S/\Ü Fen Bilimleri Enstitüsü Dergisi 9.Cilt. !.Sayı 2005 Tools For Detectıng Chaos- A.B. Ö?[

TOOLS FOR DETECTING CHAOS

•• A. Bed ri OZER*, E rhan AKIN*

Abstract - I n this study, useful tools for detecting chaos are explained. To observe the state variabtes (time series}, the phase portrait, the Poincare map, the power spectrum, the Lyapunov exponents and bifurcation diagram are used to detect chaos in dynamical systems. In this paper, the driven pendulum was choosen indicating chaos with the help of these tools. The existence of chaos in driven pendulum was shown with all methods. Simulation results obtained from all tools agree with each other. Also, control of chaos in driven pendulum was realized in this study

Key\vords - Chaos, Phase Portrait, Poincare Map, Lyapunov Exponent, Bifurcation Diagram

••

Ozet - Kaosu n gösterilmesi için yararlı araçlar bu çalışmada açıklanmıştır. Durum değişkenlerinin gözlenmesi (zaman serileri), faz portresi, Poincare haritası, güç spektrumu, Lyapunov üsteli ve çatallaşma diyagramı, dinamik sistemlerde kaosun gösterilmesi için kullanılmaktadır. Bu çalışmada, bu araçlar yardımıyla kaosun gösterilmesi için sürülen sarkaç örneği seçilmiştir. Sürülen sarkaç'da kaosun varlığı tüm yöntemlerle gösterilmiştir. Tüm yöntemlerle elde edilen benzetim sonuçları birbirleriyle uyuşmaktadır. Ayrıca, sürülen sarkaçtaki kaosun kontrolöde bu çalışmada gerçekleştirilmiştir.

Anahtar Kelimeler - Kaos, Faz Portreleri, Poincare Harita, Lyapunov Üsteli, Çatallaşma Diyagramı

* Fırat Cini'. Mlih. Fak. Bilgisayar Müh. Böl. ELAZIG.

60

1. INTRODUCTION

The irregular and unpredictable tinıe evo1ution of nı.ııı nonlinear systenıs has been called chaos. Chaos occur-:: many nonlinear systems. Main characteristic of chao� that system docs not repeat its past behavior.

In spite of their irregularity, chaotic dynanıical �.)�L�.:ıı follow detcn11inistic equations [1]. The uııiq characteristic or chaotic systems is dependence on th

initial conditions sensitively. Slightly diffcrcnt ııııLı.. conditions rcsult in very different orbits. But in nonchCln1• systenıs, these di fferences result in n early sanı c orbıts. 1 continuous tinıc systeıns, chaos may o cc ur i ıı �) ;:,Lcııı having at least three independent dynamical varicıhll'"· , , ı

system nıust be nonlinear. These are necessary conditicw for detecting chaos in continuous time systems. r 1' --:<u has sensitivity to initial conditions, then we say that sy�tc" is chaotic. After all of these, the exact defınition t) r clıJu� ı

that: A chaotic system is a deterministic systenı ı lı. exhibits random and unpredictable behavior. The definiıı feature of chaotic system is their sensitive dependcncc u

initial conditions [2].

The asynıptotic behavior of autonomous dynamic systenı uniquely specifıed by their initial conditions. The follov��in are four possiblc types of equilibrium behaviours:

- An equilibrium point, - A limit cycle, - A torus - Chaos.

There are various methods for detecting chaos. W e po ir out nıost uscful ones in this paper. These are:

-Time series (Observe the state variables), - Phase portraits, - Poincare maps, - Power spectrum, - Lyapunov exponents, - Bifurcation diagram.

:AC Fen Bilinıleri Enstitüsü Dergisi 9.Cilt, l.Sayı 2005

Therc are other nıethods for detecting chaos (Lyapunov dinıension, correlation di mension, entropy and ete.) but they are rarely used, because they are difficult for detecting chaos in prnetical systeıns.

Dri,·en pendulunı is a practical systenı for exploring nonlİnear dynanıics. First, Galileo noticed its characteristic. Since Galileo has discovered it, pcnduluın is appropriate �,·.,tenı for İıı\·cstigatc and tcach nonlİnear dynanıics. So, it . '-

is usc rul for detccting chaos. Figure. l shows a simple pendulunı .

Control ol' clıaotic systeıns is of current intercst and several techniqucs have becn proposed. One fundanıental property of chaos is the occurrence of dense orbits. When a chaotic systeııı i to be controlled, the periodic orbit is used such that the des i red trajectory of the controlled system is given .ı .ı period ı c orbit of the chaotic systeın [3].

In our researcl ı, tooL for detecting chaos are given and we lıa,-e applicd the Pyragas control nıethod to a chaotic rendul u nı.

Displacement Angle

' ' •

•

'

' ' ' '

: & �.,. . ' . . ' . . •

mass

length

Figure.l Simple Pendulunı

2. DRIVEN PENDU LUM

A driveıı danıped pendulum is deseribed by the equation: 1 d-8 d8 ? . . 1 , + b + ü) ö I s ı n e == T sı n w r t (ı )

dt- dt

\\here O İl.) position of pendulum, I is inertia torque, b is friction paranı eter, cu0 is resonance frequency, co f is

angular vclocity of drive, T is magnitude torque of drive. for casiııcss, w e need dimensionless equation [ 4]. The dinıensionlcss equation is:

d28 1 dO . -) +- +sın8==gcosco0t (2) dt'- q dt

•

\\here t == Cı) 0 t ,

61

Tools For Detectıng Chaos- A.B. ÖZER

We can split a second order differantial cquation to two fırst order equation for solving with conıputational n1ethod (Equation 3.a and 3.b). m is the angular velocity. Pendulunı is a non autononıous systenı becausc of external driving forcc. In chaotic analysis, an nth-order non autonomous systenı can always be converted into an (n+ 1 )th-order autononıous system by appending an extra state (Equation 3.c) [4]. The fınal equations are below:

dw CD . e

1-. - == ---sın +gcos<p dt q de

- == Ü) dt d� dt == ffio

(3.a)

(3.b)

(3.c)

For various g values, pendulunı cxhibits different behaviors. For exanıple at g=l, periodic behavior, g=l.4, period-2 behavior, and for g= 1.2, chaobc behavior [ 1].

3. TOOLS

a. Time Series (Trajectory Plot)

This nıethod is easiest one and it is a visual method. In this method, the state variabtes of the systeın are observed and if they exhibit irregular or unpredictablc behavior, then it is called chaotic. Otherwise (fixed point, periodic and quasi periodic) it is called nonchaotic. Figure.2.a, 2.b and 2.c show the tinıe series of periodic, period-2 and chaotic behaviors of pendulunı respectively.

b. Phase Portraits

Phase portarit is a two-dimensional projectian of the phase-space [2]. It represents each of the s ta te variable' s instantaneous state to each other. For pendulum example, angular velocity-position graph is a phase portrait. Chaotic and other motions can be distinguished visually from each other according to the Tab le. ı. A fıxed po int solution is a point in a phase portrait. A periodic solution is a closed curve in phase portrait. Chaotic solutions are distinct curves in phase portrait. Periodic and chaotic phase portrait examp1es are shown in Figure 3. Figure 3.a shows periodic behavior, Figurc 3.b shows chaotic behavior.

Tabi 1 S 1 t' e. o u ıons o Solution Fixed

Phase Point portrait

fd ıynamıc systems (Pt P ı as e

Periodic Quasi Period i c

Cl o sed To rus curve

ort ra ı t . ) C ha os

Distinct shapes

SAO Fen Bilinıleri Enstitüsü Dergisi 9.Cilt, l .Sayı 2005

c. Poincare Maps

Another nıethod is the Poincare nıap. The basic is that n'11-order continuous tinıe systcnı is replaced with (n-I )th_ order nıap. lt is con tructed by sanıpling the phase portrait stroboscopica1ly. lts ainı is to sinıp1ify the co1nplicated systenıs, and it is usefu 1 for st abi 1 i ty ana 1ysis [ 5]. Chaotic and other motions can be distinguished visually fron1 each other according to the Tab1e.2. Periodic behavior is a fıxed point in Poincare nıap. A quasi periodic behavior is closed curve or points in Poincare nıaps. Distinct set of points indicate the chaos in Poincarc nıap. In practice, one chooscs an (n-l) dinıensional Poincarc surface 2:, which divides R11 into two regions. If 2: is choosen properly, then the trajectory u nder observation \Yİl 1 repeatedly pass through 2:. The set or thcsc crossing points is a Poincare nıap. Figure 4 shovvs the basic idea of Poincare Map. Figure 5.a and 5.b show the Poincare nıap of periodic and chaotic bchavior, respectively.

Tahle.2 Solutionsor dynanıic systenıs (Poincare Maps)

Solution

Poincare Maps

2

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Tools For De tectıng Chaos- A.B. O.

Tınıe sf'rıes ol Chdotıc 8eha111or (9" 1 2)

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Figure. 2 Time scrıcs of a statc variable (angular vclocity) of pendul (n) periodic bcha\·ior (g-=1 ), (b) pcriod-2 hchavior (g=1.4), (c) chaı

behavior (g=-1.2).

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Figure 3. Phasc portraits of pendulunı (Angular velocity-position) period i c behavior (g= 1 ) (b) chaotic behavior (g= 1.2 ).

S.-\L Fen Bilinıleri Enstitüsü Dergisi 9.Cilt, !.Sayı 2005

7

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Figure. 4 Basic of Po ineare M ap

Poıncarc S�tion of Pcn()(Jıc Behrı1.1or (g= 1)

·ı - 1 - - - - - - 1

• - - - - - -ı- - - - -

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( a)

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1 Figure. 5 Poım:arc ıııaps or pendulunı (Angular velocity-Position) (a) pcrıodic hclıa\·ior (g 1 ) . (b) clıaotic belıavior (g=1.2).

d. Power Spectrum

Clıaotic signals are wideband signals, so they can be casily distinguislıed from peıiodic signals by looking at !heir frequency spcctra. Figure 6.a and 6.b show the power

63

.. Tools For Detectıng Chaos- A.B. OZER

spectrum of peıiodic behavior and chaotic behavior, respectively. If the behavior is chaotlc, thcn power spectra of system is expressed in tenııs of oscillations with a continuurn of frequencies.

e. Lyapunov Exponents

The Lyapunov exponents (li i= 1 ,2, .. n) are the nunıbers

that nıeasure the exponential attraction or separation in time of two adjacent orbits in the phase space with close in1tial conditions. n dinıensional system has n Lyapunov exponents. If original systenı is non autonomous as forced pendulunı, one Lyapunov exponent is zero. If the system has at least one positive Lyapunov exponent, it indicates the chaos. If the largest Lyapunov exponent is negative then the orbits converge in tinıe and systenı is insensitive to inibal conditions. If it is positive, then the distance betvveen adjacent orbits grows exponentially and systeııı exhibits sens1tive dependence on initial conditions, so it is chaotic we say. Main idea of Lyapunov exponent is below: Suppose x is a point at tinıe t, and consider a nearby point say x + 8. where li is a tiny separation veetar of initial length. ln nuınerical studies, one Lyapunov fınds as

b'= o0cA1. If ..i> O, neighboring trajcctories scparate

exponentially rapid. So positive ;ı indicates the sensitivity to initial conditions.

As su nı e an i n i ti al c on di tion .Yo i s c ho sen arbitraıi 1 y. The

Lyapunov exponents are

.il; = 1 i nı1�r:l) � 1 n In ı;( 1) ( 4) 1

i= 1,2, . .. , n and nı be the e1genvalues of Jacobian matrix of systenı [ 5].

Lyapunov exponents of penduluııı for periodic and chaotic behaviors are shown in Figure. 6.a and 6.b respcctlvely. The pendulunı is 3Td-order so, it has three exponents. One of thenı is zero because of equation 3 .c. as seen. In Figure 7.a system is peıiodic, so exponents are negative. But in Figure 7.b one exponent is positive, so it is chaotic [Matlab/LET Toolbox] [6] .

SAÜ Fen Bilinıleri Enstitüsü Dergisi 9.Cilt l.Sayı 2005

Power Spcclrum or Periodic Oetıavior (g=1)

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(n)

Tools For Detect1ng Chaos- A.B. ÖZ

Ly.;purıov Exponenıs of Penodıc 8etı:.:wıour (g= ı) ' ı ı ı • 1 •

1 f ı 1 1 ı ı 1 ı ı 1 1 ı • ' ı

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so o

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Figure. 6 Po" er spcctruııı of pcnduıum (1\nguıar velocity) (a) pcriodic be lım ı or (g= ı), (b) clıaotic behavior (g-1.2).

f. Bifurcation Diagram

Phase portraits. Poincare nıaps, tinıe series, po'A'er spectru1n and lyapunov exponcnts provide infornıation about the dynanıics of the pendulunı for specific values of the

raranıctcrs ( g , q , (V D ). The dynanıics 1nay also be

vie\ved nıore globally over a range of parameter values, thercby allowing sinıultaneous comparison of periodic and chaotic behavior. For some values of the parameters, a pendulunı will have only one long term motion, while for other slightly diffcrent choices; two or more motions may be possible. If several of them are stable the actual '

behRvior \vill be depending on initial conditions. In di ffcrential equations, if a change in the nuınber of solution is dcpcnding on paranıeter variation, it is called bifurcation [ 1 ] .

64

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ı• • • • ; • • • • r -... - • -.. , ... • • • -- , • - - .. -- r "" - -• - -,- - - - � - ., .. • --• - r - - - - � 1 1 ! 1 1 1 1 1 1 ı ' ı ı 1 1 ı 1 • 1 ı 1 ı 1 ı '

ı �--j__ l. o ı no ·rıo Joo 400 soo soo ?oo soo goo ı ooc

Tim e

. _ı 1 HS2 O ·O b 1 85�· ı 2 1'31 6 l

(b) Figure. 7 I )apuııov cxponcnts of pendulum for (a) periodic bcha\ı (g== ı), (h) chaotıc he ha' ı or (g= 1.2)

For the pcndulunı, bifurcations can be easily detected t exanıining a graph of cv versus the drive anıplitude g Several cxanıplcs of these graphs called bifurcatio diagranıs. A bifurcation diagram of pendulunı is shO\Yl1 ı

Figurc. 8 [7]. As shown in Figure.8, the interval [0,9 -l in horizontal axis (g), exhibits periodic behavior, inten·a [ı -1. I] and [ı .3 - 1 .4] exhibit period-2 behavior, an

intcrvals [1.13-1.3] and [1.7-1.8] exhibit clıaotı

behavior.

4. CONTROL O F CHAOS IN DRIVEN PENDU LULUM

Control of chaos is an important task for scientist. Ther are a few nıethods for chaos control: O GY ınethod, Pynıgn· nıcthods (tvvo nıethods), tanıing chaos nıethod and ete. [: Pyragas nıethods use continuous time feedback [8]. Tlı\ b lock diagranıs of these nıethods are shown in Fig.9.

.. S.\l' Fen Bilinıleri Enstitüsü Dergisi 9.Cilt, !.Sayı 2005

:-.. -:... ..

.... ... .. -

. :.

'::J. r "ı - . . _,.

. ' .J

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j

.

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/

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1 5

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1 6

F1gure. 8 Bifurcation diagram of penduiunı

•· '

.A.co�(W'� r •• 4

\,_

Chaotic y (t; .. � o ... • .... system 1

(a) Chaotic systerr ı ı ı Delay l

(b)

Y(l}

Figure. 9 B lock diagram of Pyragas nıethods.

1 T ı a

In thıs study, Pyragas control method (second one) is applied to chaotic penduhnn (g= I .2). Chaotic behaviour of peııdulunı is show n in Figure 1 O.a. A fter control, behaviour ofcoııtrolled pendulunı is shownin Figure lO.b

5. CONCLUSIONS

:\ non autononıous dynaınical system (dıiven pendulum) has been sinıulated by MATLAB. Driven pendulum has different behaviours, when g= 1, driven pendu1um exhibits periodic belıavior, and at g=l.2 it exhibits chaotic behavior. So, tools for detecting chaos are used in this system. All thcsc results of meth o ds are İn full agreement and confi mı the corrcctness of these nıethods. Also, control of chaos \\as ınıplenıented in this study.

65

Tools For Dctectıng Chaos- A.B. ÖZER

:· -·- Chaotic (g=1 2) 2.5 r---.---.-----,-----.-----...-------..

.

2 - - - - -r- -·-- _ _ _ _ _ _ _ _ _ _ _ _ _ LL_-------:- - - -- - -- -- : - -- _ _ _ _

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- -

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·

1 1 1 1 ı

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..... . .. -...... '---· .... .. .. . . .. L. .. . .. .. .... .. ....... .ı .. - - ·· - ---· ...... . .. .. ........ ... i. . .. .. .. ..... .. ... . ı ı 1 1 • ı 1 ı 1 1

··:: -2.5 ı__ _ ___..ı. ___ -..�.-__ __ı_ __ .......L_ __ ..L_ _ ___j o 50 100 150

tırna (a)

Controlled (g=1.2)

200 250 300

2.-----�--�----�-----�----�----�

1 -,Tı-ı) - T -n: -�- - - /ı----ı- - ,--: f- -- - : \ r yr- -o - IL - - J f � - - - _ _ _ _ _ _ i � ı- _ _

ı-1 �- :··· r · t ---·� -2 -r- t - ·- ! - -� : :

> _':! ı ' • ı � 4

- - - ı j , - -r -

- -.

- - - · · - - - r g> -4 • - - • : . . •

- - - ' f . - : . - J . . i � _ı

-. J

c;,( ı ı ı ı . ' . ı ' . . • • •

: ' 1 : : ' ı ı • • • ı 1 ' i

1 1 1 ' ı -7�--�-�----�----�----_J----�

o 50 100 150 200 250 300 tım e (b)

Figure. 1 O Behaviour of chaotic pendulunı (g= 1 .2) (a)- Be fo re control (b)­After contro 1

6. REFERENCES

[1] Baker, G. L., Gollub, J. P.,. Chaotlc Dynamics an Jntroduction, Cambridge University Press, 1990.

[2]Giannakopoulos, K., Dehyannis, T., Hadjidemetıiou, J.,. Means for Detecting Chaos and Hyperchaos İn Nonlinear Electronic Circuits,. DSP 2., 951 - 954, 2002.

[3] Castillo, 0., Melin, P.,. Soft Computing for Control ofNon-linear Dynamical Systems, Phys1ca-Verlag, 2001.

[4] Strogatz, S. H.,. Nonlinear Dynamics and Chaos: with Applications to Phys1cs, Biology, Chemistry, and Engineering, Cambıidge Press, 1994.

[5] Parker, T. S., Chua, L. 0.,. Practical Nunıeıical Algorithms for Chaotic Systenıs, Springer-Verlag, World Publlshing Corp. 1989.

SAÜ Fen Bilinıleri Enstitüsü Dergisi 9.Cilt, l .Sayı 2005

[6] Siu, S. W. K,. Lyapunov Exponent Toolbox, ftp ://11p.ınatlnvorks.conı/pu b/c on tr i b/v S/n1i sc/let, 1 998.

66

Tools For Detectıng Chaos- A.B. 01 '-

[7] Walter, J. M.,. The nonlİnear Pendulum Project:Pl 362 project, University Of Ogata, I 998.

[8] Pyragas, K., Continuous Control of Chaos by Se If­Control 1 ing Feedback, Physics Letter A, 1992