

−→X ),

−→F =

f1

f2

(10)

Let us consider a small increment−→δX(k), such as:

−→δX(k) = [δx(k), δy(k)] (11)

So, we may write for each components of the state:δx(k + 1) = f1 [x(k) + δx(k), y(k) + δy(k)] − f1 [x(k), y(k)]δy(k + 1) = f2 [x(k) + δx(k), y(k) + δy(k)] − f2 [x(k), y(k)]

(12)

Thus, approximately: ⎧⎪⎨⎪⎩δx(k + 1) =

∂f1

∂xδx(k) +

∂f1

∂yδy(k)

δy(k + 1) =∂f2

∂xδx(k) +

∂f2

∂yδy(k)

(13)

Or:

−→δX(k + 1) = J · −→δX(k) with J =

⎡⎢⎣δf1

δx

δf1

δyδf2

δx

δf2

δy

⎤⎥⎦ (14)

Where J is the Jacobian matrix of the linearized transformation.


Figure 3: Dynamical behavior.

The nature of stability depends then on the eigenvalue of the jacobian matrix J eval-uated at the equilibrium state. In our case the characteristic equation can be write:

λ2 − Tr · λ + ∆ = 0 avec

⎧⎪⎨⎪⎩Tr =

δf1

δx+

δf2

δy

∆ =δf1

δx· δf2

δy− δf1

δx· δf2

δy

(15)

where λ is a eigenvalue, Tr is the trace of J and ∆ its determinant.

The evolution becomes stable if and only if the each eigenvalue module is inferior tothe unit.

By using the matrix (7), one can show that the equilibrium point of the system becomes

stable for b− 1

4< a < b +

1

4, and unstable outside this interval (a and b are the synaptic

matrix elements). To obtain the oscillation in the system it is necessary to choose a andb such manner that previous inequality is not verified, in the opposite case the systemconverges to the unstable equilibrium point.

5 Dynamical behavior

So as to undertake a qualitative study allowing the measure of chaos in our network, weconsider the system (1) with e1 = e2 = 0. As is shown in the figure 3, by using differentvalues of synaptic weights the system represents oscillations between 0 and 1 [12, 23] (Fig.3).

Accordingly, in the two-dimensional phase space (y vs x) an attractor can be observed, as is shown in figure 4 (left) for the start vectors (x0, y0) = (0.4, 0.3999). However, fora slightly different choice of start vector, (x0, y0) = (0.3999, 0.4), the system convergesto another attractor, as shown in figure 4 (right). These two attractors divide phasespace into two separate regions, their respective basins of attraction. These are separatedalong the line y = x. Start vectors (x0, y0) = (a, b) with a > b converge to the lowerleft attractor, whereas those with a < b converge to the attractor on the upper right.Start vectors (x0, y0) = (a, a) lying on the line y = x converge to the unstable fixedpoint (0.5, 0.5). This sensitivity to initial conditions (start vectors) which we notice in thebehavior of the system characterizes generally chaotic dynamical systems [3].


Figure 4: Attractors.

5.1 Bifurcation

To be able to analyze more precisely the behavior of the system, when a parametervaries, we have calculated the bifurcation diagrams, for different values of neuron gainand synaptic weights. The principle and the meaning of these diagrams are explainedhereafter [14].

Considering a discrete system of order n, xn(k + 1) = f(xn(k), α) with a parameterα ∈ R. When α varies, the limit sets of the system change equally. Typically, a weak mod-ification of α product a weak qualitative change of the limit set. For example to disturbα could change slightly the position of the limits set, and if this set is not a equilibriumpoint, its form or its size could modify equally. A such change is called bifurcation, andthe value of α for which this bifurcation appears is called bifurcation value. To study theappearance of bifurcation in a system, one draws then a bifurcation diagram. A bifur-cation diagram is a graph representing the position of system’s equilibrium points, whenthe parameter α varies.

The bifurcation diagrams which are present here, have been calculated by report to theneuron gain and to synaptic weights. For each σ (resp wij) we have taken 600 iterationsfor each point with initial vector (x0, y0) = (0, 3999, 0, 4) and we have obtained somegraphs represented in the figure 5.

5.2 Spectral analysis

An insufficiency of data can bring to consider that a temporal serie is chaotic, while ifone had a complete serie, this last it would not be inevitably chaotic. Consequently, aspectral analysis becomes an interesting and important aspect. This last is a tool allowingto analyze properties of frequencies in a system which varies in the time [13].

The spectral study consists to calculate and to draw Fourier transform of the autocor-relation function of x and y. The correspondent spectrums are represented in the figure6. One can observe easily that the spectrum is continuous and contains many frequentialcomponents. A disturbed spectrum like this is characteristic of a chaotic behavior.


5.3 Lyapunov exponents

The determination of the Lyapunov exponents [25] of the dynamical systems is not in-evitably easy. One knows that all dynamical system having at least a positive exponent isdefined as being chaotic and that the magnitude of this exponent reflects the scale of timeon which this system becomes unpredictable. One of the most efficient methods consiststo measure the average exponential rate of divergence or convergence of neighbor orbitsin a phase space.

For a one dimensional discrete dynamical system xk+1 = f(xk), with initial conditionx0, the Lyapunouv exponent λ(x0) is defined as:

λ(x0) = limN→∞

1

N

N∑k=1

lg |f ′(xk)| = limN→∞

1

Nlg(

N∏k=1

|f ′(xk)|) (16)

Of course, the limit in question may be not easy to calculate. In practise, it is oftenhelpful to use a ”finite time” approximation to get some idea of what the Lyapunouvexponent may be for a given orbit. Explicitly, for an orbit, the finite time analog of theLyapunov exponent λ(x0) is :

λ(x0, N) =1

N

N∑k=1

lg |f ′(xk)| (17)

In this sum the number λ depends on several parameters, notably the synaptic weights.Taking into account the matrix W , one can demonstrate easily that for a > 1 the Lya-punov exponent is strictly positive, it is equally possible to find the condition on b, butwe know well that an alone positive exponent allows us to deduce the chaotic behaviourin the system (see table I).

5.4 Fractal dimension

The instantaneous state of a dynamic system is characterized by a point in the phasespace. A sequence of these states subsequently to the time defines the path in the phasespace. If the dynamics of the studied system is reducible to a system of determinist laws,then the system arrives to a permanent state regime. This fact reflect by the convergenceof the totality of paths of the phase space. It is the attractor of the system.

Now, we can test the existence of an attractor and evaluate its dimensions [20]. Anattractor becomes strange if it has a fractal dimension and therefore, the dynamics systemrelative to this attractor is chaotic.

There exists several fractal dimensions, we have chosen two dimensions: the box count-ing dimension and the correlation dimension. This choice is founded on the possibleapplication to a dynamical systems and their implementation is relatively easy.

5.4.1 Box-counting dimension

One obtains it roughly by seeking the number N(ε), which represents the boxes numberof diameter ε which is necessary to cover the set to evaluate. This dimension Df is givenby the following equation:


Df = limε→0

ln( 1

N(ε))

ln(ε)(18)

Generally, the computation of this dimension is made by counting the number ofspheres not empty for each ε. To calculate the limit when ε → 0, we make an interpolationof a points cloud having as ordinates ln(N(ε)), corresponding to abscissas ln(ε). From itsslope, we calculate then the fractal dimension.

This algorithm of box-counting gives good results, but possesses nevertheless twonumerical disadvantages:

• The convergence is enough slow and it is necessary to possess a very great numberof points.

• To count the number of boxes necessitates a computation enough expensive in timeand in space.

5.4.2 Correlation dimension

To evaluate this dimension, we apply the Grassberger-Procaccia method [9]. It consiststo reconstruct the space phase and more particularly the points which the coordinates aredefined by:

Xn(ti) = [X(ti), X(ti + τ), ..., X(t + (n − 1)τ)]T (19)

A point of reference x(ti) is chosen and all the distances between this point and theother points are calculated according to the formula:

|X(ti) − X(tj)| (20)

This allows us to calculate the number of present points to a distance r from thechosen point in the phase space. If we repeat this for all the points, (for all values ofi) and we calculate the sum, we obtain the correlation integral which its formula is thefollowing:

C(r) =1

N2

N∑i,j=1

φ(r − |X(ti) − X(tj)|) (21)

Where φ is a function of Heaviside with φ(x) = 0 for x < 0 and φ(x) = 1 for x > 0.In a certain dimension of r, C(r) behaves as if:

C(r) = rd (22)

The attractor dimension d can be determined from the computation of the slope(ln C(r)/ ln(r)) [2, 9].

The results of the simulation of these algorithms (Lyapunov exponent, fractal dimen-sion) are recapitulated in the table I.


Figure 5: Fractal dimension from the graph Ln(C(r)) vs. Ln(r). (With embeddingdimensions m = 1 through m = 8).Equilibrium state.

6 Discussion of results

The different simulations have allowed us to obtain two Lyapunov exponents (See table I),the first is positive and the second is negative, asserting both the chaotic state of systemand the attractors divergence in the phase space (See figure 4), thus a fractal dimensionclose to the value 1, which corresponds to:

• The attractors having a close structure of line curved in the space phase.

• Quasi-periodic evolution in the time space (See figure 3 and 4).

From these verifications, thus of the qualitative and quantitative study undertakenabove, we can conclude that the system of neuron which its dynamic follows the equationsof the system 1 evolves chaotically.

7 Conclusion

In this work, one has studied dynamic properties of a small discrete neural network. Wehave determined the equilibrium states as well as their nature of stability by varying thenetwork parameters which are the synaptic weights wij and the neuron gain σ. In a secondtime we have seen the system bifurcation which has shown that the system can becomechaotic through the period-doubling route as the parameter varies. This has incited us tomeasure the chaos appearing in this system by computing the Lyapunov exponents andfractal dimension.

In this way one finds great perspectives such us, the application of the chaotic neu-ral network in the different areas (information processing, associative memories, patternrecognition and classification, combinatorial optimization, segmentation and bending ofobjects, etc.), without forgetting to perform experiments with larger networks.


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