TOOLS FOR DETECTING CHAOS

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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,

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: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 uscrul 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 researclı, 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 ı as e P Periodic Quasi Period i c Cl o sed To rus curve ort ra ı . t ) C ha os Distinct shapes

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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 1 5 -_c: ll> � -g 05 cı: -.> � _o V _o ai ;;>

- ·O

₅ t: :ı Ol � ·l ·1 5 o 5 -c: � o u nı cı: - >-::: -0 5 u E _'ll > B _, ::ı Ol c <{ ·1 5 ·2 o

Fixcd Pcriodic Quasi C ha os

Period i c

- Po int Cl o sed

Curve

Tım e series of PerıOdıc Beha1.4or (g= 1)

• • .. . ' t -. ı • •• . . ı ıoo 200 300 1100 500 600 Tım e (a)

Tım e serı es of Perıod-2 BehaiAor (g= ı 4)

. .

.

' 100 200 ·, 300 Tım e (b) ' •ı.- _{• •} 400 •' . . . 500 Distinct QOints . . . ., 700 ı ı -ı ·ı ı ı ı • i ' 1 • • 1 600 62 c "' -1 5 . � o 5 cı: -·1 ₅ı ·2 ·2 5 o

Tools For Detectıng Chaos- A.B. O.

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

" ' ı . . ı • 200 1 _{' ı} • .ı • •• ,. 1 • • ·-• ı ' _ı ı 1 400 .

.

• 600 Tıml' (c) .ı . . ' ı . • soo j ' 1000

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). � c 1/) -'O ro 0:: - > -u o Q) > .... ro :ı Ol c <( 2.5 2 1.5 1 ' os o. -0.5 ·1 ı -1.5 1 1 - 2' ·2.5 ·3 2 5 2 1 5 :, - ₁_• c 1/) - ' •• 'O ₍ <1) ₀₅ a: ı - i � f ' u o. ! o _\_{ Qi ı > 1 • .... -0 5 \. � _.\ :ı _. ·2 • . .

Phase Porıraıt of Periodic Behavior (g= 1)

• 1 ' o Pos1lion (a) 1 2

Phase Porıraıı of Chaotıc _B_eha_vıo_{r (g=}₁ ₂₎

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Figure 3. Phasc portraits of pendulunı (Angular velocity-position)

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S.-\L Fen Bilinıleri Enstitüsü Dergisi 9.Cilt, !.Sayı 2005 7 •' .. rr ' :;ı <l 1 . . 1 . . ' ' '• _" ' ·� _' _' .: + / . / i> � / . . Poıııcıır c Suı facc / / / ı / / . . . / , , , 1 ' 1 1 • ' . ' . ' •

Figure. 4 Basic of _Poineare M ap

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

·ı - - - 1 1 • - - - -ı- - -1 ₁ 1

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flııırır',ıı• ı.;,,,·ıııııı ol Ch.ıotıc 8F-havıor (y= ı 2)

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(b)

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

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

Power Spcclrum or Periodic Oetıavior (g=1)

t 1 45 ' : ' 40 >-1.) o _.J5 (ii > -"' JO O> c < - _LS o E J � ₂₀ u ... c.. J ı'· Cil � o 10 Q_ s . -'2 4 6 8 ı o 12 ı 4 16 18 20 F requeııcy (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 o.a ---:

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Power Spec.trunı ol Chaotıc 8eha\10r (g= _{1 21} lırne

( J i� 705 1 • :--•• , ı o ı ı!ıo (a) -V .9 � 200 � � ::> o c � _o 150 E 2 V � 100 (/) so o ı • •• • o 1 o 1s :ı o ıs :ı o 35 40 4 s Frequency (b) ..

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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o [ı - ... 1 ... ,, .. ·- 1 -... -• .,.. .� •. >"'·"' .!_.._----� .... ---�,.e.� --.. -- ·'- -... - .. - ! . -.. ... -.. _.__.. •c.:..; · -� -• 1 ı ı ı r---.... --._,.--- -�-..-� ·08 ı ' • _ı ' . 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 OF 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\

(6)

..

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

:-.. -:... _.. .... ... .. -. .

:

'::J. r _"ı - . . _,. . ' _.J - _..J j ... . , • ı ı / t

�

1 3 . 4 g ' ., . , ) � ; 1. · . . .. . �: 1 • . !:{ $ ' .. ı . • · .. �\! l. r '' ;"e' � .. '' , 1 1 .,... ı ' " .!-··'' ... .. ' .;. . � 1 i\; ;' ' _._,. )i. 1 1 � ,l... • . .f � . 1 5 .· 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

- -·-- _ _ _ _ _ _ _ _ _ _ _ _ _

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-

----

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j

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l

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ı

-

r

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-

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.

--

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.

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-

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--

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--

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l-t

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-

-ı

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�

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\

ı

:

/- ----

:

1

- - ---

i

f

-.

{

L

_ · 1 1 1 1 ı ·2 ... . .. -... '---· .... .. .. . . .. 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

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IL

-

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f

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--

- _ _ _ _ _ _

i

�

ı-

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ı-1

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_:···

_r

·

t

-

--·�

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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.

(7)

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, ₁998.

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 If

TOOLS FOR DETECTING CHAOS