THE AN

Technical Note

AN INVESTIGATION OF CHAOS IN THE RL-DIODE CIRCUIT USING THE BDS TEST

R. KASAP

DepartmentofStatistics, Gazi University,Ankara, Turkey.

E. KURT

Departmentof^{PhysicsEducation, Gazi} University,Ankara, Turkey.

Abstract. Inthispaper, RL-diodecircuit drivenbya sinusoidalvoltageisemployedto obtain nonlinearexperimental data. TheBDSteststatistic isusedtoanalyse these data. Accordingto theresultsoftheanalysis for thefirstdifferenced orderdata,chaoticstructurehas beenfound for each evalues.

Keywords: Chaos,RL-diode,BDStest.

1. Introduction

Therecan be aperiodic and complex behaviors in many simple deterministic sys- tems, a pendulum, a fluid under convection or some chemical reactions

(Gleick,

1987

Keener

and

Tyson, 1986). ^In

1981, P.

A.

Linsay from the Massachusetts InstituteofTechnology carried outthe first rigorousstudyof the chaotic behavior ofanelectricalcircuit

(Smith, 1992).

Experimentallyanelectricalcircuitpossesses anumberofadvantagesoveran opticalor achemical system

(Newell,

^et

al.., 1996).

One

suchadvantage’^{isthat the}experimenterhas controlover many of the param- eters which influence the behavior.

In

the past two

decades,

many observations ofchaotic behaviour in electrical circuitshave been reported, the

Van

der Pol os- cillator, the RL-diode and Chua’s circuits can be mentioned as examples

(Hasler, 9S).

An

important reason for carrying out a research on nonlinear systems is that they can potentially explain the variations that seem to be random. Scheinkman and

LeBaron (1989),

^{and Hsieh}

(1991)

^usethe

^BDS

^statistic

(Brock, Dechert,

and Scheinkman,

1987)

^{to testfor}independenceinstock marketdata. Hsieh

(1989)

^uses

the

BDS

statistic to detect non-linearity and chaos ⁱⁿ monetary exchange rates.

The use of

BDS

statistic for testing other financial time series data in economics fornonlinear structureisnow afairly well establishedpractice.

In

this paper, time seriesdata obtained from adriven RL-diode circuit istested for chaotic structure using the

BDS.

^{This is the}first time that the

BDS

test has been used to identify chaotic structurein electricalstudies.

Section 2covers abrief explanation of the

BDS

test. Section 3 displays the results ofthe experimentontheelectricalcircuit. Section 4 summarizesthe results ofthe

BDS

test. Finally, themainfindings ofthisstudy ^arepresented in section5.

2. The BDS Test

The

BDS

approach tests the null hypothesis that the variable ofinterest is inde- pendently and identically distributed

(IID).

^This test is more powerful than the alternative of deterministic chaos or stochastic non-linear models

(Brock,

et al.,

1991). Now,

^{let usbrieflyconsider}the test statistics-BDS itself.

It

isbasedonthe so-called integral introducedby

Grasberger

and Procaccia

(1983).

Thetime seriesto be analysed

(Xt

^-t 1, 2,

T)

^{is usedto form} the so-called N-histories

Xt

^N

^(Xt Xt-t-l Xt+N-1)

Each N-history ^can be considered to be point in an N-dimensional space, where N is called the embedding dimension. These N-histories can be used to define a correlationintegral

CN(e)

2

TN(TN 1)t<s ^Z Ie(xtN’xsN)’

where

TN ^{T N} +

^{1 and}

I

^{is the indicatorfunction}of the event

[Xt+ X+ 1<

^{e, 0, 1,}

^...,N-

^1.

i.e.

r(xN, X ^N)

^{is unity if}

^lXg X I<

^{e and zerootherwise.} The correlation integral,

CN(e),

^canbe interpretedas an estimateof the probability that

X

^and

X

^{N arewithin a distance e. Given}this interpretation, we can see that under the independence hypothesis

CN(e)

^--+

C1 (e) N,

^as

T

--4co

holds. That is,

P(I Xt+i Xs+i [< e), (i

0, 1,

..., N 1)

is, dueto independence, equal to

1-I.N,. P(I Xt+i-Xs+i 1< ^e),

^{whichisestimatedby}

^{CI (e)}

^{yasthevariables}

are identically distributed

(Brock,

et al., 1991 and Chappell, et al.,

1996).

Thus, the

BDS

statisticreduces to

WN(e,) [r(CN(e) C1 (e,)N]/N(e,),

where

rN(e)

^iSanestimateofthe standard devisionunderthe null hypothesis. The distributionof

Wg(e)

convergestoastandard normal withexpectationzero anda varianceunity, as

T

approachesinfinity. Thus,one can now calculate the statistic that has astandard normal asymptotic distribution under the independence hy- pothesis. Ifthe absolute values ofthe test statistic are large, the null hypothesis of

IID (randomness)

^{is to be} rejected. The critical values reported by Brock, et

a/.(1991)

for significancelevels of0.05 and 0.01 are2.22and 3.40 respectively.

Table 1. The Pesults of theBDSTest

w

22.375

0.1 2 3 4 5 6 7 8 9 10 20 0.5 2

28.354 48.363 63.306 85.462 114.240 146.171 201.328 275.680 0.519

10.070

3 4 5 6 7 8 9 10 2O

9.628 9.538 9.334 9.030 8.906 8.774 8.325 8.051 5.340 0.9 2

3 4 5 6 7 8 9 10 20

33.548 44.931 49.905 54.972 60.556 65.792 70.334 76.659 114.840 3. Obtaining the Experimental Signal

The data were collectedfrom the circuit shown in Figure 1. This circuit hasonly one nonlinear element the silicon diffused rectifier diode.

We

can modelthe diode as a nonlinear capacitorin parallelwith anonlinear resistoras discussed by

Mat-

sumoto

(1987).

The behaviorofthe circuithas beenthoroughly studiedinseveral papersduringthelast two decades

(see

Linsay, 1981 and

Matsumoto,

et

al., 1984).

Therefore,^{we will}not gointo detailshere.

It

has been rigorously proven that if the

circuitparametersand the external drive (input

voltage)

are chosen appropriately, the system admits nonlinear oscillations.

GENERATOR,,,.

OUTPUT ^/----

GENERATOR GROUND

PROBE 1 0 UT) R

PROBE 2

(OUTPUT)

’L

;t_ ^{D scoP}

GROUND

Figure 1. The diagram ofcircuit: R 220 fl,L 2mH,D 1N4001

In

our experiment, we useda1N4001silicon diffused rectifierdiode, but anydiode with alarge capacitance can be used

(Smith, 1992).

^Thedriven sinusoidalvoltage was obtained from a function generator; the experimental results were obtained from an oscilloscopewhose probeswere attached asinFigure 1.

Thenature of the output signal depends onthe value ofthe input voltage.

We

have observed period doubling for (input

voltage)

Vin 160

mV

and for f 80 kHz. After anumber ofperioddoublings, a chaotic signal wasobservedfor Vin 120

mV

andf

=

^{6kHz as inFigure2.}

4. Results of the Chaotic Analysis

Chaotic analysis has been realised in two

forms;

analysis of the raw data and analysis of the first differenced series whichwasobtainedfromthe

(Xt Xt-1). ^In

thispaper,wewilldiscussonlythe results that

belong

tothe first differenced data seriesin detail.

Because,

autocorrelation canaffectsometests of chaos,so thatwe must remove it from data. This is typically done by taking the first difference of the data

(Hsieh, 1991).

The first differenced of the output data are plot ⁱⁿ Figure 3.

6 5 4 3 2-

20 40 60 80 100 120 140 160

Time (ms)

Figure2. Theoutput signal takenatVin-120mVandf--6 kHz

Table 1 gives thetheresults of the

BDS

test applied to the first differenced data.

Theembedding dimension,

N

is asvaried from2 to 10 and ewaschosen to be 0.1 ,0.5and 0.9for eachseries. Clearly, the

BDS

statisticslies withinthepositive tail of the standard normal distribution for thesedata.

Hence

wehave rejected the null hypothesis that the data are

IID. However,

for a very high embedding dimension like20 the null hypothesis can not be rejected. This result also supports that the data are chaotic.

5. Conclusion

We

haveinvestigatedchaos using the

BDS

teststatistic onanelectricalsignal taken from the RL-diodeseries connection.

In

anearlier study changing the embedding dimension

N,

from 2 to 10 and choosing^{eto be}

0.1,

0.5 or 0.9 indicated that the data manifested chaotic behavior for e being 0.1 and 0.9 but not for e being 0.5.

Results of this early analysis are not given ⁱⁿ this paper,

however,

they can be obtained from the authors.

6 5 4

2 0

-4 -5 -7

20 40 60 80 100 120 140 160

Time (ms)

Figure 3. The plot of the first differenced series.

According to theresults for thefirst differenced

data,

chaoticstructurehas been found foreach evalues. These resultsindicate that chaotic structure in electrical circuits canbe investigated byusing

BDS

test.

References

1. W.A. Brock.; W.D. Dechert andJ.A. Scheinkman. Atest for independence based on the correlationdimension,WorkingPaper,University of Wisconsin, 1987.

2. W.A. Brock D.A. Hsieh and B. LeBaron. Non-linear Dynamics, Chaos, and Instability.

Cambridge, Massachusetts: TheMITPress, USA,1991.

3. D. Chappell J. Padmore and C. Ellis.A note on the distribution ofBDSstatistics for a realexchange rateseries.Oxford Bulletin ofEconomics andStatistics,58,3, 561-566, 1996.

4. J.Gleick.Chaos: Making aNewScience. VikingPenguin, New York, 1987.

5. P. GrassbergerandI. Procaccia. Measuring the strangeness ofstrange attractors. Physica, 9D,189-208, 1983.

6. M.J. Hasler. Electrical circuits with chaotic behavior. Proceedings of the IEEE, 75, 1009- 1021,1987.

7. D.A.Hsieh. Testing fornonlineardependenceinforeignexchange rate,Journal of Business, 62, 3, 339-369, 1989.

8. D.A.Hsieh. Chaosand nonlineardynamics: anapplication to financialmarkets, Journalof Finance,46, 5, 1839-1877 ^1991.

9. J.P Keenerand J.J.Tyson. Spiralwaves inthe Belousov-Zhabotinskii reaction.PhysicaD, 21,307-324, 1986.

10. P.A.Linsay.Perioddoublingand chaotic behavior inadriven anharmonic oscillator. Phys.

Rev.Lett.,47, 1349-1352,1981.

11. T.Matsumoto.Chaosin electronic circuits.Proceeding of theIEEE,75, 1033-157, 1987.

12. T.Matsumoto L.O. Chuaand S.Tanaka. Simplest chaotic nonautonomous circuit. Phys.

l:tev.,A30, 1155-1158,1984.

13. T.C. Newell V. Kovanis A. Gavrielides and P. Bennett. Observation of the concurrent creation and annihilation of periodic orbits in a nonlinear RLC circuit. Phys. tev., E54, 3581-3590, 1996.

14. J.A.ScheinkmanandB.LeBaron.Nonlinear dynamics and stockreturns, Journal of Business, 62, 3, 311-337, ^1989.

15. D.Smith.Howtogeneratechaosathome. Scientific American,January, 121-123, 1992.

Special Issue on

Modeling Experimental Nonlinear Dynamics and Chaotic Scenarios

Call for Papers

Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system. Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision. In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.

Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from “Qualitative Theory of Differential Equations,”

allowing more precise analysis and synthesis, in order to produce new vital products and services. Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.

This proposed special edition of the Mathematical Prob- lems in Engineering aims to provide a picture of the impor- tance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.

Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophis- ticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.

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