Delay-Differential Equation Versus 1D-Map

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E L S E V I E R P.hysica D 104 (1997) 127-14 7

PH YS I C A

D e l a y - d i f fe r e n t i a l e q u a t i o n v e r s u s 1 D - m a p :

A p p l i c a t i o n t o c h a o s c o n t r o l *

P . C e l k a 1

Department of Electrical Engineering, Signal Processing Laboratory, Swiss Federal Institute of Technology,

1015 Lausanne, Switzerland

Received 10 June 1996; accepted 18 November 1996

Comm unicated by P. Kolodner

A b s t r a c t

Theore t i ca l and expe r imental r e su l ts on t he co n t ro l o f chao t i c m ot ion i n an e l ec t ron i c c i rcu i t mode led by a de l ay-d if fe ren ti a l

equa t ion a re p resent ed . The mod e l i s de rived f rom a non l inea r op t i cal sys t em w i th t ime-de l ay feedback . O ur con t ro l me thod

i s based on Pyragas appro ach and app l i ed t o t he i n fin i te -d imens iona l sys t em. Gene ra l i zed m ean Floque t mul ti pl ie rs and m ean

Lyap unov exponen t s r e l at ed t o de l ay-d i f fe ren t ia l equa t ions a re i n t roduced in o rde r t o com pare a d i sc re t e -t ime app rox ima t ion

wi th t he con t inuous- t ime equa t ion . Th i s co n t ro l me thod i s a l so used to f ind t he so -ca l led i somers re la t ed t o ha rmonic pe r iod i c

so lu t ions p red i c t ed b y Ikeda and Mat sum oto .

K ~ w o r d s : Chao s; Delay-differential equations; C ontrol; Lyapuno v expon ents; Floquet multipliers

1 . I n t r o d u c t i o n

I t h a s b e e n r e p o r t e d t h a t n o n l i n e a r t i m e - d e l a y e d s y s t e m s c a n p r o d u c e a w i d e v a r i e ty o f b e h a v i o r s : s t a b le e q u i l ib -

r i u m p o i n t s , p e r i o d i c s o l u t i o n s [ 1 ,2 ] a s w e l l a s c h a o t i c s o l u t i o n s [ 3 - 1 2 ] . M a n y e x p e r i m e n t a l a n a l y s e s o f c o m p l e x

d y n a m i c a l r e g i m e s h a v e b e e n c o n d u c t e d a n d e s p e c i a l l y i n t h e c o n t e x t o f i n t e r f e r o m e t e r - b a s e d o p t i c a l s y s t e m s

[ 9 , 1 3 - 1 5 ] . T h e f u n d a m e n t a l p e r i o d i c s o l u t i o n w h i c h a p p e a r s a f t e r a H o p f b i fu r c a t i o n h a s a p e r i o d o f a b o u t T o

2 ( r + T ) , w h e r e r i s t h e d e l a y i n t r o d u c e d i n th e f e e d b a c k p a t h a n d T i s t h e ti m e r e s p o n s e o f t h e d e v ic e . W h i l e

t h e m o d e l i s i n f in i t e - d im e n s i o n a l a n d e v o l v e s o n a n i n f i n it e - d i m e n s ! o n a l m a n i f o l d , t h e a t t r a c to r d i m e n s i o n i s fi n it e

d u e t o t h e c o n t r a c t io n e f f e c t o f th e n o n l i n e a r m a p p i n g [ 3 , 1 6 ]. I k e d a a n d M a t s u m o t o [ 8 , 1 7 ] h a v e g iv e n a n e s ti -

m a t e o f th e a t tr a c t o r L y a p u n o v d i m e n s i o n D E f o r t h e I k e d a m a p , a n d i t r a n g e s a p p r o x i m a t e l y f r o m 2 t o 1 3 w h e n

s o m e b i f u r c a t i o n p a r a m e t e r i s v a r i e d . L e p r i e t a l . [ 5 ] h a v e c o m p u t e d L y a p u n o v s p e c t r a a n d D L f o r B e r n o u l l i a n d

l o g i s ti c m a p - b a s e d d e l a y - d i f f er e n t i a l e q u a t i o n s ( D D E s ) i n t h e c a s e o f a la r g e d e la y . B o t h I k e d a a n d L e p r i f o u n d

* This w ork w as partially done at the Chaire des C ircuits et Sy st~mes at the Swiss Federal Institute of Technology.

J Corresp onding autho r. Tel.: 4-41 21 693 26 05; fa x: 4-41 21 693 76 00 ; e-mail: [email protected].

0167-2789/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved

P II S 0 1 6 7 - 2 7 8 9 ( 9 6 ) 0 0 2 9 9 - 0



128 P. Celka/Physica D 104 (1997) 127--147

h i g h - d i me n s i o n a l a t t ra c t o r s f o r l a r g e d e l a y r a n d v a r i a t io n w i t h a g a i n f a c t o r / z . 2 A s s u m i n g t h e K a p l a n - Y o r k e

c o n j e c t u r e i s v a l id [ 8 , 1 8] , w e c a n s a y t h a t th e H a u s d o r f f d ime n s i o n o f t he a t t ra c t o r s e t i s g r e a t e r o r e q u a l t o D E a n d

t h u s t h e n u mb e r o f d e g r e e s o f f r e e d o m o f t h e s y s t e m m a y b e h i g h . I t is th e r e a s o n w h y d e l a y - d i f f e r e n t i a l s y s t e ms

a r e r e f e r r e d t o a s h i g h - d i me n s i o n a l a n d n o t i n f i n i te - d i me n s i o n a l . I n Se c t i o n 3 , Ch a r a c t e r i s ti c L y a p u n o v s p e c t r a a n d

d i me n s i o n s r e l a t e d t o o u r s y s t e m a r e c o mp u t e d .

I t h a s b e e n me n t i o n e d t h a t s e v e r a l c o - e x i s ti n g a t t r a c to r s , b e l o n g i n g t o d i f f e r e n t b i f u r c a t i o n b r a n c h e s , c a n b e f o u n din these sy s t em s [7 ,8 ,14]. T he fund am enta l so lu t ion evo lves th rough a per iod do ub l ing To ~ 2T0 • - • - -+ 2 N To , as

o n e p a r a m e t e r i s v a r ie d , a n d w e w i l l c a ll t h is t h e f u n d a m e n t a l b r a nc h . T h e s e s o l u t i o n s a r e c a l l e d 2N r - - p e r i o d i c a n d

t h e c a s c a d e a c c u m u l a t e s a t th e Fe i g e n b a u m p o i n t . I k e d a a n d Ma t s u m o t o [ 8] h a v e d i s c o v e r e d t h e e x is t e n c e o f o t h e r

c o - e x i s t i n g p e r i o d i c s o l u t io n s a n d c a l l e d t h e m h a r m o n i c s a n d i s o m e r s o f t h e s e h a r mo n i c s . Se v e r a l t y p e o f i so me r s

e x i s t a n d t h e y a r e a l l d i ff e r e n t in s h a p e . E a c h b r a n c h f o l l o w s a p e r i o d - d o u b l i n g c a s c a d e . T h e s e s o l u ti o n s a r e mo s t l y

p e r i o d i c b e l o w t h e Fe i g e n b a u m p o i n t , b u t w h e n t h e b i f u r c a t i o n p a r a me t e r i s v a r i e d t o w a r d s t h e f u l l y d e v e l o p e d

c h a o t i c r e g i o n , e a c h b r a n c h e v o l v e s i n d e p e n d e n t l y w i t h i n a g i v e n p a r a me t e r r e g io n , a n d g i v e s r is e t o d i f f e re n t s t a b le

o r u n s t a b l e p e r i o d i c o r c h a o t i c s o l u t io n s . A me r g i n g p r o c e s s o c c u r s f o r h i g h e r v a l u e s o f t h e p a r a me t e r , y i e l d in g a

f ina l merger and a un ique chao t i c a t t rac to r [7 ,8 ,13 ,14] .

Fo r s e v e r a l y e a r s , r e s e a r c h e r s h a v e p r o p o s e d n u m e r o u s c o n t r o l 3 t e c h n i q u e s [ 1 9 - 2 6 ] a n d a p p l i e d t h e m i n d i f f e r e n tc o n t e x t s: e l e c t r o n i c c ir c u i ts [ 2 7 ,2 8 ] , l a s e r - b a s e d ( N M R o r CO 2 l a s e r s ) [ 2 9 - 3 3 ] o r c h e mi c a l s y s t e ms [ 3 4 ]. T h e c o n t r o l

c a n b e p e r f o r m e d o f f l in e w i t h t h e h e l p o f c o mp u t e r s , b u t it r e q u ir e s l o t s o f c o mp u t a t i o n a n d o f t e n a p r i o r i k n o w l e d g e

o f t h e s y s t e m mo d e l ( o r o f a d u p l i c a t e o f t h e s y s t e m ) [ 1 9 , 21 , 2 5, 2 6] . O n t h e o t h e r h a n d , t h e c o n t r o l c a n b e c o n s i d e r e d

u s e f u l i f i t c a n b e p e r f o r me d o n l i n e , w h i l e c o n s i d e r i n g t h e s y s t e m mo r e o r l e s s a s a b l a c k b o x . T h e c o n t i n u o u s

c o n t r o l me t h o d w h i c h w e h a v e i n v e s t i g a t e d w a s f i r s t p r o p o s e d b y Py r a g a s [ 2 3 ] a n d h a s b e e n a p p l i e d l a t e r i n

d i f f e r e n t c o n t e x t s [ 2 8 , 3 4 - 3 8 ] . T h i s c o n t r o l me t h o d , b a s e d o n t h e e r r o r f e e d b a c k c o n tr o l s c h e m e u s i n g d e l a y e d

o u t p u t s i g n a l s , w a s e mp l o y e d t o s t a b i l i z e u n s t a b l e p e r i o d i c o r b i t s ( U PO s ) e mb e d d e d i n t h e c h a o t i c a t t r a c t o r t o

b e c o n t r o l l e d [ 2 3 , 3 9 , 4 0 ] . W h i l e c h a o s c o n t r o l o f l o w - d i me n s i o n a l s y s t e ms h a s b e e n d e v e l o p e d f o r ma n y y e a r s ,

c o n t r o l l in g h i g h - d i me n s i o n a l c h a o t i c s y s t e ms i s a r a t h e r n e w f i el d o f r e s e a r c h [ 3 7, 3 9- - 42 ] .

W e d e v e l o p t h e s a m e c o n t r o l t e c h n i q u e a s Py r a g a s b u t f o r a r a t h e r d i f f e r e n t o b j e c ti v e a n d i n t h e c a s e o f a n o n l i n e a r

M a c h - Z e h n d e r o p t i c a l t i me - d e l a y e d f e e d b a c k s y s t e m [ 1 5,3 6] . W i t h in t h i s i n f in i t e - d ime n s i o n a l c o n t e x t , th e me a n i n g

o f c h a o s c o n t r o l h a s t o b e r e v i s it e d . I n f a c t , a s s e v e r a l k i n d s o f a t t ra c t o r s c o - e x i s t f o r a g i v e n s e t o f p a r a m e t e r v a l u e s ,

t h e c o n t r o l p r o c e s s ma y b e u n d e r s t o o d a s f o ll o w s :

- Co n t r o l t h e s y t e m t o w a r d s o n e o f t h e s t a b le f u n d a m e n t a l p e r i o d i c s o l u t i o n s o r e q u i li b r iu m p o i n t s o f th e u n c o n -

t r o l le d s y s t e m.

- Co n t r o l t h e s y s t e m t o w a r d s o n e o f th e s t a b l e p e r i o d i c b r a n c h e s , e i th e r a h a r mo n i c o n e o r a n i s o me r o n e .

- Co n t r o l t h e s y s t e m t o w a r d s o n e o f t h e u n s t a b l e p e r i o d i c o r b i t s e mb e d d e d i n th e c h a o t i c a t t ra c t o r.

W e wi l l focus on the f i r s t two po in t s a s the th i rd one h as a l read y been the sub jec t o f severa l s tud ies [23 ,28 ,39 ,43 ,44] .

I t h a s b e e n s h o w n t h a t i n s o me s i t u a ti o n s a d i s c r e t e - t ime a p p r o x i m a t i o n o f th e D D E m o d e l q u i t e a c c u r a t e l y mo d e l s

e q u i l i b r i u m p o i n t s a n d p e r i o d i c s o l u t i o n s [ 8 , 1 5 , 3 6 ] . W e w i l l e x t e n d t h i s a p p r o a c h a n d d e f i n e g e n e r a l i z e d me a n

F l o q u e t mu l t i p l i e r s ( MFMs ) a n d me a n L y a p u n o v E x p o n e n t s ( ML E s ) r e l a t e d t o t h e D D E s , a n d s h o w h o w t h e s en e w c o n c e p t s h e l p u s t o d e s i g n a c o n t r o l s c h e me . W e w i l l mo v e b e t w e e n c o n t i n u o u s a n d d i s c r e t e - t i me c o n c e p t s

t h r o u g h o u t t h i s p a p e r , s o, i n o r d e r to d i s ti n g u i s h b e t w e e n t h e t w o c a s e s , w e u s e r s u b s c r i p ts f o r c o n t i n u o u s - t i me

q u a n t i ti e s w h e n n e c e s s a r y .

Se c t i o n 2 i s d e v o t e d t o a g e n e r a l d e s c r i p t i o n o f t h e s y s t e m s t a te e q u a t i o n a n d i t s e x p e r i me n t a l r e a l i z a ti o n , Se c t i o n 3

i n t r o d u c e s a d e e p e r u n d e r s t a n d i n g o f t h e r e l a ti o n s h i p b e t w e e n t h e c o n t i n u o u s - a n d d i s c r e t e - t i me e q u a t io n s . S e c t i o n 4

2 Ikeda et al. ha ve founded the empirical expression D L ~ ( 1 . 1 -t- 0.05) r/z.

3 We w ill use the word con trol and stabilization indifferently throughout this paper.



P Celka/Physica D 104 (1997) 127-147 129

p r e s e n t s th e o r e t i c a l r es u l t s o n th e c o n t r o l me t h o d r e l a t e d t o t he p r e v i o u s l y d e f in e d MF M a n d ML E , a n d e x p e r i me n t a l

r e s u l t s o n c o n t r o l l i n g a n i n f i n i t e - d i me n s i o n a l s y s t e m i mp l e me n t e d w i t h a n e l e c t r o n i c c i r c u i t . W e e mp h a s i z e t h i s

e x p e r i me n t a l a p p r o a c h b e c a u s e n u me r i c a l s i mu l a t i o n s o n c h a o s c o n t r o l ma y n o t b e r e l i a b l e e n o u g h , t h e mo r e

s o i n t h e c a s e o f a n i n f i n i t e - d i me n s i o n a l s y s t e m w h e r e n u me r i c a l s i mu l a t i o n r e d u c e s i t t o a f i n i t e - d i me n s i o n a l

s i mu l a t i o n . T h e d y n a mi c a l s t a t e e q u a t i o n o f t h e e l e c t r o n i c c i r c u i t i s v e r y s i m i l a r t o t h e o p t i c a l s y s t e m w e h a v e

a l ready s tud ied ] 15 ,36] bu t the var ious para me ters a re eas i ly acces s ib le in th i s e l ec t ron ic m odel , m ak in g exper im en t smore f l ex ib le .

2 . T h e d y n a m i c a l s y s t e m

The n on l inear e l em ent in an op t i ca l r ing cav i ty has a s inuso ida l t rans fe r charac te r i s t i c f (x ) = ½ 1 - s in (x - cp0)) .

I n t h e c a s e o f t h e M a c h - Z e h n d e r i n t e r f e r o me t e r [ 15 ], t h e s t a te v a r i a b l e x i s e i t h e r t he i n t e n s i t y o f t he e l e c t r o m a g n e t i c

w a v e v e c t o r o r e q u i v a l e n tl y t h e p h a s e i n d u c e d b y a n e l e c t r o - o p t i c a l e f f e c t v i a t he e l e c t r o d e o f t h e d e v i c e , a s t h ey

a r e b o t h r e l a t e d b y a l i n e a r t r a n s f o r m. T h e d y n a mi c a l s y s t e m i s u s u a l l y d e s c r i b e d b y t h e n o n l i n e a r ma p f ( x ) , a n

n t h - o r d e r l i n e a r l o w - p a s s f il te r mo d e l e d b y t h e t r a n s f e r f u n c t i o n H ( p ) i n th e L a p l a c e d o ma i n , a n d a s e t o f b if u r c a t i o n

p a r a m e t e r s {m, ~P0, r , - . . }

1H ( p ) - - l )

1 + P n ( P )

with Pn (P) = Y~'~= a k p k an d ak = otk T k . In op t i ca l sys t em s , the time cons tan t T o f the low-pass f i l t e r i s de te rm ined

by the sma l l es t bandwid th va lue o f a ll the dev ic es in the sys t e m and the o rde r n i s usua l ly t aken to be equa l to

one . W i th in th i s op t i ca l con tex t , i t m ay b e eas i e r t o use the pha se qg0 par am eter as a b i fu rca t ion fa c to r and ke ep

t h e g a in m f ix e d , w h i c h i s p r o d u c e d e i t h e r b y a n e l e c t r o n i c c i rc u i t ( i n tr o d u c i n g a s u p p l e m e n t a r y l o w - p a s s e l e me n t )

o r b y i n c r e a s i n g t h e l a s e r p o w e r ( i n t ro d u c i n g m o r e c o mp l e x s o l u t i o n s a s t h e l a s e r o r t h e n o n l i n e a r o p t i c a l e l e me n t

c h a n g e s t h e i r f u n d a me n t a l c h a r a c t e r i s t ic w i t h p o w e r ) . I n t h e c a s e o f a n e l e c t r o n ic c i r c u it , t h e t i me c o n s t a n t T i s

p ropor t iona l to the R C charac te r i s t i c and the parameter m i s an eas i ly e l ec t ron ica l ly ad jus t ab le ga in fac to r . Thus ,

th i s l as t so lu t ion i s p referab le to the op t i ca l one i f we a re on ly in t e res t ed in exp lo r ing exper imen ta l ly the behav io r

o f a D D E . L e t x b e t h e s ta t e v a r i a b le o f th e s y s t e m a n d X i ts L a p l a c e t r a n s f o r m; t h e d y n a mi c a l e q u a t i o n i s g i v e n b y

X ( p t = m H ( p ) Y ( p ) e - r p , (2 )

w h e r e Y ( p ) = £ [ y ( t ) ] i s t h e L a p l a c e t ra n s f o r m o f th e o u tp u t v a r i a b le y ( t ) = f ( x ( t ) ) o f t h e n o n l i n e a r e l e me n t f .

The non l inear func t ion i s usua l ly a un imodal map [45] such as the Ikeda , l og i s t i c o r t en t map . We have chosen to

s y n t h e t i z e th e l i n e a r p a r t o f th e s y s t e m a s a s e c o n d - o r d e r ( n = 2 ) l o w - p a s s f i l te r a n d t o i mp l e m e n t a t e n t ma p a s

t h e n o n l i n e a r e l e me n t . T h e s t a te e q u a t i o n o f t h e s y s t e m i s t h u s g i v e n b y

T 2 d 2 x ( t ) d x ( t )c e 2 - - d ~ + T oil -dt + x ( t ) = m f ( x ( t - r ) ) ( 3 )

and Fig . 1 show s a schema t i c c i rcu i t rea l i za t ion o f (3 ) toge the r wi th the non l inear su bsys tem wh ich i s supposed to

i mp l e me n t t h e n o n l i n e a r ma p .

T h e p a r a m e t e r s o f t h e c i r c u it a r e r e la t e d t o t h o s e o f th e n o r m a l i z e d e q u a t i o n ( 3 ) b y : T 2 ~2 = L C a n d T o q = R C .

The va lue o f or2 wi l l be f ixed to un i ty th rough ou t the paper . Th e t im e cons tan t T i s con t ro l l ab le by the va lues o f

t h e i n d u c t a n c e L , c a p a c i t a n c e C a n d l i n e a r p o s it i v e r e s i s t a n c e R . T h e d e l a y r i s o b t a i n e d w i t h a n a n a l o g d e l a y l i n e

RD I 0 7 , w h i c h c a n p r o d u c e d e l a y s w i t h in t h e r a n g e 1 ms t o mo r e t h a n 2 s , a n d r = 1 02 4/ Vc w h e r e V c is th e c l o c k



130 P C e l k a / P h y s i c a D 1 0 4 ( 1 9 9 7 ) 1 2 7 - 1 4 7

L R

~ + + 0

m f s ( V ( t - x ) ) V ( t ) v

_ I _ - - L _

a , b ,

R1 T

R2Re

Re~ +

- - R , f s ( V )

2RI "

F i g . 1 . ( a ) C i r c u i t r e a l i z a t io n o f ( 3 ) w i t h a v o l t a g e c o n t r o l l e d v o l t a g e s o u r c e . ( b ) T h e n o n l i n e a r s u b s y s t e m w i t h Re = 1O0 k~2.

T a b l e 1

P a r a m e t e r v a l u e s

R (92 ) L (m H) C (nF ) T ( l~ s ) r (ms ) a 2 c t 1

303 18 330 76.8 1 .28 1 1 .3

f r e q u e n c y o f th e RD 1 0 7 c i r cu i t. T h e a c t u a l v a l u e s f o r t h e c i rc u i t a n d n o r ma l i z e d p a r a me t e r o f ( 3 ) a re s u m ma r i z e d

in Tab le 1 .

T h e a c t u a l v a l u e s o f r a n d T g i v e u s r / T ~ . 1 6. 6. T h e b i f u r c a t i o n p a r a me t e r m i s f i x e d b y a n a d j u s ta b l e o p e r a t io n a l

a mp l i f i e r ( L M7 4 1 ) b a s e d g a i n c i r c u i t f o r w h i c h m = 1 + Rl/ R2 . This s imp le c i rcu i t is no t shown in F ig . 1 . Du e to

t h e l o w - p a s s e f f e c t o f t h e o p e r a t i o n a l a mp l i f ie r s , th e h a t o f th e t e n t ma p f ( x ) - - 1 - 0 .5 1 x l, w h i c h w e w i l l r e f e r to

as the standard tent map, i s s mo o t h e d . A c t u a l ly , t h e h a r d w a r e i m p l e m e n t a t i o n o f t he t e n t ma p c a n b e mo d e l e d b y

f s ( V ) =

VDc -- f l (3 j v j ° )( V / 8 )

~ A +

VDC + c ~ e - - I V l ) + l V l

i f V > 0 ,

i f V < 0 ,

(4 )

w h e r e A + a n d A - a r e t h e a b s o l u te v a l u e s o f t h e g a i n f a c t o rs o f th e o p e r a t i o n a l a m p l i f ie r ( L M7 4 1 ) g a i n c ir c u it s :

A + ---- R + / R ~ a n d A - = R [ / R 2 . T h e s e g a i n f a c t o r s a r e n o t a d ju s t a b l e a n d A + ~ A - ~ 0 . 5 . T h e D C b i a s V DC i s

se t t o 1 .25 V and i s no t shown in F ig . 1 fo r purpo se o f c la r i ty . The o r, f l , 6 and v fac to rs t ake in to acco un t the fac t t ha t

the c i rcu i t rea l i za t ion in t roduce a smoo th ing o f the ha t resu l t ing in f s i ns t ead o f f as the non l inear func t ion . A SPI CE

s i mu l a t i o n o f t h e n o n l i n e a r i ty s u b s y s t e m h a s e n a b l e d u s t o d e t e r mi n e ot = 0 . 7 8 3 9 9 , / ~ = 0 . 5 6 8 7 6 , 8 = 1 . 39 7 9and v = 1 .5005 wi th a l i near reg ress ion coef f i c i en t o f 0 .99994 . The sh ape o f the ma p (4 ) i s qu i t e d i f fe ren t f ro m

t h e s t a n d a r d t e n t ma p , a n d F i g . 2 s h o w s t h a t o u r mo d e l ( 4 ) i s i n a g r e e me n t w i t h o u r me a s u r e me n t s a n d SPI CE

s imula t ion on the non l inear i ty .

F i g . 2 ( b ) w a s r e a l i z e d b y i n j e c t i n g a t r i a n g u l a r w a v e v o l t a g e o f a mp l i t u d e 3 V a t t he i n p u t o f t h e n o n l i n e a r e l e me n t

a n d m e a s u r i n g t h e v o l t a g e a t t h e o u t p u t o f th i s n o n l i n e a r b l o c k . T h e f r e q u e n c y o f t h e i n p u t s ig n a l w a s a b o u t 3 k H z .

W e h a v e o b s e r v e d a h y s t e r e s i s - l i k e p h e n o me n a f o r h i g h e r f r e q u e n c i e s w h i c h i s d u e t o p h a s e d e l a y b e t w e e n i n p u t

a n d o u t p u t s i g n a ls . A s w e w i l l de a l w i t h f r e q u e n c i e s u p t o a b o u t 6 / T o = 2 . 3 4 k H z w e c a n c o n s i d e r t h a t t h e s ta t ic

non l inea r i ty m ode l (4 ) i s s ti l l va l id .



P. Ce lka /Phys ica D 104 (1997) 127-147 131

-0 .1 ~ - a .

- 0 . 2

~.~ -0.3

~ -0.4

~ -0.5

' ~ - 0 . 6

-0.7

- O . 8 I I ~ . ~ 1 _ ~ ~ _ ~ L I

- 2 - 1 . 5 - 1 - 0 . 5 0 0 . 5 1 1 . 5 2

[ - - r I I I I | l • l T I 1

v [Vo#] b.

F i g . 2 . ( a ) T h e s t a n d a r d t e n t m a p f ( c o n t i n u o u s l i n e w i t h ( ¢ ,) ) a n d t h e s m o o t h t e n t m a p f s ( d a s h e d l i n e ) t o g e t h e r w i t h S P I C E s i m u l a t i o n

( o ) f o r V o c = 0 ( b ) T h e e x p e r i m e n t a l m e a s u r e m e n t s o n th e n o n l i n e a r f u n c t i o n f s ( V ) : h o r i z o n ta l a x i s V ( I V / D i v ) , v e r ti c a l a x i s

( 5 00 m V / D i v ) .

3 . R e l a t i on s b e tw e e n c on t i n u ou s - an d d i sc r e te - t im e e q u at i o n s

When the time constant T is lower than about 15 times the delay r, it has been shown [7,8] that the DDE (3) can

be approximated by the discrete-time equation x ~-* mf(x) in the stable regimes, and the dynamics of the discrete-

time equation reproduce quite accurately the behavior of the continuous-time system. In the standard tent map f,

there is no period-doubling route to chaos [46]. The standard tent map is well known to produce stable fixed point

solutions if m < 2, and chaotic solutions for m _> 2. Correspondingly, the continuous-time equation (3) exhibits

stable equilibrium points and chaotic solutions in the respective regions. The discrete-time system with the smooth

tent map x w, mfs(x) is characterized by a limited period-doubling route to chaos (Fig. 4) giving rise to 2N-cycle

periodic solutions X u = { X ' l , x 2 , " " • , X 2 N } - In the continuous-time system (3) with the nonlinearity being fs, these

cycles correspond to 2Ur-periodic solut ions X u which will be defined later, It is not surprising to find the same

appearance of harmonics, isomers and also of different types of chaotic attractors in the electronic circuit as well as

in interferometer optical systems because they are modeled by the same kind of DDE (with unimodal map). When

m ~ [mr, m,[, m f standing for the value of m that ends the period doubling in the map x ~-* mf~(x)(in the standard

tent map f , m f = 2 ) , the chaotic attractors of the smoothed or standard tent map belonging to the fundamental

branch are composed of two sets of unconnected chaotic intervals and the solutions of (3) are square-wave-like [2,71.

The value of m, van be computed with the so-called critical lines associated with the map f~ [47,48]. Ifm > mMax

where mM~x >_ m, is the solution of f~(m) + f ( -~( -m) = 0 (for the standard tent map mM~ = 4), the solution

becomes unbounded. Throughout this section, we will use f as a generic unimodal map.

MFMs associated with stable solutions of (3) will be introduced in order to make the comparison with the map

x ~ mf (x ) . Furthermore, we will show that within a given [m f, m , [ bifurcation interval where chaotic solutions

appear, the discrete-time approximation can also be used if we introduce a novel definition for the Lyapunov exponent

LE associated with the DDE (3). This generalized LE will be very close to the LE of discrete map and thus both

approaches give similar chaotic-like solutions from a statistical point of view. This point is very important for the

meaning of the control scheme we will present, so we will show the relation between the LE of the map x ~ mf (x )

and the new definition of the LE of the DDE (3). This comparison between the two LEs is motivated by two facts:



132 P. Ce lk a /Ph ys ica D 104 (1997) 127-14 7

¢:

I¢

0 . 5

0

- 0 . 5

-1

- 1 . 5

- 2

- 2 . 5

2 0 4 0 6 0 8 0

Num ber of Exponent1 0 0

2 0

1 8

1 6

1 4

1 2

1 0

8

1 4

1 2

1 0

8

6

4

3

6

4

1 0

b ' . . . . . . , , , , . . . . . . .

i i I , , , I , , , I , , , I , , , I , , , I , ,

3 . 2 3 . 4 3 . 6 3 . 8 4 4 .2 4 ,4

m

. ' ' ' ' 1 ' ~ ' ' 1 ' ' ' ' 1 . . . . I ' ' ' ~ ' 1 ' ' ' ' i . . . . I ' ' '

2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0

~ / T

Fig. 3. (a) Typical Lyapunovspectra (N b = 100). (b) Lyapunov dimension for r / T = 100/6. (c) Lyapunov dimension for m = 3.2. Thecharacteristic numbers were computed on 1000 × r length time series.

f i r s t ly , con t ro l l ing chao t i c t ra j ec to r i es wi th h igh LE i s more d i f f i cu l t t han wi th low LE; second , con t ro l l ing two

t r a j e c to r i e s w h i c h h a v e t h e s a me L E , e v e n f r o m t w o d i f f e r e n t s y s te ms , w i t h t h e s a m e c o n t r o l me t h o d , i s o f t h e s a me

o r d e r o f d i f f ic u l ty . W e t h u s c o n c l u d e t h a t i f t h e c o n t r o l s c h e m e w e p r e s e n t w o r k s f o r t h e d i s c r e t e - t i me s y s t e m , t h e n

i t w i l l g i v e g o o d r e s u l t s f o r t h e c o n t i n u o u s - t i me s y s t e m i n t h e s a m e r a n g e o f t h e p a r a m e t e r s e t v a l u e s. L e t u s f i r st

reca s t Eq . (3 ) , reca l l ing tha t or2 = 1 , i n the fo l lowing w ay :

d x ~ m A ---- ( ? T 2 - 1 )d t + A x = h ( t ) 1 O t l / T

(5 )

wi th x = ( x d x / d t ) T a n d h ( t ) = ( O f ( x ( t - r ) ) ) T E q . ( 5 ) c a n b e r e w r i t t e n a s x = - A x + F , n ( X r ) , w i t h x r ( t ) =

x ( t - r ) , a n d t h e r e l a t e d v a r i a ti o n a l e q u a t i o n i s g i v e n b y

~ Fm(x~)3 ± = - A ~ x + - - ~ x r . (6 )

O Xr

I n o r d e r t o h a v e a n i d e a o f th e c o m p l e x i t y o f t h e s e t y p e s o f D D E , i t i s in t e r e st i n g t o e s t i ma t e t h e L y a p u n o v d i me n s i o n ,

w h i c h g i v e s a g o o d a p p r o x i m a t i o n o f t h e f r a c t a l d i me n s i o n o f t h e a t tr a c t in g s e t . U s i n g n u me r i c a l i n t e g r a ti o n ( f o u r t h -

o r d e r R u n g e - K u t t a , A t = r / ( N b - - 1 )) o f ( 5 ) a n d ( 6 ) t o g e t h e r w i t h th e G r a m - S c h m i d t p r o c e d u r e a s d e s c r ib e d b y

Fa r me r [ 3 ] , a n d I k e d a a n d Ma t s u mo t o [ 1 7 ] , w e w e r e a b l e t o c o mp u t e t h e L y a p u n o v s p e c t r u m o f ( 5 ) , a n d u s i n g

K a p l a n - Y o r k e c o n j e c t u r e [ 8 ] t o c o mp u t e D L . V a r y i n g m a n d r , w e d i s p l a y i n F i g . 3 s o me c h a r a c t e r i s t i c L y a p u n o v

s p e c t r a a n d L y a p u n o v d i me n s i o n s .



P . C e lka /P h ys i ca D 1 0 4 (1 9 9 7 ) 1 2 7 -1 4 7 133

F i g . 3 ( a ) d i s p l a y s L y a p u n o v s p e c t r u m f o r s t a b le ( m = 2 ) a n d c h a o t ic s o l u ti o n s ( m = 3 . 2 a n d m = 4 . 3 ) . W e

o b s e r v e t h e e x p o n e n t i al - l ik e b e h a v i o r o f th e s e s p e c t r a a n d t h e e x is t e n c e o f an " i n f in i t e n u m b e r " 4 o f n e g a t iv e L E s .

T h e L y a p u n o v d i m e n s i o n D L r a n g e s f r o m 4 t o m o r e t h a n 1 9 w h e n m a n d r / T a r e v a r ie d . T h e l in e a r r e l a ti o n b e t w e e n

D L a n d m r / T o b s e r v e d e x p e r i m e n t a l l y b y I k e d a [ 3 , 1 7 ] s e e m s t o b e e ff e c t iv e f o r t h e m p a r t b u t n o t f o r t h e r / T o n e

(F ig . 3 (c ) ) .

E q . ( 5 ) c a n b e s o l v e d o n e a c h i n t e r v a l [ ( n + 1 ) r , ( n + 2 ) r ] f o r 5 n 6 ~ U { - 1 } , g i v i n g r i s e t o t h e s o l u t i o n x ( . ) ={ x~ ( . )} ~ j w i t h t h e i n it i al c o n d i t i o n X _ l ( t ) = ( ~ p ( t ) d ~ o ( t ) / d t ) T = q ~ ( t) o n [0 , r ] . T h e 2 - v e c t o r v a l u e d f u n c t i o n

x ( . ) b e l o n g s to t h e s p a c e C 2 ( [ 0 , + ~ ] , I ) × c l ( [ 0 , + ~ ] , I ) , w h e r e 1 is a l e n g t h m M a x i n te r v a l. T h e s o lu t i o n x ( . )

w i t h r e s p e c t t o t h e i n i t i a l c o n d i t i o n q t , ( . ) i s c o m p u t e d i t e r a t i v e l y o n e a c h r - t i m e i n t e r v a l v i a t h e e v o l u t i o n o p e t a t o r

5Vm . W i th in t he g ive n r an ge m ~ [0 , m Ma x , t h e n o n l i n e a r o p e r a t o r )t-m i s b o u n d e d , a n d d u e t o th e l o w - p a s s b e h a v i o r

o f (5 ) , 5Vm w i l l b e a c o n t i n u o u s f u n c t i o n a l s p a c e v a l u e d o p e r a t o r . I n t r o d u c i n g t h e t i m e n o r m a l i z a t i o n t ~ ( n + 1 ) r ÷ / "

w i t h ~ ~ 1 0 , r ] a n d n ~ ~ W { - 1 }, w e i n t r o d u c e 5 rm i n th e f o l l o w i n g d e f i n i t i o n :

D e f i n i t i o n 1 . W e w il l ca l l.T 'rn t h e e v o l u t io n o p e r a t o r o f E q . (5 ) i fU r n : C 2 ( [ 0 , r ] , I ) × c l ( [ 0 , r ] , I ) --+ C 2 ( [ 0 , r ] , I )

× C I ( 1 0 , r l , 1 ) " x , , _ j ( t ) ~ F m ( X n - l ( t ) ) = x n ( t ) i s a c o n t i n u o u s l y d i f f e r e n t i a b l e m a p . ( /', { x ~ ( t )} ~ = 0 ) i s t h e

t r a j e c t o r y a s s o c i a t e d t o o n e s o l u t i o n o f ( 5 ) f o r w h i c h x , , ( t ) = , 2 , ( t" ) + e - - a t x n - 1 r ) , wi th x _ l ( t ) = ~ ( t" ) t he i n i t ia lc o n d i t i o n a n d

~ , , ( [ ) ~ - ( m / T 2) f d t ' e - a ~ [ - t ' ) h n ( t ' ) ,

o

wi th h , , ( t ' ) - -- ( O f ( x , , _ I ( t ' ) ) ) T f o r X n ( 0 ) = X n - l ( r ) t h e c o n t i n u i t y c o n d i ti o n .

W e n o w g i v e a l e m m a t h a t w il l b e u s e f u l in t h e p r o o f s o f t h e m a i n t h e o r e m s . T h i s l e m m a s t a te s t h a t u n d e r t h e

a s s u m p t i o n t h a t r / T > > 1 t h e k e r n e l e - A r - t ' ) i n th e e v o l u t i o n o p e r a t o r c a n b e a p p r o x i m a t e d b y a D i r a c d i s t r ib u t i o n

u p t o a m a t r i x m u l t i p l i c a t i o n f a c t o r . T h i s o b s e r v a t i o n w a s a l r e a d y m e n t i o n e d b y I k e d a e t a l . [ 8 ] .

L e m m a 2 . L e t 3 ( t - t o ) b e t h e s c a l a r v a l u e d D i r a c d i s t r i b u t i o n c e n t e r e d a t t o , d ~ ( t , to ) - d t e - A ( t - t ° ) a m a t r i x

v a l u e d m e a s u r e d e f i n e d o n [ 0, r ] w i t h A d e f i n e d in (5 ) . I f w e i m p o s e d ~ ( t , t o) = B ( t , t o ) 3 ( t - t o ) d t t h e n B ( t , t o ) , ~

A - J e a { f -t ~) i f r / T > > 1 a n d T > 0 .

P r o o f N o t e t h a t t h e m a t r i x A i s n o t s i n g u l a r b e c a u s e T > 0 . W e t h e n h a v e t o i d e n t i f y t h e m a t r i x B ( t , t o ) . F i r s t o f

a l l w r i t e t h e f o l l o w i n g e x p r e s s i o n :

i

f s ( t - t o ) d t = e (/ " - t o ) ,

o

w h e r e t h e d i s tr i b u t i o n ~ ( t ) i s 0 i f t < 0 a n d 1 i f t > 0 . L e t u s i n t e g r a t e th e m e a s u r e d se (t, to ) o v e r [ 0 , / ']

i i

f d ~ ( t , t o ) = f d t e A " - t o ),

o o

4 Num erical simulations l imit this number to N b .

5 ~ is the set of natural numbers.



13 4 P. Celka/Physica D 104 (1997) 127-147

w h i c h g i v e s u s i m m e d i a t e l y

f dse ( t , t o ) = A - l [ e A J -t ° ) - e - A t ° ] .

0

I f t < t o t h e n f ~ d ~ ( t , to ) ~ 0 a s t h e c o e f f i c i e n t s in m a t r i x A d e p e n d o n I / T a n d r / T

f o dse ( t , t 0 ) - ~ A - l e a ( f - t ° ) , a n d w e c a n r e w r i t e fo d ~ ( t , t o ) ~ A - l e a ( t - t ° ) E ( t - - t o) . F i n a l l y

B ( t ' , to ) ~ ( t " - t o ) ~ A - l e a ( t - t ° ) E ( t - - to ) a n d B ( t , t o) ~ A - l e a ( f- t ° ) .

> > 1 . I f / " > t o t h e n

T h i s c o m p l e t e s t h e p r o o f . [ ]

F r o m t h is l e m m a , w e h a v e t h e f o l l o w i n g t r i v ia l b u t i m p o r t a n t r e s u lt o n th e m a t r i x B ( t , t o ) .

C o r o l l a r y 3 . W i t h t h e s a m e a s s u m p t i o n a s i n L e m m a 2 , l i m t - - , t 0 B ( t , t o ) = A - 1 .

T h e p r o o f o f C o r o l l a r y 3 is s t r a ig h t f o r w a r d f r o m L e m m a 2 . W e d e f i n e a 2 N r - p e r i o d i c s o l u t io n o f (5 ) in r e l a ti o n

w i t h t h e e v o l u t i o n o p e r a t o r U , n a s f o ll o w s :

D e f i n i t i o n 4 . W e w i l l c a l l X N = { ~'1 t" ), x 2 ( ~ . . . . . x 2 N ( ; )} a 2 N r - p e r i o d i c s o l u t i o n o f (5 ) i f 5 2N (Xn (7) ) = -~'n t ' ) ,

V n E { 1 , 2 . . . . . 2 g } an d .T 'm ~ 2 ~ ( t" ) ) = X l (t " ).

I n o r d e r t o i n t r o d u c e M F M a n d M L E q u a n t i t ie s r e l a t e d t o ( 5 ), d e p e n d i n g o n t h e ty p e o f s o l u t io n w e h a v e , w e

i n t r o d u c e a d i f f e r e n ti a l o p e r a t o r 7 9 o n C 2 ( [ 0 , r ] , I ) x C 1 [0 , r ] , I ) .

D e f i n i t i o n 5 . W e d e f i n e t h e d i f f e r e n t i a l o p e r a t o r D a s

fmf9 . T 'm ( X n ( ~ ) =-- " ~ d t ' e - A ( t - t ' ) 7 9 h n ( t ' ) ,

o

w h e r e 7 9 h = - ( 0 d f ( x ) / d x ) T .

I t i s c l e a r t h a t t h i s d i f f e r e n t i a l o p e r a t o r e x i s t s o n l y i t f ( x ) i s d i f f e r e n ti a b l e e x c e p t o n a s e t o f z e ro L e b e s g u e

m e a s u r e . T h e s t a n d a r d t e n t m a p f h a s a n u n d e f i n e d d e r i v a t i v e at {0} b u t th i s s et is o f n u ll m e a s u r e . W e c o u l d

a l s o u s e th e t h e o r y o f d is t r ib u t i o n s a n d c o n s i d e r t h a t i f f i s l o c a l l y i n te g r a b l e , t h e n w e c a n d e f i n e d f ( x ) / d x a s a

d i s tr i b u ti o n . T h e M F M / Z r , p a s s o c i a t e d w i t h a 2 N r - p e r i o d i c s o l u t io n i s d e f in e d a s :

N N . . . . 2g - - ^ 2ND e f i n i t i o n 6 . L e t X r b e a 2 r - p e r i o d i c s o l u n o n , t h e M F M i s /z r, o = ( I - In = j I l D . ~ m ( X n ( t ) ) l l p ) / l ~ l w h e r e

I I " l i p : L P x L p ~ R + : ( f l ( x ) , f 2 ( x ) ) -"+ I I ( f l ( x ) , f 2 (x ) l ip - - - - -( ~ i = 1 f a I f / ( x ) l p d x ) l / P i s t h e s e m i n o r m o n t h e

s p a c e L p x L p o f L e b e s g u e m e a s u r a b l e f u n c t i o n s o n th e t im e i n t e rv a l ~ , a n d I S 2 1 i s i t s l e n g t h .

T h e t i m e i n t e r v a l f 2 i s u s u a l l y [ 0 , r ] b e c a u s e t h e t r a j e c t o r y ( ~ ', {Xn 7 ) }n°~=0 i s c o m p u t e d o n i t , b u t , a s t h e f u n d a m e n t a l

p e r i o d o f t h e 2 r - p e r i o d i c s o l u t io n i s To ~ 2 ( r + T ) , 1 2 c o u l d b e t a k e n t o b e ½ T0 i f w e r e c o m p u t e t h e i te r a t iv e

s o l u t io n o n t h is i n t e rv a l . F i n a l l y w e i n t r o d u c e a M L E A r , p o f o n e t r a je c t o r y (~ ', {x n ( 7)} n~ _0 a s s o c i a t e d t o E q . ( 5 ) a s :

D e f i n i t i o n 7 . L e t ( t ' , { x n ( ~ }n~_0 b e a t r a j e c t o r y a s s o c i a t e d t o o n e s o l u t i o n ' o f ( 5 ) , t h e M L E i s



P. Ce lka /Ph ysica D 104 (1997) 127 -147

A t , p = lim l n ( lz u p ) / N w h e r e l z u ( F I 2 ) /N -+ oO r ,p ~ i [ ~ m ( ~ n ( ; ) ) l l p ] ~ Q I N .\ n = - - I

13 5

L e t u s n o w d e r i v e t he d i s c r e t e - t i me a p p r o x i ma t i o n o f ( 5 ). I n f a c t , w e w i l l d e d u c e f r o m ( 5 ) a d i ff e r e n c e e q u a t i o n b u t ,

a s s u m i n g L e m m a 2 , th i s c o n t i n u o u s - t i me e q u a t i o n w i l l b e n o t h in g b u t a d i s c r e t e - ti me o n e .

Lemma 8 . I n t h e c a s e r / T - -~ ~ ( 5 ) r e d u c e s t o t h e d i f f e r e n c e e q u a t i o n x ( t ) . ~ m f ( x ( t - r ) ) w i th 2 ( t ) ~ 0 'v' t > 0 .

P r o o f I f aga in / " 6 [0 , r ] , t he so lu t ion o f Eq . (5 ) i s o f the fo rm

Xn (; ) = a~'m X n_l ( t ) ) ,

wh ere 5Vm i s def ined in Def in i t ion 1 . Le t us wr i t e the r igh t -han d s ide o f the p rev ious equ at ion as

f

g n ( t ) = ~ 5 dse(t, t ) h n ( t ) .

0

U s i n g L e m m a 2 , w e c a n w r i te

im~g ~ 6 ) ~ . - r-

0

mdt B (t, t")8(t - t )hn(t ) "~ ~-~-B (t, t")hn (t)

w i t h Co r o l l a r y 3 w e g e t

m A - 1 ^g n(i) "~ - - ~ h n ( t ) ,

and w i th the def in i tion o f the vec to r hn ( t ) a n d t h e ma t r i x

A - ' { ° q / T l o )= ~ _ I / T 2 r 2 ,

w e o b t a i n

y ( ; ) J 0 "

T h i s l e m m a i mp l i e s t h a t u s i n g L e m m a 2, E q. ( 5 ) r e d u c e s t o a d i s c r e t e - t i me e q u a t i o n w h e n r / T > > 1 , a s th e

s o l u ti o n mu s t b e c o n s t a n t o n e a c h t i me i n te r v a l [ n r , ( n + 1 ) r ] . O b s e r v e th a t t h e MFM / Z r , p a n d ML E Ar,p d e p e n d

o n t h e d e l a y r . W i l l sh o w t h a t in th e l i m i t i n g c a s e r / T ~ ~x~ t h e F l o q u e t mu l t i p l ie r a n d t he L E o f t h e d is c r e t e ma p

x w-~ m f( x ) w i l l b e c l o s e to , r e s p e c ti v e l y , t h e MF M a n d M L E o f t h e c o n t i n u o u s - t i me e q u a t i o n ( 5 ) . I n p r a c t ic e , f o r

a f ini te rat io r / T ~ 20 there ex i s t s a va lue o f p such tha t A r , p w i l l b e a s c l o s e a s w e w a n t t o t h e L E o f t h e ma p

x w-~ m f( x ) , a n d t h e s a me h o l d s f o r t h e F l o q u e t mu l t i p l i e r . W e h a v e n o t h e o r e m a b o u t t h e e x i s t e n c e o f s u c h a p ,

g i v e n t h e p r e c i s i o n o n t h e L E o r F l o q u e t m u l t ip l i e r, b u t n u m e r i c a l s i mu l a t io n s i n Se c t i o n 4 w i l l s u g g e s t th a t s u c h a

c o n d i t i o n e x i s ts . T h i s w i l l l e a d t o t h e s t u d y o f t h e c o n t r o l s c h e m e o n l y i n t h e d i s c r e t e - t i me a p p r o x i m a t i o n . I n t h is

s i tua t ion , w here the ra t io r / T - -- r ~x~, we get the fo l lowing tw o theore ms :



1 3 6 P. Celka/Physica D 104 (1997) 127-147

Theorem 9. F o r T > 0 f ix e d a n d f o r a g i v e n v a l u e o f m = m s , le t X ~ b e a 2 N r - p e r i o d i c s o l u t io n o f (5 ) , a n d # r , p2 N

t he c o r re s p o n d in g M F M . F o r a n y p 6 ] 0 , + c ~ [ w e h a v e l i m r ~ / Z r , p = I -I n= l I d f ( - £ n ) / d x l, w h e r e X N = {-£~ 2~=1

i s t h e s t a b l e o r b i t o f th e m a p x ~ m s f ( x ) .

P r o o f U s i n g L e m m a 2 w e c a n w r it e

m f m f mn( t ) = ~ d ~ ( t , ?)7)hn(t ) ~ ~-~ dt B( t , t")~(t - ~79hn( t ) ~ - ~ B ( i , ~79hn(t )

~2 ~2

a n d t h u s

m 1 ^

gn( t ) ~ ~ -~A- 7 )hn( t ) ,

w h i c h g i v e s

R e c a l l i n g t h a t b y d e f i n it i o n t h e M F M i s

n = l

D u e t o t h e f a c t t h a t i n t h e c a s e r / T --+ ~ t h e D D E r e d u c e s t o t h e d i f f e r e n c e e q u a t i o n a s s ta t e d b y L e m m a 8 , w i t h

Xn (} ') c o n s t a n t f o r n ~ ~ tO { - 1 }, t h e r e i s a u n i q u e a n d b i j e c ti v e c o r r e s p o n d e n c e b e t w e e n t h e 2 N r - p e r i o d i c s o l u t io n

X ~ o f th e D D E a n d t h e 2 N - c y c l e X Iv o f t he m a p x ~+ m f ( x ) : X ~ ~ X N = { ~ 1 , " " , X 2N } . T h e d i f f e r e n t i a l

o p e r a t o r c a n b e r e c a s t i n t h e f o l l o w i n g f o r m :

D ' ~ m ( X n ( t ) ) ~ ( m d f ( - E n ) / d x0

a n d

I I~D~mCXn(t) )l lp ~ ]df( -£n ) /dxlP dt ~ lY2 l ldf ( -£n)d x I.

F i n a l l y

l i m / z r , p ~ IS211 d f ( ~ n ) / d x l ~ I d f ( -~n) /dx l . []r - - -~ oo ~kn = l n = l

Theorem 10. F o r T > 0 f i x e d a n d f o r a g i v e n v a l u e o f m = m c l e t ( ~ , { Xn (t)} n~ __ 0 b e a t r a j e c t o r y a s s o c i a t e d t o

o n e s o l u t i o n o f ( 5 ), a n d Ar,p th e c o r re s p o n d in g M L E . F o r a n y p E ] 0, + ~ [ w e h av e l i m r ~ At, p = l i m N ~

l n ( l - I ~ = l [ d f ( x n ) / d x b ) N w h e r e {Xn}~_~ i s t h e t ra j e c t o r y a s s o c i a t e d w i t h t h e m a p x ~-~ m cf (X ) .

. T h e p r o o f o f T h e o r e m 1 0 i s t h e s a m e a s th a t o f T h e o r e m 9 . In th e p r a c t i c a l c a s e w h e r e r / T i s f in i te a n d i n th e

r a n g e [ 1 5 , 8 0 ], w e h a v e o b s e r v e d t h a t t h e r e e x i st s a p v a l u e s u c h t h a t Ar,p i s v e r y c l o s e d t o th e L E a n d / Z r , p v e r y

c l o s e t o th e F l o q u e t m u l t i p l ie r o f th e d i s c r e t e m a p f o r g i v e n v a l u e s o f m . W e w i l l g i v e a n u m e r i c a l e x a m p l e i n

S e c t i o n 4 , b u t t h is m o t i v a t e s t h e n e x t t w o c o n j e c t u r e s .



P . C e l k a / P h y s i c a D 1 0 4 ( 1 9 9 7 ) 1 2 7 - 1 4 7 1 3 7

C o n j e c t u r e 1 1. F o r T > 0 , r / T > 1 5 f ix e d , a n d f o r a g i v e n v a l u e o f m = m s , le t X u b e a 2 N r - p e r i o d i c s o l u t io n

o f ( 5 ) , a n d lZr,p t h e c o r r e s p o n d i n g M F M . 3 p 6 ] 0 , o o [ su c h th a t l U r . p - ( I- lZ U l I d f ( - £ n ) / d x l ) l i s m i n i m u m w h e r e

X N - 2 N= {Xn}n=j i s th e s t a b l e o r b i t o f th e m a p x ~ m s f ( x ) .

C o n j e c t u r e 1 2 . F o r T > 0 , r / T > 15 , and fo r a g iven va lue o f m = m c , l e t ( /', {xn(t )}n~_0 ) be a t r a j e c to ry

a s s o c i a t e d t o o n e s o l u t io n o f (5 ) , a n d A t , p t h e c o r r e s p o n d i n g M L E . 3 p 6 ] 0 , <x ~[ s u c h t h a t: J A r . p - l i m N ~

1 Nn( l - In=2 I d f ( x n ) / d x D / N I i s m i n i m u m w h e r e { xn } ~_ 1 i s a t r a j e c t o r y a s s o c i a t e d w i t h t h e m a p x ~ -~ m c f ( x ) .

O b s e r v e t h a t , a t t h is p o i n t , t h e r e a re n o r e s t r ic t io n s a b o u t t h e v a l u e o f m e i n T h e o r e m 1 0 a n d C o n j e c t u r e 1 2. T h e o r e m

1 0 t el ls u s th a t, i n t h e m e a n , t h e M L E w i l l c o n v e r g e t o i ts d i s cr e t e c o u n t e r p a r t a s r / T ~ o o . T h i s t h e o r e m d e a l s

w i t h m e a n v a l u e s o f t h e d i f f e r e n t ia l o p e r a t o r D b u t it c o u l d a l s o b e v i e w e d a s a d i f f e r e n t i a l p r o c e s s o n a m e a n

s o l u t i o n {<x) n }~_- w i thin S2 wh er e

( 7 )

I f m 6 [ m f , m , [ a n d r / T > > 1 , t h e m e a n t r a j e c to r y m a t c h a s a d i s c r e t e -t i m e t r a j e c t o r y b e c a u s e t h e d i s c r e te - t im e

a t t r a c t o r li e s o n t w o u n c o n n e c t e d s e ts . I f m > m , , t h e c o n t i n u o u s - t i m e t r a j e c t o r y fi ll s t h e e n t i r e i n t e rv a l I a n d

t h e c o n n e c t i o n b e t w e e n t h e m e a n t r a j e c t o r y a n d t h e d i s c r e t e - t i m e a p p r o x i m a t i o n i s c o m p l e t e l y l o s t . S u r p r i s i n g l y ,

i n th i s r e g i o n , t h e M L E a s d e f i n e d i n D e f i n i t i o n 5 w i l l s t il l b e v e r y c l o s e t o t h e d i s c r e t e - t i m e v a l u e , b u t t h e t i m e

c o r r e s p o n d e n c e i s n o m o r e v a l id . W e h a v e t h u s e s t a b l i s h e d t h e d i f f e re n t c o n t i n u o u s - d i s c r e t e c o n n e c t i o n s w i th

M F M , M L E a n d m e a n t r a je c t o r ie s D D E . I n th e n e x t s e c t io n , w e w i l l a p p r e c i a t e t h e d i f f e re n t a p p r o x i m a t i o n s a n d

t h e ir r e s p e c t iv e r e s t ri c ti o n s w h e n a p p l y i n g t h e c o n t r o l m e t h o d .

4 . C o n t r o l s c h e m e

4 . 1 . C o n t r o l t o w a r d s s t a b l e p e r i o d i c o r b i t s o f t h e f u n d a m e n t a l b r a n c h

P y r a g a s h a s i n t r o d u c e d a v e r y s im p l e t i m e - d e l a y e r r o r f e e d b a c k c o n t r o l m e t h o d i n o r d e r t o s ta b i li z e U P O e m b e d -

d e d i n a c h a o t i c a t t ra c t o r [ 2 3 ] . S o m e e x p e r i m e n t a l a n d t h e o r e t ic a l a p p r o a c h e s b a s e d o n t h i s m e t h o d h a v e b e e n t h e

s u b j e c t o f n u m e r o u s p a p e r s [ 2 7 - 2 9 , 3 9 , 4 2 , 4 3 ] . T h i s s c h e m e h a s b e e n a p p l i e d t o a n o p t ic a l b i s t a b le s y s t e m r u n n i n g

i n c h a o t i c m o d e , p r e v i o u s l y . H o w e v e r , t h e s t a b i li z e d o r b i t s w e r e n o t U P O s , b u t t h e o n e s c l o s e t o s t a b le u n c o n t r o l l e d

s y s t e m o r b i t s w e r e [ 3 6 ]. A s m e n t i o n e d i n s e c ti o n l , t h e fi rs t p o s s i b l e w a y o f c o n t r o l l in g t h e D D E ( 3 ) w i t h t h e m a p

( 4 ) i s to s t a b i li z e c h a o t i c t r a je c t o r ie s t o w a r d s s o m e p e r i o d i c o r b i t s w h o s e b e h a v i o r s a r e c l o s e t o s t a b le 2 N r - p e r i o d i c

s o l u t io n s o f t h e o r i g in a l s y s t e m .

T h e c o n t r o l w i l l b e d o n e i n re a l t im e a s s u m i n g t h e s t a te x i s o b s e r v a b l e . L e t u s s u p p o s e t h a t t h e o ri g i n a l s y s t e m

i s i n a Chao t i c r eg io n wi th m - -- - m c c [ m y , m,[ a n d t h a t r / T > > l ( i n o u r c a s e 1 6 . 6) . N o t i c e t h a t th e i n t e r v a l

[ m y , m,[ c o r r e s p o n d s t o c h a o t i c t ra j e c to r i e s t h a t a r e s q u a r e - w a v e - l i k e s o l u t i o n s o f ( 3 ). W e w i l l f o r c e t h e s y s t e m t o

b e c l o s e t o o n e o f it s 2 N r - p e r i o d i c s o l u t i o n s b y u s i n g a n e r r o r f e e d b a c k s c h e m e w i t h a c o n t r o l s i g n a l o f th e f o r m

c ( t ) = K ( x ( t - n T o ) - x ( t ) ) w i t h K 6 ~ , n 6 { 1 , 1} a n d T o = 2 r . T h e c o n t r o l l e d s y s t e m c a n b e m o d e l e d b y

T 2 d 2 x ( t ) ~ d x ( t )o t 2 ~ + t o t l - - - - - ~ + x ( t ) ---- m f ( x ( t - r ) ) + c ( t ) . (8 )



1 3 8 P. Ce lka /P hys ic a D 104 (1997) 12 7-147

0

-1

' ' ' I ' ' ' ' I ' ' ' ' I . . . . I ' ' ' ' I '

a . "4

,,----

.... ~i' i!

K = O, , , I . , . , l , , . , i . . . . I , , , , l ,

1 . 5 2 2 . 5

-1

b .

........ l~

........ ~~ i]"~

. . . . . . . . . . . . . : ' ~ ! i v ~ . -

, . r - - :

0 0 . 5 1 3 . 3 . 5 0 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5

m + K m + K

F i g . 4 . C o m p a r i s o n b e t w e e n n u m e r i c a l s im u l a t i o n s o f c o n t r o l l e d ( K # 0 ) a n d u n c o n t r o l l e d ( K = 0 ) sy s t e m s m o d e l e d b y Fro, k w i t h :

( a ) f a n d ( b ) f s .

T h e u s e o f o t h e r v a l u e s o f n w i l l b e t h e s u b j e c t o f Se c t i o n 4 . 2 . T h e e v o l u t i o n o p e r a t o r o f ( 8 ) w i l l b e n o t e d ~ m , Ka s i t d e p e n d s o n K . I t is e a s y t o d e r iv e t h e d i s c r e te m a p t h a t c o r r e s p o n d s t o ( 8 ) w i th L e m m a 8 a n d i n t h e c a s e w h e r e

1

x k = ( m f ( x k -1 ) + K X k - l ) / ( 1 + K ) = F m , r ( X k - l ) (9 )

K e e p i n g m = mc f i x e d a n d v a r y i n g K i n t h e ma p ( 9 ) r e s u l ts i n a r e v e r s e p e r i o d - d o u b l i n g e n d i n g w i t h f ix e d p o i n t

s o l u t io n s i n d e p e n d e n t o f t h e K v a l u e s . F i g . 4 s h o w s t w o b i f u r c a t i o n d i a g r a m s c o r r e s p o n d i n g t o th e ma p s f a n d

f s . T h e s e t w o b i f u r c a t i o n d i a g r a ms h a v e b e e n c o mp u t e d w i t h ( 9 ) , a n d t h e c o r r e s p o n d i n g ma p , v a r y i n g m w h i l e

K = 0 u n t il m ---- mc , a n d th e n v a r y i n g K w h i l e m = mc w a s k e p t u n c h a n g e d . W e o b s e r v e t h a t t h e s mo o t h ma p

g i v e s e f f e c t iv e l y a l i m i t e d p e r i o d - d o u b l i n g e n d i n g a t a p p r o x i ma t e l y m y - - 2 . 3 b e f o r e e n t e r i n g t h e c h a o t i c r e g i o n .

R i g h t a f t e r m y t h e so lu t ion evo lves to chao t i c in t e rva l s as in the s t andard t en t map where m y = 2 . T h e v f a c t o rd e t e r mi n e s t h e v a l u e o f m y a n d a l s o t h e n u mb e r N o f d i f fe r e n t s t a b le c y c l e s . T h e K f a c t o r d e t e r mi n e s s t a b il i ty

in t e rva l s one wa n t s to access , ca l l ed in [36] the target in tervals . W h e n K i s c h o s e n i n s u c h a w a y t h a t it c o r r e s p o n d s

t o a 2 N - c y c l e , t h e c o n t i n u o u s - t i me s y s t e m ( 8 ) w i l l e v o l v e to a 2 N r - p e r i o d i c s o l u ti o n t h a t is c l o s e t o th e o r i g i n a l

2 N z - p e r i o d i c s o l u t i o n .

I n o r d e r t o c o n f i r m t h e c o r r e s p o n d e n c e b e t w e e n c o n t r o l l e d d i s c r e t e - a n d c o n t i n u o u s - t i me s o l u t i o n s , w e h a v e

c o m p u t e d t h e L E o f ( 9 ) a n d t h e ML E o f ( 8 ) u s in g a n o p t i m a l t i me i n t e r v a l l e n g t h 1 12 ] = ½To. Para me ters fo r (8 ) a re

g i v e n i n T a b l e 1 . W e c a n c o m p u t e t h e e v o l u t i o n o p e r a t o r 3Cm, o f ( 8 ) u s in g D e f i n i t io n 1 a n d c h a n g i n g t h e v e c t o r

/In to

( o )~ In (t ) = m f ( x n - l ( t ) ) + K ( X n - l ( t ) ) • ( 1 0 )

m ( 1 + K )

F r o m c o mp u t e d t r a j e c t o r i e s 0 " , {x n ( t ) }~= - l ) o f ( 8 ) w i t h t h e e v o l u t i o n o p e r a t o r Urn,K,, w e h a v e c a l c u l a t e d t h e

M LE in d i f fe ren t s itua t ions wi th mc = 3 .2 f ixed : (1 ) kee p ing r / T f i x e d a n d v a r y i n g K f o r t h r e e v a l u e s o f p ;

( 2 ) k e e p i n g K f i x e d a n d v a r y i n g r / T f o r t h r e e v a l u e s o f p ; ( 3 ) k e e p i n g r / T , K f i x e d , a n d v a r y i n g p . Fo r a l l

n u m e r i c a l s i mu l a t i o n s w e h a v e t a k e n 1 0 00 12 i n t e rv a l s t o c o m p u t e MF M a n d ML E .

F i g . 5 ( a ) s h o w s t h a t t h e ML E a n d MFM a r e c l o s e t o t h e i r d i s c r e t e e q u i v a l e n t v a l u e s c o mp u t e d w i t h Fro,K,

a n d t h a t t h e r e a r e s e v e r a l t a r g e t i n t e r v a l s f o r t h e c o n t r o l l e d s y s t e m. W e o b s e r v e i n F i g . 5 ( b ) t h a t i n b o t h c a s e s



0 . 3

0 . 2

, ~ 0 . 1

, , ~ - 0 . 1

- 0 . 2

- 0 . 3

P. Ce l ka /P hys ica D 104 (1997) 1 27-14 7

• G . ' 1= O .

~ , I L y a -~, p = l "

i i , i i ' t i 1 '

0 0 . 0 5 0 . 1 0 . 1 5 0 . 2 0 . 2 5 0 . 3

K

13 9

0. 3

0.25

0. 2

0.15

0.1

, i I ' , ' I , ' , I ' , , I ' ' '

b . - - Lya-p=0.3 ]

Lya-p= 1 I........ Lya-p=2

0 , . 0 4 0 ,

0.10364~i , I i , i I i ~ ~ I i i L I i , i _

20 4 0 60 80 100

Tau/T

o

-0.05

-o.1

- 0 . 1 5

0 .1 ~ r - ~ - - - , I ~ ' ' ~ ' ' ' I ' ' '

, I L . - p = 0 3 I\ I Lya-p=l I

o . o I . . . . . . . . . I

4 . 1 7 2 2 1" 0 . 2 L ~ ~ ~ 2 - - ~ ~ , _ I , ~ , J ~ ~ J I i i i

0 20 40 60 80 100

T a u / T

0 .17

0 .16

0 .15

[ '~ 0.14

0 .13

0 .12

0.11

-0.132

-0.133

-0 .134

-0.135

,~ -0 .136

-0.137

-0.138

-0.139

-0.14

0.1

C .

0.1 0.2 0.3 0.4 0.5 0.6

P. . . . ~ . . . . i . . . . I . . . . ~ . . . .

J

0.2 0.3 0.4 0.5 0.6

P

F i g . 5. V a l i d a ti o n o f T h e o r e m s 9 a n d 1 0 a n d C o n j e c t u r e s 11 a n d 1 2 b y c o m p u t a t i o n o f M L E a n d M F M : ( a ) f i n it e r a t io r / T = ~ fo r

t h r e e v a l u e s o f p w i t h a b o l d l i n e i n d i c a t in g d i s c r e t e v al u e s ; ( b ) c o n v e r g e n c e o f M L E a n d M F M t o t h e i r d i s c r e t e c o u n t e r p a r t ( l in e m a r k e r ) ;

( c ) e x i s t e n c e o f a n o p t i m a l p v a l u e .



14 0 P C e l k a / P h y s i c a D 1 0 4 ( 1 9 9 7 ) 1 2 7 - 1 4 7

IIIIIIII

(a) (b)

(c )

Fig. 6. (a) Phase plane representation o f a chaotic solution. (b) Control towards a 2r-p erio dic for K = 0.1704 4- 0.0085. (c) Control

towards a 4r-p erio dic for K = 0.080 5: 0.004. Horizontal axis V (500 mV /Div ), vertical axis d V / d t (1 V/Div ).

o f M L E a n d M F M , t h e c o n v e r g e n c e i s fa s t e r i f p = 0 . 3 . I n f a c t, F i g . 5 (c ) s h o w s t h a t t h e r e e x i s t s a n o p t i m a l

v a l u e o f p f o r w h i c h M L E a n d M F M a r e c lo s e s t t o th e i r d i s c r e t e e q u i v a l e n t s , t h i s l a s t n u m e r i c a l e x p e r i m e n t

s u p p o r t s C o n j e c t u r e s 1 1 a n d 1 2. E a c h r e g i o n w h e r e M L E i s n e g a t i v e c o r r e s p o n d s t o a s t a b l e e q u i l i b r i u m p o i n t

o r p e r i o d i c s o l u t i o n . T h e s e s o l u t i o n s o f ( 8 ) a r e c l o s e to o r i g i n a l s t a b l e s o l u t i o n s o f (5 ) o r ( 3 ) o n t h e f u n d a m e n t a l

b r a n c h .

T h e b i f u r c a t i o n d i a g r a m s o f F i g . 4 s h o w t h a t th e r e i s a r e v e rs e p e r i o d - d o u b l i n g c a s c a d e 2 N T o - -+ 2 N - 1 T o . • • - ->

T o a s p a r a m e t e r K i s v a ri e d . I n s t e a d o f u s i n g a c o n t r o l s i g n a l o f t h e f o r m c ( t ) = K ( x ( t - n T o ) - x ( t ) ) w i t h

K ~ • , n ~ { 1 , 1 }, w e c a n a l s o c o n s i d e r a w e i g h t e d l i n e a r c o m b i n a t i o n o f x ( t - n r ) s u c h a s c ( t ) - -- K ( x ( t ) - ( 1 -

R oo) ~V~,n= R n - l x ( t - n r ) ) . T h i s m e t h o d h a s b e e n u s e d b y S o c o l a r e t a l. [ 3 9 ,4 2 ] i n t h e c a s e o f c o n t r o l t o w a r d s U P O s .

I n c e r t a in c a s e s , t h e y h a v e s h o w n t h a t u s in g R 5~ 0 c o u l d l e a d t o th e s t a b i l i z a t i o n o f U P O s t h a t w e r e u n c o n t r o l l a b l e

i n th e c a s e R = 0 a s in P y r a g a s ' s m e t h o d . F i g . 6 s h o w s a n e x p e r i m e n t a l r e p r e s e n t a t io n o f th e c h a o t i c t r a je c t o r y fo r

m -----3 . 2 0 - t- 0 . 1 6 a n d t w o p e r i o d i c o r b i t s o n t h e f u n d a m e n t a l b r a n c h i n t h e p l a n e x - d x / d t . T h e l a r g e ra n g e o f

p o s s i b l e v a l u e s f o r n s h o w s a f i rs t l i m i t a t i o n o f th e p r e s e n t m e t h o d , b e c a u s e w e c a n n o t p r e d i c t w h a t k i n d o f s o l u t io n

w e w i l l c o n t r o l w i t h s u c h a c o n t r o l s i g n a l.



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F i g . 7. ( a) T i m e s e r i e s r e p r e s e n t a t i o n o f a c h a o t i c s o l u t i o n . ( b ) C o n t r o l t o w a r d s a 8 r - p e r i o d i c t r a j e c t o r y ~b r K = 0 . 0 5 + 0 . 0 0 2 5 .

P . C e l k a / P h y s i c a D 1 0 4 ( 1 9 9 7 ) 1 2 7 - 1 4 7 141

( a ) ( b )

F i g . 8 . ( a ) C o n t r o l t o w a r d s a s ta b l e p e r i o d i c o r b i t f o r K = 0 . 0 9 2 q - 0 . 0 0 4 6 a n d F = 1 . 0 1 3 4 , a n d ( b ) it s p e r i o d d o u b l e d v e r s i o n f o r

K = 0 . 0 8 0 + 0 . 0 0 4 a n d y = 1 .0 1 34 . H o r i z o n ta l a x i s V (5 0 0 m V / D i v ) , v e r ti c a l a x i s d V / d t ( 1 V / D i v ) .

R e m a r k 1 3. A l l t i m e s e r i e s a n d p h a s e p l a n e p i c t u r e s a r e e i t h e r p h o t o g r a p h s t a k e n f r o m a n a n a l o g o s c i l l o s c o p e o r

d i g i t al l y a c q u i r e d o n a c o m p u t e r . T h e K v a l u e s a re a p p r o x i m a t e b e c a u s e o f a 5 % t o l e r a n c e o n t h e c i rc u i t e l e m e n t s

R , R I , R 2 , R ~ , R ~ , R e , L a n d C . T h e m g a i n f a c t o r w a s 1 + R 1 / R 2 a n d a l s o s u b j e c t t o 5 % t o l e ra n c e . P h a s e p l a n e

t r a je c t o ri e s i n p h o t o g r a p h s a r e w i t h o u t t h e D C b i a s a n d c e n t e r e d a t ( V = 0 , d V / d t = 0 ).

T h e 4 r - p e r i o d i c o r b i t w a s c o n t r o l la b l e f o r K = 0 . 0 8 0 4 - 0 . 0 0 4 . I f w e h a v e a l o o k a t F i g . 5 (a ) , it i s f a r f r o m b e i n g

o b v i o u s t h a t i t w a s p o s s i b l e b e c a u s e t h e M L E h a s a s u d d e n a n d i n e x p l i c a b l e j u m p a r r o u n d t h i s K v a l u e f o r a ll p

v a l u es . F i g . 7 s h o w s a c o n t r o l l e d c h a o t i c tr a j e c t o r y to w a r d s s o m e 8 r - p e r i o d i c s o l u t io n a s p r e d i c te d b y F i g . 5 (a ) f o r

m = 3 . 2 0 4 - 0 . 1 6 .

A n o t h e r d r a w b a c k o f th i s m e t h o d i s t h a t t h e c o n t r o l m a y g e n e r a t e n e w p e r i o d i c s o l u t i o n s t h a t ar e n o t a t a ll r e la t e d

t o s t ab l e p e r i o d i c s o l u t i o n s o f t h e o r ig i n a l s y s t e m . F i g . 8 s h o w s t w o e x a m p l e s o f s u c h a s it u a ti o n w h e r e s t a b l e



142 P. Celka/Physica D 104 (1997) 127-147

p e r i o d i c o r b i t s a r e f o u n d u s i n g t h e c o n t r o l s i g n a l c ( t ) = K ( x ( t - y T o ) - x ( t ) ) , w i t h y i r r a t i o n a l a n d a r b i t a r i l y

c l o s e t o a n i n t e g e r v a l u e f o r m = 3 , 2 0 ± 0 . 1 6 .

4 . 2 . C o n t r o l to w a r d s s t a b l e p e r i o d i c h a r m o n i c s a n d i s o m e r s

A l r e a d y s t a t e d i n S e c t i o n 1 , I k e d a e t al . h a s p r e d i c t e d t h e e x i s t e n c e o f h a r m o n i c s a n d i s o m e r s o f t h e f u n d a m e n t a l

b r a n c h [ 7 , 8 ] . D e r s t i n e e t a l . [ 1 4 ] h a v e v e r i f i e d e x p e r i m e n t a l l y t h e s e p r e d i c t i o n s w i t h a h y b r i d o p t i c a l l y b i s t a b l e

s y s t e m . O n e p r o b l e m i n o b s e r v i n g t h e s e s o l u t i o n s i s th e i r re l a ti v e r a n d o m a p p e a r e n c e a s a f u n c t i o n o f t h e p a r a m e t e r

m a n d t h e s p e e d a t w h i c h w e v a r y i t b e c a u s e t h e y u s u a l l y c o - e x i s t f o r a g i v e n m . M o r e o v e r , s o m e o f th e s e i s o m e r s

a r e u n s t a b l e a n d u n o b s e r v a b l e u n l e s s w e u s e s o m e c o n t r o l o n t h e m . D e r s t i n e e t a l . [ 1 4 ] h a v e p r o v i d e d a s i m p l e

a n d e f f i c i e n t l o c k i n g m e t h o d i n o r d e r t o s t a b i l i z e t h e w a v e f o r m s . O u r a p p r o a c h i s q u i t e d i f f e r e n t s i n c e w e s t a r t

f r o m a c h a o t i c t r a j e c to r y , N p , w h i c h i s a c h a o t i c s q u a r e - w a v e - l i k e s o l u t i o n , o r L ~ w h i c h i s a c h a o t i c t r a j e c t o r y

w i t h a p r e d o m i n a n t p o w e r s p e c tr a l p e a k l o c a t e d a t f = p ~ T o , a n d c o n t r o l i t t o w a r d s p e r i o d i c i s o m e r s . U s i n g o u r

t i m e - d e l a y c o n t r o l s c h e m e , w e w e r e a b l e t o co n t r o l s u c h i s o m e r s a n d o b s e r v e m o r e e f f ic i e n tl y t h e b i f u r c a t io n r o u t e

t o f u l ly d e v e l o p e d c h a o s . W e h a v e u s e d t h e f o r m u l a t i o n o f D e r s t i n e [ 1 4 ] f o r t h e w a v e f o r m s : L 1 i s th e f u n d a m e n t a l

m

(a) (b)

( c )

Fig. 9. (a) Phase p lane representation of a chao tic solution, (b) control tow ards a L34/3 orbit (c) control towards a L3/3 orbit. Horizontal

axis V (500mV/Div), vertical axis d V / d t (1V/Div).



P. Celka/Physica D 104 (1997) 127-147

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.5

nb r a n c h , L n i s a n n t h - o r d e r h a r m o n i c b r a n c h w h i l e L m / n f o r n = 3 , 5 , 7 . . . a n d m = 2 , 4 , 8 , . . . a r e i t s v a r i o u s

i s o m e r s w i t h p e r i o d 6 m T o / n . F i g . 9 s h o w s e x p e r i m e n t a l L 3 / 3 a n d L 3 / 3 c o n t r o l l e d i s o m e r s f r o m a c h a o t i c s o l u t i o n

f o r m e d f r o m t h e L 3 b r a n c h . T h e L38/3 i s t h e p e r i o d - d o u b l e d s o l u t i o n o f L34/3 T h e b i f u r c a t i o n p a r a m e t e r m w a s

s l i g h t l y g r e a t e r t h a n 3 . 2 a n d k e p t f i x e d d u r i n g t h e c o n t r o l. W e w e r e t h u s a b l e t o c o n t r o l L34/3 a n d L ~ / 3 u s i n g a c o n t r o l

s i g n a l o f t h e f o r m c ( t ) = K ( x ( t - T ( m ,n ) - x ( t ) ) , w h e r e T (4 ,3 ) ~,~ 1 . 8 m s a n d T ( 8, 3) ~ 3 . 8 m s . T h e c o n t r o l d e l a y

T (4 ,3 ) c o r r e s p o n d s t o 2 T o a n d T ~8 ,3 ) t o ~ -T o w i t h l e s s t h a n 5 % e r r o r. T h i s e r r o r c o u l d b e d u e t o e i t h e r t h e i n a c c u r a c y

o n t h e t im e c o n s t a n t T ( p r e c i s i o n o v e r R , L a n d C ) a n d o n r ( p r e c i s o n o n u c i n th e a n a l o g d e l a y l i n e R D 1 0 7 s e t u p ),

e i t h e r t h e m i s m a t c h b e t w e e n t h e e f f e c ti v e p e r i o d o f t h e s q u a r e - w a v e a n d o u r a p p r o x i m a t i o n T o ~ 2 ( r + T ) .T h e n u m b e r o f e x i s t in g h a r m o n i c s d e p e n d s o n t h e r a t io r / T : t h e g r e a t e r t h i s r a t io , t h e g r e a t e r t h e h a r m o n i c a n d

i s o m e r s n u m b e r . W e h a v e m e n t i o n e d a b o v e t h a t th e c o n t r o l w a s m a d e o n a L ~ a t tr a c to r . W e c a n a c c e s s i t b y a p p l y i n g

t h e c o n t r o l w i th a h i g h v a l u e o f K a n d a c o n t r o l d e l a y T ( m , n ) w i t h n = 3 , 5 , 7 . • • a n d m = 2 , 4 , 8 . - • t h a t r e s u l t s

i n " f i s s u r i n g " N P i n t o L ~ . I f w e p r o c e e d d i r e c t l y f r o m N p t h e c o n t r o l s i g n a l i s l a r g e a n d t h e e v e n t u a l r e s u l t i n g

p e r i o d i c o r b i t d o e s n o t c o r r e s p o n d a t a ll w i th e i t h e r o n e h a r m o n i c o r i s o m e r s o l u t i o n . T h i s c o u l d i n t r o d u c e n e w

p e r i o d i c s o l u t i o n s t h a t a r e o n l y r e l a t e d t o t h e c o n t r o l l e d s y s t e m s t a te e q u a t i o n s a s a l r e a d y m e n t i o n e d i n S e c t i o n 4 .1 .

6 Recalling that TO ~ 2( r + T) .



144 P. Ce lka/P hys ica D 104 (1997) 127-147

i i l l I i i 1 1 1 . . . . I , , , , I , , , 1 ~ , , , , I . . . .i i l l l l l l l l l l l l l l l l l l l l l l l l l l l l l" - - I . . . . I . . . . I . . . . I . . . . I . . . . I . . . . I . . . .

1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

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t i m e

Fig. 11. Effect of additive Gaussian noise on the control trajectories for K = 0.05, T / T = L~_ and m = 3.2 with (a) SN R 4 0 d B ;(b) SNR = 30dB ; (c) SNR = 20dB .

F i g . 1 0 s h o w s s t a b l e p e r i o d i c o r b i ts t h a t a re n o t s t a n d a r d i s o m e r s . W e c a n c h e c k t h is b y c o m p a r i n g t h e s h a p e o f th ec h a o t i c a t t r a c t o r a n d t h o s e o f t h e s t a b i l i z e d o r b i t s .

4 . 3 . N o i s e i n f l u e n c e o n t h e c o n t r o l l a b i l i t y

W h i l e t h e e l e c t r o n i c s e t u p w e h a v e u s e d d o e s n o t i n t r o d u c e m u c h n o i s e , i n m a n y p r a c t i c a l s i t u a t i o n s ( b i o l o g y ,

m e c h a n i c s , c h e m i s t r y ) , n o i s e i s o f m a j o r i m p o r t a n c e . I n t h is s e c t i o n , w e e x a m i n e t h e i n f lu e n c e o f a d d it i v e G a u s s i a n

n o i s e n ( t ) o n t h e c o n t r o l l a b i li t y o f c h a o t i c t r a j ec t o r i es w i t h i n o u r c o n t i n u o u s - t i m e c o n t r o l s c h e m e . I n o r d e r t o a s s e ss

t h is i n f l u e n c e, w e h a v e m o n i t o r e d t h e c o n t r o l l e d t r aj e c to r i e s f o r v a r i o u s s i g n a l t o n o i s e r a ti o s ( S N R ) . 7 W e c a n n o t ,

o n t h e o r e t i c a l g r o u n d s , i n f e r a b o u t a d d i ti v e n o i s e i n f l u e n ce , a n d w e h a v e t h e r e f o r e a s s e s s e d i t b y u s i n g n u m e r i c a l

s i m u l a t i o n s . F i g . 1 1 d i s p l a y s b o t h t h e s t a t e x ( t ) a n d t h e b i a s e d c o n t r o l s i g n a l c ( t ) - l . 5 f o r c l a r it y p u r p o s e s .D e p e n d i n g o n d e f i n it i o n o f c o n t r o ll a b il i ty , w e o b s e r v e t h a t u p to S N R = 3 0 d B t h e r e s u l ti n g c o n t r o l l e d t ra j e c to r i e s

a r e s ti ll c l o s e t o th e n o i s e l e s s o n e s . B u t o b v i o u s l y , f o r S N R a p p r o x i m a t e l y s m a l l e r t h a n 2 0 d B , s t a b i l i z at i o n d o e s

n o t t a k e p l a c e. L o o k i n g a t t h e c o n t r o l s ig n a l c ( t ) w e o b s e r v e , a s m e n t i o n e d p r e v i o u s l y , th a t i t d o e s n o t g o t o z e r o

b u t r e m a i n s p e r i o d i c ( i n n o i s e f r e e e n v i r o n m e n t s ) . T h i s c l e a r l y e s t a b li s h e s e v i d e n c e o f th e f a c t t h a t o u r c o n t r o l le d

t r a je c t o r ie s a r e n o t U P O s o f t h e o r ig i n a l s y s t e m , w h i c h i n th a t c a s e s h o u l d r e s u l t i n c (t ) ---> 0 a s t --+ oo. W h e n

c o n s i d e r i n g t h e l a rg e f e e d b a c k g a i n s i t u a t i o n K = 1 , t h e c o n t r o l l e d o r b i t s h o u l d b e a s t a b l e e q u i l i b r i u m s t a te ( s e e

7 SNR = 10 l og (Px /Pn) , whe re Px is the power o f the state x (t) and Pn is the p ower of the additive noise n (t).



P. Ce lka /P hys ic a D 104 (1997) 1 27-14 7 1 4 5

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t i m e

F i g . 1 2 . E f f e c t o f a d d i t i v e G a u s s i a n n o i s e o n t h e c o n t r o l t r a j e c t o r i e s f o r K = 1 , r / T L ~ _ a n d m = 3 2 w i t h S N R = 2 0 d B .

Fig . 4 ) , bu t in the p resence o f no ise , t he con t ro l resu l t s i n some w ay in the ex t rac t ion o f the no i se co m pon en t as

seen in Fig. 12.

5 . C o n c l u s i o n

W e h a v e a p p l i e d a c o n t i n u o u s - t i me c o n t r o l me t h o d t o a n i n f i n i t e - d i me n s i o n a l s y s t e m mo d e l e d b y a s e c o n d -

o r d e r n o n l i n e a r d e l a y - d i f f e r e n t i a l e q u a t io n ( D D E ) i n o r d e r t o s t a b il iz e t h e s y s t e m o n t o d i f f e r e n t s t a b le s o l u ti o n s . A

s mo o t h e d n o n l i n e a r t e n t ma p w a s i mp l e m e n t e d w i t h o p e r a t i o n a l a mp l i f i e r s in e l e c t ro n i c c i r cu i t, a n d e x p e r i m e n t a l

e v i d e n c e o f t h e e x i s te n c e o f d if f e r e n t s o l u ti o n s b e l o n g i n g t o d i f f e re n t b i f u r c a t io n b r a n c h e s h a s b e e n p e r f o r me d .

A d i s c r e t e - t i me a p p r o x i ma t i o n o f t h e D D E h a s l e d u s t o c o n s i d e r r e l a t i o n s h i p s b e t w e e n D D E a n d d i s c r e t e - t i me

q u a n t i t ie s a n d t o i n t r o d u c e g e n e r a l iz e d m e a n F l o q u e t mu l t i p l ie r s ( MFM s ) a n d me a n L y a p u n o v e x p o n e n t s ( ML E s )

f o r n o n l i n e a r d e l a y - d i f f e r e n t i a l ty p e e q u a t i o n s . W e w e r e a b l e t o c o n f i r m t h e v a l i d i t y o f t w o i n t r o d u c e d c o n j e c t u r e s

c o n c e r n i n g t h e c o n v e r g e n c e o f M FM a n d M L E t o th e i r d i s c r e te c o u n t e r p a r t s a n d t he e x i s te n c e o f a n o p t i ma l L P × L P

s e m i n o r m f o r t h is a p p r o x i ma t i o n . T h e s t a b i l it y r e g i o n s f o r c o n t r o ll e d 2 m- c y c l e s c o mp u t e d f r o m t h e d i sc r e t e m a p

h a v e p r o v e d a g a i n t o b e i n g o o d a g r e e m e n t w i t h e x p e r i m e n t a l l y me a s u r e d s t a b il i ty r e g i o n s f o r c o n t r o ll e d 2 m r -p e r i o d i c s o l u t io n s . Se v e r a l i s o me r s o f h a r mo n i c s o l u ti o n s h a v e b e e n s u c c e s s f u l l y c o n t ro l l e d w i t h o u r m e t h o d .

Co n t r o l l e d s o l u t i o n s w e r e n o t u n s t a b l e p e r i o d i c o r b i t s ( U PO s ) b u t r a t h e r s o me c l o s e l y r e l a t e d s t a b l e o r u n s t a b l e

per iod ic so lu t ions o f the o r ig ina l sys t em. In the case where the sys t em behaves in a chao t i c way , due to no i se , t he

e x p e r i m e n t a l c o n t r o l s ig n a l c ( t ) = K ( x ( t - T ( m , n ) ) - x ( 1 ) ) c o u l d n e v e r b e z e r o a n d t h e c o n t r o ll e d p e r i o d i c o r b i t s

a re l imi t ed to be as c lose as poss ib le to the o r ig ina l . A f i r st d raw bac k o f th i s con t ro l me thod i s t he imp oss ib i l i t y to

d e t e r mi n e i n a d v a n c e a p e r i o d i c o r i b i t t o w h i c h w e w a n t t h e c o n t ro l l e d s y s t e m t o g o , a n d a s e c o n d o n e i s t h e p o s s i b le

g e n e r a t i o n o f n e w s t a b le p e r i o d i c s o l u t i o n s t h a t a r e n e i th e r U PO s n o r c l o s e t o s t a b l e s o l u t io n s o f t h e u n c o n t r o l l e d

s y s t e m. Se v e r a l p r o p o s i t i o n s h a v e b e e n ma d e t o u s e D D E s f o r i n f o r m a t i o n s t o r a g e i n p e r i o d ic s o l u ti o n s [ 8 ,3 71 , b u t ,



14 6 P. Ce l ka /P hys ica D 104 (1997) 12 7-147

d u e t o n o i s e c o n s i d e ra t i o n s, t h e d e v i c e m u s t b e u n d e r c o n t ro l . A s w e h a v e e m p h a s i z e d , t h is i m p l i e s t o c o n s i d e r n e w

s o l u t i o n s g e n e r a t e d b y t h e c o n t r o l l e d s y s t e m a n d t o t a k e in t o a c c o u n t t h e p r e v i o u s l y s t a t ed d r a w b a c k w h e n u s i n g

o u r c o n t r o l m e t h o d .

A c k n o w l e d g e m e n t s

T h e a u t h o r i s g r a t e f u l o f P r o f . M . O g o r z a l e k f o r m a n y f r u i t f u l d i s c u s s i o n s , a n d t o D r . J . - M . V e s i n f o r t h e s u g g e s t i o n s

a n d c o r r e c t i o n s i n t h e m a n u s c r i p t .

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