J. Math. Biology (1982) 14:1 -2 3 Journal of Mathematical Biology 9 Springer-Verlag1982 Phase Locking, Period Doubling Bifurcations and Chaos in a Mathematical Model of a Periodically Driven Oscillator: A Theory for the Entrainment of Biological Oscillators and the Generation of Cardiac Dysrhythmias Michael R . Guevara and Leon Glass Department of Physiology, 3655 D rumm ond Street, McGill University, Montreal, Quebec H3G 1Y6, Canada Abstract. A mathematical model for the perturbation of a biological oscillator by single and periodic impulses is analyzed. In response to a single stimulus the phase of the oscillator is changed. If the new phase following a stimulus is plotted against the old phase the resulting curve is called the phase transition curve or PTC (Pavlidis, 1973). There are two qualitatively different types of phase resetting. Using the terminology of Winfree (1977, 1980), large per- turbations give a type 0 PTC (average slope of the PTC equals zero), whereas small perturbations give a type 1 PTC. The effects of periodic inputs can be analyzed by using the PTC to construct the Poincar6 or phase advance map. Over a limited range of stimulation frequency and amplitude, the Poincar6 map can be reduced to an interval map possessing a single maximum. Over this range there are period doubling bifurcations as well as chaotic dynamics. Numerical and analytical studies of the Poincar6 map show that both phase locked and non-phase locked dynamics occur. We propose that cardiac dysrhythmias may arise from desynchronization of two or more spontaneously oscillating regions of the heart. This hypothesis serves to account for the various forms of atrio- ventricular (AV) block clinicall y observed. In particular 2 : 2 and 4 : 2 AV block can arise by period doubling bifurcations, and intermittent or variable AV block may be due to the complex irregular behavior associated with chaotic dynamics. Key words: Cardiac dysrhythmias - Phase locking - Chaos - Period doubling bifurcations I. Introduction The rhythm of autonomous biological oscillators can be markedly affected by periodic perturbation. Studies of the effects of periodic electrical stimulation of neural and cardiac oscillators have revealed that the intrinsic rhythm may become entrained or phase locked to the periodic stimulus (Perkel et al., 1964;
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J . Ma th . B iolog y (1 9 8 2 ) 1 4 :1 - 2 3J o u r n a l o f
Ma t h ema t i c a l
B i o l o g y9 Springer-Verlag 1982
Ph ase L ock ing, Period D oubl ing Bifurcat ions and Chaos in a
M athem at ica l M odel o f a Per iodica lly Driven Osci lla tor: A
Theo ry for the Entrainmen t of Biolog ical O sci l lators and the
Generat ion of Cardiac Dysrhythmias
M i c h a e l R . G u e v a r a a n d L e o n G l a s s
Dep ar tmen t o f Phy siolog y , 3 6 5 5 D r u m m o n d S tr ee t, McG i l l Un iv e r s i ty , Mo n t r ea l, Qu eb ec H3 G 1 Y6 ,
C a n a d a
A b s t r a c t . A m a t h e m a t i c a l m o d e l f o r t h e p e r t u r b a t i o n o f a b i o lo g i c al o s c il la t o r
b y s i n g le a n d p e r i o d i c i m p u l s e s i s a n a l y z e d . I n r e s p o n s e t o a s i n gl e s t i m u l u s t h e
p h a s e o f t h e o s c i l l a t o r is c h a n g e d . I f th e n e w p h a s e f o l l o w i n g a s t i m u l u s i s
p l o t t e d a g a i n s t t h e o l d p h a s e t h e r e s u l t i n g c u r v e i s c a l l e d t h e p h a s e t r a n s i t i o n
c u r v e o r P T C ( P a v l id i s , 1 9 73 ). T h e r e a r e t w o q u a l i t a t i v e l y d i f f e r e n t ty p e s o f
p h a s e r e s e t ti n g . U s i n g t h e t e r m i n o l o g y o f W i n f r e e ( 19 77 , 1 98 0) , la r g e p e r -
t u r b a t i o n s g i ve a t y p e 0 P T C ( a v e ra g e s l o p e o f t h e P T C e q u a l s ze r o ), w h e r e a s
s m a l l p e r t u r b a t i o n s g i v e a t y p e 1 P T C . T h e e f f ec t s o f p e r io d i c i n p u t s c a n b e
a n a l y z e d b y u s i n g t h e P T C t o c o n s t r u c t t h e P o i n c a r 6 o r p h a s e a d v a n c e m a p .
O v e r a l i m i t e d r a n g e o f s t im u l a t i o n f r e q u e n c y a n d a m p l i t u d e , t h e P o i n c a r 6 m a p
c a n b e r e d u c e d t o a n i n t e r v a l m a p p o s s e ss i n g a s in g le m a x i m u m . O v e r t h is r a n g e
t h e re a r e p e r i o d d o u b l i n g b i f u r c a t io n s a s w e ll as c h a o t i c d y n a m i c s . N u m e r i c a l
a n d a n a l y t i c a l s t u d ie s o f t h e P o i n c a r6 m a p s h o w t h a t b o t h p h a s e l o c k e d a n d
n o n - p h a s e l o c k e d d y n a m i c s o c cu r . W e p r o p o s e t h a t c a r d ia c d y s r h y t h m i a s m a y
a r is e f r o m d e s y n c h r o n i z a t i o n o f t w o o r m o r e s p o n t a n e o u s l y o s c i ll a ti n g r e g io n s
o f t h e h e a r t. T h i s h y p o t h e s i s s e r ve s t o a c c o u n t f o r t h e v a r i o u s f o r m s o f a tr i o -v e n t r i c u l a r ( A V ) b l o c k c l i n ic a l ly o b s e r v e d . I n p a r t i c u l a r 2 : 2 a n d 4 : 2 A V b l o c k
c a n a r i s e b y p e r i o d d o u b l i n g b i f u r c a t i o n s , a n d i n t e r m i t t e n t o r v a r i a b l e A V
b l o c k m a y b e d u e t o t h e c o m p l e x i r r e g u l a r b e h a v i o r a s s o c i a t e d w i t h c h a o t i c
d y n a m i c s .
K e y w o r d s : C a r d i a c d y s r h y t h m i a s - P h a s e l o c k i n g - C h a o s - P e r i o d
d o u b l i n g b i f u r c a t i o n s
I . I n t r o d u c t i o n
T h e r h y t h m o f a u t o n o m o u s b i o l o g i c a l o s c i l l a t o r s c a n b e m a r k e d l y a f f e c t e d b y
p e r i o d i c p e r t u r b a t i o n . S t u d i e s o f t h e e f f e c t s o f p e r i o d i c e l e c t r i c a l s t i m u l a t i o n o f
n e u r a l a n d c a r d i a c o s c i l la t o rs h a v e r e v e a l ed t h a t t h e i n tr i n si c r h y t h m m a y b e c o m e
e n t r a i n e d o r p h a s e l o c k e d t o t h e p e r i o d i c s t i m u l u s ( P e r k e l e t a l . , 1 9 6 4 ;
F i g . l . Sc h e m a t i c r e p r e s e n t a t i o n o f t h e e f f e c t o f
p e r t u r b a t i o n w i t h a n i m p u l s e o f m a g n i t u d e b o n t h e
m o d e l o s c i l l a t o r . T h e u n i t c i r c l e fo r m s a l i m i t c y cl e
w h i c h i s g l o b a l l y a t t r a c t i n g f o r a l l p o i n t s e x c e p t t h e
o r ig i n . T h e p e r t u r b a t i o n i n s t a n t a n e o u s l y re s e ts t h e
p h a s e o f t h e o s c i l l a t o r f r o m p h a s e q5 p r i o r t o t h e
p e r t u r b a t i o n t o p h a s e 0 a f t e r th e p e r t u r b a t i o n . E v e r y
t i m e t h e o s c i l l a t o r p a s s e s t h r o u g h p h a s e q~ = 0 , w e
a s s o c i a t e t h i s w i t h a n o b s e r v a b l e e v e n t s u c h a s a
n e u r a l o r c a r d i a c a c t i o n p o t e n t i a l . I d e n t i f y i n g t h e x -
a x is v a r i a b l e w i t h m e m b r a n e p o t e n t ia l , p e r t u r b a t i o n s
d i r e c t e d a l o n g t h e x - a x i s ( b > 0 ) a r e a n a l o g o u s t o
d e p o l a r i z a t i o n s , w h e r e a s p e r t u r b a t i o n s d i r e c t e d i n t h e
o p p o s i t e d i r e c t i o n ( b < 0 ) a r e a n a l o g o u s t o h y p e r -
p o l a r i z a t i o n s
where the funct ionfi s called the phase transition curve (PTC). In Section III of this
paper we analytically compute the PTC for this model and discuss its properties.
Our choice of the model in Fig. 1 has been motivated by the fact that an analytic
expression can be found for the PTC. This is in sharp contrast to other simple
models of limit cycle oscillations such as the van der Pol oscillator for which
analytical expressions for the PTC are not available (Pavlidis, 1973; Winfree, 1980).
In addit ion to its analytic simplicity, the PTC of (1) displays certain qualitative
features which resemble some experimental observations. Consider a perturbation
directed along the positive x-axis (b > 0). Applying the per turbation during the firsthalf of the cycle (0 < q5 < 0.5) leads to a delay in the phase of the oscillation,
whereas application of the same perturbation in the second half of the cycle
(0.5 < q5 < 1.0) leads to an advance of phase. Also, for this model, the average slope
of the PTC is 1 (type 1 PTC) at low amplitudes of perturbation and 0 (type 0 PTC) at
higher amplitudes. Winfree (1977, 1980) reviewed data resulting from different
experimental preparations and showed that in many situations the PTC's are
biphasic and either type 1 or type 0.
Despite these parallels with experiment, the mathematical model is unrealistic
for many reasons. Since most realistic mathematical models for biological
oscillations are systems of differential equations of dimension greater than 2, thetopological dimension of the proposed system of equations is too low.
Furthermore, it is unrealistic to assume that d~b/d t is independent of r, and that the
relaxation back to the limit cycle is instantaneous. Finally, due to the rotational
symmetry inherent in the model, the PTC shows symmetries (see Section III) which
are no t observed experimentally (Pavlidis, 1973 ; Jalife and M oe, 1979; Scott, 1979;
Guevara et al., 1981). Possible changes in the behaviour of the system which arise
from relaxing one or more of our assumptions to make them more realistic have not
yet been studied.The PTC describes the response of the oscillator to an isolated impulse. The
PTC can be used to predict the response to periodic stimulation. In Section IV weshow how the PTC can be used to derive a mathematical function called the
Poincar6 map. Iteration of the Poincar6 map allows us to study the dynamics of the
model in response to periodic input. Although equivalent procedures have been
previously employed (Perkel et al., 1964; Moe et al., 1977; Pinsker, 1977; Glass
Phase Locking , Per iod Dou bl ing Bi furcat ions and Chaos 5
an d M ack ey , 1979 ; Sco t t , 1979; 1980 ; Lev i , 1981) , t he re ha s no t been a tho ro ug h
a n a l y s is o f t h e d i f f e r e n t p h a s e l o c k i n g b e h a v io r s d i s p l a y e d b y a m a t h e m a t i c a l
m o d e l d i s p l ay i n g b o t h t y p e 0 a n d t y p e 1 P T C ' s .
T h e m a j o r r e s u l t s o f t h i s p a p e r a r e g i v e n i n F i g . 2 w h i c h s h o w s t h e p r i n c i p a l
p h a s e l o c k i n g zo n e s a s a f u n c t i o n o f th e m a g n i t u d e b o f t h e p e r t u r b a t i o n a n d t h e
t i m e r b e t w e e n s u c c e s s iv e s t i m u l i . F i g u r e 2 a s h o w s t h e z o n e s o f N : M p h a s e l o c k i n g
fo r N ~> M , N ~ 3 . F ig u re 2b show s so m e o f the se zon es a s w e l l a s a l l t he 4 : M
l o c k i n g z o n e s o v e r a m o r e l i m i t e d r e g i o n o f ( r , b ) p a r a m e t e r s p a c e. I n t h e u n l a b e l l e d
r e g i o n s o f F i g . 2 , t h e r e a r e o t h e r p h a s e l o c k e d z o n e s , a s w e l l a s p o i n t s a t w h i c h p h a s e
l o c k e d d y n a m i c s d o e s n o t o c c u r ( se e S e c t i o n V ).
2 . 5
2 - 0
1 .5
I b l
I -0
0 - 5
( a )
I:0
i0 - 2 5"0 0 - 5 0
l"
I:1
-2:2
i
0 . 7 5 1 . 0
Fig . 2 . Phase locking zones resu l t ing f romperiodic pulsat i le inputs of magn i tude band f requency r - i . The areas not l abelledcon t a in bo t h phas e l ocked and non -phas elocked dynam ics (see Sect ion V) . (a) Phaselocking zones of the fo rm N ~> M , N ~< 3 .
(b) Phase locking zones of the formN i> M, N ~< 4 ove r a mo re l imi ted reg ionof (z , b) parameter space. Us ing the sym -met ry re la t ions in Sect ion IVB, phaselocking pat terns for o the r values of z canbe generated
In o rde r to i l lus t r a te the d i f f e ren t phase lock ing pa t te rns , we a ssume tha t
crossing of the po si t ive x-axis (i .e . ~b passing t hro ug h 0) corre spo nd s to a n
obse rvab le even t ( fo r exam ple , an ac t ion po ten t ia l) . In F igs. 3 and 4 we show som e
o f t h e d i f fe r e n t c o u p l in g p a t t e rn s b e tw e e n th e p e r io d ic p e r tu r b a t i o n a n d t h e mo d e l
osc i l la to r a t seve ra l d i f f e ren t va lues o f f r equen cy and am pl i tude o f the pe r -
tu rba t ion . The pe r iod ic inpu t pu lses a re r epresen ted in F igs . 3 and 4 by heavy da rk
l ines and the t im es when the m ode l osc i lla to r passes th roug h the phase q5 = 0 ( ca lled
the f i r ing t imes) a re r epresen ted by l igh te r , shor te r l ines. A l l the pa t te rns show n a re
phase lock ed excep t fo r those in F ig . 3b and Fig . 4d . The r hy thm in Fig . 3b is an
exam ple o f quas ipe r iod ic dyn am ics (Glass e t a l. , 1980), and the rhy th m in Fig . 4d i s
a n e x a mp le o f a n i r re g u l a r p a t t e r n a r is in g o u t o f c h a o t i c d y n a m ic s (s e e Se c t io n V ) .
Th ere is a s t r ik ing qua l i ta t ive s imi la r i ty be twe en the pa t te rns obse rved in F igs . 3
a n d 4 a n d c o u p l in g p a t te r n s o b s e r v e d b e tw e e n p e rio d i c i n p u t a n d d r iv e n a c ti v it y i nexp er im enta l pre pa ra t io ns (P erkel e t a l. , 1964; Reid, 1969; Pinsk er , 1977; Ja l i fe an d
M oe , 1979; G ut tm an e t al ., 1980). M an y of the pa t te rn s in F igs. 3 and 4 r e semb le
c l in ica l ly obse rved c a rd iac a r rhy th m ias such a s AV b lock ( see Sec t ion VI) .
(a )
h h I , h L I ,0:0 3 :0
(r
t h I , I i i , l L I0 . 0 3: 0
( b )
I , i h i , I , I , I L i , h I , I I L I I I i , i6:0 0:0 3:0 6 : 0 9 : 0 12.0
(d)
I h I , I i o l o l , I I I , I I I , I I h i I I i6 : 0 9 : 0 - 3 : 0 6 : 0 9 : 0
T i m e
F i g . 3 . S c h e m a t i c r e p r e s e n t a t i o n o f t h e e f f e c ts o f p e r io d i c s t i m u l a t i o n o n t h e m o d e l o s c i l la t o r f o r
b = 0 .9 5 . T h e h e a v y d a r k b a r s h o w s th e p e r i o d i c p u ls a t il e i n p u t a n d t h e s h o r t e r l ig h t b a r s h o w s t h e f ir i n g
t i m e s ( t i m e w h e n o s c i l l a t o r p a s s e s t h r o u g h p h a s e ~b = 0 ) o f t h e m o d e l o s c i l l a t o r . ( a) ~ = 0 . 9 0, 1 : 1 p h a s e
l o c k i n g ; ( b ) T = 0 . 7 5, q u a s i p e r i o d i c d y n a m i c s ; (c ) z = 0 . 7 0, 4 : 3 p h a s e l o c k i n g ; ( d ) z = 0 . 6 5, 3 : 2 p h a s e
l o c k i ng . T h e p a t t e r n s o f ( c) a n d ( d ) d i sp l a y W e n c k e b a c h p e r i o d i c it y , s in c e t h e i n te r v a l b e t w e e n s t i m u l u s
a n d s u c c e e d i n g re s p o n s e g e t s p r o g re s s i v e ly l o n g e r u n t i l t h e d r i v e n o s c i l l a to r s k i p s o r m i s s e s a b e a t
(a) Ibl (e)
h I , h I , I , I , I , [ , I h I , h I i i , I , i~ I , I i h I I h i l , i , i , l h0:0 3 0 6:0 0:0 3:0 6:0 0-0 3:0 6:0
Id)
] ! l I I h l I , I , I h I i ~ L h l k I , i I , i I , I h I h I h I h l i ,0.0 3:0 6:0 9:0 IP.O 15.0 18.0
t e l ( f ) (g )
I , l I , I h L i , I I , L I , I I I I , I I I h l t I , l i , I t , l i , i L I I ,0:0 3:0 6:0 0.0 3:0 6:0 0 0 3:0 6.0
Time
Fi g . 4 . Sa m e as F i g . 3 bu t w i t h b = 1 .13 . ( a ) z = 0 .75 , 1 : 1 pha se l ock i ng ; (b ) ~ = 0 .69 , 2 : 2 ph as e l ock i n g :n o t e t h e a l t e r n a t i o n o f f i r i n g t i m e s ; ( c) 9 = 0 . 6 8, 4 : 4 p h a s e l o c k i n g : n o t e t h a t t h e r e a r e f o u r d i f f e r e n t
f i r in g t im e s ; ( d ) ~ = 0 .6 5 , i rr e g u l a r c o u p l i n g a r is ! n g f r o m c h a o t i c d y n a m i c s : n o t e t h e n a r r o w r a n g e o f
f i r in g t im e s a n d t h e i r r e g u l a r l y s k i p p e d b e a t s o f t h e d r i v e n o s c i l l a t o r ; ( e) ~ = 0 . 6 07 , 4 : 3 p h a s e l o c k i n g :
n o t e t h e a t y p i c a l W e n c k e b a c h p e r i o d i c it y ; (f ) z = 0 .6 0 , 4 : 2 p h a s e l o c k i n g : n o t e a g a i n t h e a l t e r n a t i o n o f
f i r in g t i m e s ; ( g ) ~ = 0 . 55 , 2 : 1 p h a s e l o c k i n g
P h a s e L o c k i n g , P e r i o d D o u b l i n g B i f u rc a t io n s a n d C h a o s 7
I l L T h e P h a s e T r a n s i t io n C u r v e ( P T C ) a n d I ts P r o p e r ti e s
T h e P T C g iv e s t h e n e w p h a s e 0 a s a fu n c t i o n o f t h e o l d p h a s e 4 ', f o l l o w i n g a
p e r t u r b a t i o n o f a m p l i t u d e b (F i g . 1 ). T h e c o m p u t a t i o n s w h i c h f o l l o w r e q u i r ea n a l y t ic a l e x p r e s s i o n s f o r t h e P T C . T h e P T C is w r i t te n u s i n g t h e p r in c i p a l v a l u e s o f
t h e i n v e rs e t a n g e n t a n d i n v e rs e c o s i n e f u n c t i o n s , d e n o t e d b y t a n - 1 x a n d c o s - 1 x
r e s p e c t i v e l y . F o r - ~ < x < ~ , - z ~ / 2 < t a n l x < ~ / 2 a n d f o r - 1 < x < 1 ,
0 < c o s - i x < z . F r o m t h e c o n s t r u c t i o n i n F ig . 1 t h e P T C is r e a d i ly c o m p u t e d .
D e f i n e
( s i nZ ~ z4 ' ) ( 4a )= z ~ z t a n - l \ c ~ o s 2 ~ - b '
]~ = l c o s - l ( - b ). ( 4 b )2 ~
W e f i n d
f( 4 , , b) = ~, 0 ~ 4, ~</?, ]b[ ~< 1, (5 a)
0 ~< 4, ~< 0.5, b > l ,
f( 4, , b) = 0.5 + c~, f l < 4 ' < 1 - / ~ , Ibl ~< 1, (5 b)
0 - . .< 4 ' < 1 , b < - 1 ,
f (4 ' , b) = 1.0 + c~, 1 - fl ~< 4' < 1, Ibl ~< 1, (5 c)
0 . 5 < 4 ' < 1 , b > l .
F i g u r e 5 s h o w s e x a m p l e s o f t y p e 1 P T C s ( w h i c h a r e f o u n d f o r Ib[ < 1 ) a n d t y p e 0
P T C s ( w h i c h a r e f o u n d f o r [bl > 1 ). T h e P T C s a r e c o n t i n u o u s o n t h e u n i t c ir c le a n d
h a v e c o n t i n u o u s d e r i v a ti v e s o f al l o r d e r s o n t h e u n i t c ir cle . M a t h e m a t i c a l m o d e l s
d i s p l a y i n g P T C s w h i c h a r e d i s c o n t i n u o u s o n t h e u n i t c i r c l e h a v e b e e n d e s c r i b e d
( G l a s s a n d M a c k e y , i 9 7 9 ; K e e n e r , 1 9 8 0 ; K e e n e r e t a l. , 1 9 8 1) .
T h e P T C s g i v en b y ( 5) a n d s h o w n i n F i g . 5, d i s p l a y th e f o l lo w i n g s y m m e t r ie s
f ( 1 - 4 ,, b ) = 1 - f ( 4 , , b ) , ( 6 a )f ( 4 , + 0 . 5 , - b ) = f ( 4 , , b ) + 0 . 5 , 4 , ~< 0 . 5 , ( 6 b )
f (4 , - 0 . 5 , - b ) = f ( 4 , , b ) - 0 . 5 , 4 ) ~> 0 . 5 . (6c )
F i g . 5. T h e p h a s e t r a n s i t i o n c u r v e ( P T C ) o f t h e m o d e l o s c i ll a t o r
f o r s e v e r a l v a l u e s o f b. F o r Ib < 1 t h e P T C h a s a v e r a g e s l o p e 1
a n d i s c a l l e d a t y p e 1 P T C , w h e r e a s f o r Ib[ > 1 t h e P T C h a s
a v e r a g e s l o p e 0 a n d i s c a l l e d a t y p e 0 P T C
Phase Lock ing, Period Dou bling Bifurcations and Chaos 9
Fig. 6. The Poincar6 map for b = - 1.30 and threevalues of
1.0
( • L + I0 ' 5
0 .0
c> /
0 . 5 1 .0
d e f i n e d b y ( 1 1 ) is c a l le d t h e P o i n c a r 6 o r p h a s e a d v a n c e m a p . F i g u r e 6 sh o w s t h e
P o i n c a r 6 m a p f o r b = - 1 .3 0 a n d f o r t h r e e v a l u e s o f r . T h i s f i g u r e i l lu s t r a t e s t h a t
v e r ti c a l t r a n s l a t i o n o f t h e P T C b y a n a m o u n t r g iv e s t h e P o i n c a r 6 m a p .
I t e r a ti o n o f th e P o i n c a r 6 m a p c a n b e u s e d t o c o m p u t e t h e e v o l u t io n o f t h e p h a s e
i n r e s p o n s e t o p e r i o d i c i n p u t ( A r n o l d , 1 9 73 ; G l a s s a n d M a c k e y , 1 9 79 ;
G u c k e n h e i m e r , 1 9 8 0 ; L e v i , 1 9 8 1 )
= T ( 4 ' 0 ,
~b,+m = T (q ~ , +z -1 )= T ' (~b , ) . ( 13 )
A p h a s e 0 * i s c a l le d a f ix e d p o i n t o f p e r i o d N i f
TN(dP*) = q~*' (14 )
T'(q~*) # q~*, 1 < i < N .
A f i x e d p o i n t 4 )* o f p e r i o d N is s t a b le i f
0TN(qS')~?~, *,=** < 1. (1 5 )
A c o n c e p t w h i c h h a s b e e n u s e f u l i n t h e s t u d y o f p h a s e l o c k i n g i s t h e r o t a t i o n
n u m b e r , O , d e f i n e d a s t h e a v e r a g e a d v a n c e i n p h a s e o f th e d r i v e n o s c i l l a to r f o r e a c hc y c le o f t h e p e r io d i c i n p u t . T h e r o t a t i o n n u m b e r is d e f i n e d
1 s
p = l im ~ aqS,, (16)J~azJ i=l
w he re A~b, i s g ive n in (10) .
S t a b l e f i x e d p o i n t s o n t h e P o i n c a r 6 m a p a r e a s s o c i a t e d w i t h p h a s e l o c k e d
d y n a m i c s . A s s u m e t h a t f o r s o m e v a l u e o f (7 , b ), t h e r e a r e N s t a b l e f ix e d p o i n t s o f
pe r i od N , q~o ,01 . . . . , q~N = q~o w he re ~bi+l = T (O i ) a s b e f o r e . D e f i n e
N
M = ~ A49,. (17), =1
Si nce ~bN = ~bo , M m us t be an i n t eg e r an d t he s equ enc e is a s soc i a t ed w i t h N : M
p h a s e l o c k i n g w i t h r o t a t i o n n u m b e r p = M / N . C o n s e q u e n t l y , f o r p h a s e l o c k e d
p a t t e r n s , t h e r o t a t i o n n u m b e r i s r a t i o n a l . N o t e t h a t p d o e s n o t s p e c i f y a u n i q u e
p h a s e l o c k i n g p a t t e r n . F o r e x a m p l e , b o t h 1 : 1 a n d 2 : 2 l o c k i n g h a v e p - - 1 a n d b o t h
2 : 1 a n d 4 : 2 p h a s e l o c k i n g h a v e p = 89 A l s o , f o r f i x e d v a l u e s o f b a n d z , p m a y n o t
b e u n i q u e a n d m a y d e p e n d o n t h e in i t i a l p h a s e .
I V B . S y m m e t r i e s o f t he P h a s e L o c k i n g Z o n e s
A s a c o n s e q u e n c e o f t h e s y m m e t r i e s o f t h e P T C d e t a i le d i n (6 ), t h e r e a r e s y m m e t r i e s
in th e p h a s e l o c k in g z o n e s . T h e s e s y m m e t r i e s a r e s u m m a r i z e d b y t h e f o l lo w i n g
p r o p o s i t i o n s , t h e f i r s t o f w h i c h is p r o v e n i n A p p e n d i x A .
P r o p o s i t i o n 1 . A s s u m e t h a t t h e re i s a s t a b le p e r i o d N c y c le w i th f i x e d p o i n t s
~bo , ~b1 . . . , ~ bN - i f o r b = Y , z = 0 .5 - 6 ( 0 < 6 < 0 .5 ) a s s o ci a te d w it h a n N : M p h a s el o c k in g p a t t e r n . T h e n , f o r z = 0 .5 + 3 , t here w i l l be a s t ab l e cyc l e o f per iod N
a s s o ci a te d w i th a n N : N - M p h a s e lo c k in g r a ti o. T h e N f i x e d p o i n t s
~o, ~ i , . . . , ~bN- i o f th i s cy c le are given by
qh = 1 - ~bi. (1 8)
Proposition 2 . S u p p o s e t h a t f o r b = 7 , z = 6 (0 < 6 < 1) there i s a s table cyc le o f
p e r i o d N w i t h f i x e d p o i n t s (a o, ~ i , . . . , ~ n - i a s s o c i a t e d w i t h a n N : M p h a s e l o c k in g
pa t t e rn . Th en , f or b = - 7 , z = 3 , t here w i l l be a st ab l e cyc l e o f per iod N as soc ia t ed
w i t h a n N : M p h a s e l o c k in g p a t t e r n w i t h N f i x e d p o i n t s ~ b o, ~ bi . . . . , ~ n - 1 w here
~ i = q~i + 0 .5 ,
~ i = q~i - 0 .5 ,
0 < qSi < 0. 5,
0. 5 ~< qSi < 1.0 .( 1 9 )
Proposition 3 . S u p p o s e t he r e i s a s t a b le c y c le o f p e r i o d N w i t h f i x e d p o i n t s
(ao, ~ 1 , . . . , ~ N - i f o r b = 7 , z = 6 (0 < 6 < 1) a s s o ci a te d w it h a n N : M p h a s e l o ck i ng
pa t t e rn . Then t here w i l l be a st ab l e cyc le o f per io d N w i th f i x ed p o in t s
q~ o, ~ b l , . . . , ~ N - i fo r b = 7 , z = 6 + K (where K i s any pos i t i ve in teger) ass ociate d
w i t h a n N : M + K N p h a s e l o c k in g r a t io .
I V C . T h e P o in c a rb M a p a s a n In t e rv a l M a p w i th a S in g le M a x i m u m
O v e r c e r t a in r e g i o n s o f t h e (z , b ) p a r a m e t e r s p a c e , th e P o i n c a r 6 m a p r e d u c e s t o a
m a p d e f in e d o n a n i n t e rv a l in w h i c h t h e r e is a s i ng le m a x i m u m o f t h e m a p . I n t e r v a l
m a p s w i t h a s in g le m a x i m u m h a v e b e e n t h e f o c u s o f c o n s i d e r a b le a n a l y s i s r e ce n t ly
( L i a n d Y o r k e , 1 9 7 5; M a y , 1 9 7 6 ; G u c k e n h e i m e r , 1 9 77 ; g t e f a n , 1 9 7 7) . T h e
b o u n d a r i e s o f t h e p h a s e l o c k i n g z o n e s s h o w n i n F i g . 2 c a n b e p a r t i a l l y u n d e r s t o o d
b y c o n s i d e r i n g t h e b i f u r c a t i o n s o f i n t e r v a l m a p s .
T h e r e d u c t i o n o f t h e P o i n c a r 6 m a p t o a n in t e r v a l m a p w i t h a s in g le m a x i m u m
c a n b e i l lu s t r a te d b y c o n s i d e r i n g a n e x a m p l e . F i g u r e 6 s h o w s t h e P o i n c a r 6 m a p f o r= 0 . 3 5 , b = - 1 .3 0 . F o r t h i s s i t u a t i o n , T : [ 0 , 1 ] ~ I -0 . 71 0 3 , 0 . 9 8 9 7 ] a n d i n t h i s
i n t e r v a l T h a s a s in g l e m a x i m u m ( F i g . 7 ). I n g e n e r a l , f o r b < - 1 a n d 0 ~< z ~< 0 .5
t h e f ir s t i t e ra t e o f t h e P o i n c a r 6 m a p w i ll b e a n i n v a r i a n t i n t e r v a l m a p w i t h a s in g l e
m a x i m u m p r o v i d e d t h e f o l lo w i n g i n eq u a l i t i e s h o l d :
P h a s e L o c k i n g , P e r i o d D o u b l i n g B i f u r c a t i o n s a n d C h a o s 11
I '0
0 " 9
C j~ L + I
0 - 8
0.7
0 . 7 0 ' 8 0 " 9 I -0
4 , ,
F i g . 7 . T h e P o i n c a r 6 m a p f o r ~ = 0 . 3 5 a n d
t h r e e v a l u e s o f b . F o r e a c h v a l u e o f b al li n i ti a l p h a s e s b e t w e e n 0 a n d 1 w i ll m a p
i n t o t h e i n v a r i a n t r e g i o n s h o w n f o l l o w i ng
o n e i te r a t io n o f t h e P o i n c a r 6 m a p . F o r
e a s e o f p r e s e n t a t i o n , q ~i+ l i s n o t g i v e n
m o d u l o 1
Z -5
2" 0
1.5
I b l
I- 0
0 . 5
i i
0 " 0 0 " 2 5 0 " 5 0 0 " 7 5 I ' 0
T
F i g . 8 . R e p r e s e n t a t i o n o f th e t h r e e r e g i o n s i n w h i c h q u a l i t a t iv e l y d i f f e r en t d y n a m i c s o c c u r . I n t h e
d i a g o n a l ly s t r i p ed a r e a t h e P o i n c a r 6 m a p r e d u c e s to a n i n t er v a l m a p w i t h a s i n gl e m a x i m u m ( S e c ti o n
I V C ) . F o r IbJ < 1 ( r e g i o n I) t h e P o i n c a r 6 m a p i s m o n o t o n i c , c o n t i n u o u s a n d d i f f e r e n t ia b l e o n t h e u n i t
c ir c le . I n t h i s re g i o n t h e r e w i l l e i t h e r b e q u a s i p e r i o d i c o r p h a s e l o c k e d d y n a m i c s . F o r [b] > 1 t h e r e a r e t w oq u a l i t a t i v e l y d i f f e r e n t r e g i o n s w i t h r e s p e c t t o t h e t y p e s o f t r a n s i t i o n s b e t w e e n n e i g h b o r i n g p h a s e l o c k i n g
z o n e s . R e g i o n I I I i s c o m p r i s e d o f t h e s m a l l s t ip p l e d a r e a s a s w e l l a s t h e 3 : 1 a n d 3 : 2 z o n e s , w h e r e a s
r e g i o n I I i s t h e r e m a i n d e r o f t h e a r e a f o r Ib] > 1. I n r e g i o n I I a r e f o u n d p e r i o d d o u b l i n g b i f u r c a t i o n s
l e a d i n g to p h a s e l o c k i n g r a t i o s o f t h e f o r m 2 i : M . T h e c o m p l e x d y n a m i c s o f r eg i o n I I I a r e d i s c u s s e d in t h e
A p p l y i n g s o m e s i m p l e t r ig o n o m e t r y , t h e i n e q u a li ti e s c a n b e r e w r i t t e n u s in g ( 8 ) a n d
( 9 )
z > 0 .25, (21a )
1 1 s i n - 1 b2 - 2< 2 - 2~ b2 , (21b)
w h e r e s i n - i x d e n o t e s t h e p r i n c i p a l v a lu e o f t h e in v e r s e s in e f u n c t i o n
( - n / 2 < s i n - 1 x < + r e /2 f o r - 1 < x < 1 ). F r o m t h e s y m m e t r i e s i n ( 6 ) , t h e
P o i n c a r 6 m a p i n o t h e r r e g i o n s o f (z , b ) p a r a m e t e r s p a c e c a n a l so b e r e d u c e d t o a n
i n t e rv a l m a p w i t h a s i n g le m a x i m u m . I n F i g . 8 t h e r e g i o n s o f (z , b ) p a r a m e t e r s p a c e
i n w h i c h t h e P o i n c a r 6 m a p r e d u c e s to a n i n t e r v a l m a p w i t h a si ng le m a x i m u m a r e
s h o w n b y d i a g o n a l s t r i p e s .
In F i g . 7 w e sho w t he Po i nc a r6 m ap fo r z = 0 . 35 , b = - 1 .1 , - 1 .3 , - 1 .5 ov e r
t h e re g i o n in w h i c h t h e m a p i s a n i n t e r v a l m a p w i t h a si ng le m a x i m u m . N o t e t h a t a s
Ibl a p p r o a c h e s 1 , t h e m a p b e c o m e s p r o g r e s s iv e l y st e e p e r i n t h e n e i g h b o r h o o d o f th e
f i x e d p o i n t ~bi+ 1 = ~bi. N o t e f r o m F i g . 6 t h a t c h a n g e s i n r w h i l e k e e p i n g b c o n s t a n t
a l s o c a n l e a d t o c h a n g e s i n t h e s l o p e o f t h e m a p a t t h e f i x e d p o i n t . A s w e w i ll s h o w
b e l o w , a p r o g r e s s i v e s t e e p e n i n g o f t h e s l o p e a t t h e f i x e d p o i n t i s a s s o c i a t e d w i t h a
s e q u e n c e o f p e r i o d d o u b l i n g b i f u r c a t i o n s t h a t l e a d to c h a o t i c d y n a m i c s .
V . P h a s e L o c k i n g Z o n e s i n D i f fe r e n t R e g i o n s o f ( ~ , b ) P a r a m e t e r S p a c e
O u r m a j o r g o a l h a s b e e n t o p r o v i d e q u a l i t a t iv e a n d q u a n t i t a t i v e a n a l y s is o f t h e
d i f f e r e n t t y p e s o f d y n a m i c s t h a t a r is e f r o m p u l s at il e p e r i o d i c i n p u t s t o a s i m p l e t w o
d i m e n s i o n a l l i m i t c y c l e o s c i l l a t o r ( F ig . 1 ). T h e a n a l y s i s h a s i n v o l v e d n u m e r i c a l
s t u d y o f t h e e f f e c ts o f i te r a t i o n o f t h e P o i n c a r ~ m a p a t d i f f e re n t v a l u e s o f ( z, b ) a s
w e l l a s a n a l y t i c d e r i v a t i o n o f b i f u r c a t i o n b o u n d a r i e s w h e n t h is h a s b e e n f e a s ib l e.T h e a n a l y t ic c o m p u t a t i o n s a r e d i sc u s s ed in A p p e n d i x B . T h e s e c o m p u t a t i o n s w e r e
c r o s s - c h e c k e d w i t h n u m e r i c a l c o m p u t a t i o n . F o r a g i ve n v M u e o f b , t h e v a lu e s o f t a t
t h e b o u n d a r i e s o f t h e p h a s e l o c k i n g z o n e s a r e a c c u r a t e to + 0 .0 1 i n F ig . 2 a a n d t o
+ 0 . 003 i n F i g . 2b .
T h e a n a l y si s s h o w s t h a t t h e r e a r e t h r e e d i f fe r e n t re g i o n s o f (z , b ) p a r a m e t e r
s p a c e t h a t c a n b e i d e n t if i e d w i t h r e s p e c t t o t h e t y p e s o f t r a n s i ti o n s b e t w e e n
n e i g h b o r i n g p h a s e l o c k i n g z o n e s . T h e t h r e e r e g i o n s a r e s h o w n i n F ig . 8 . T h e t y p e s
o f b i f u r c a t i o n s t h a t o c c u r i n e a c h r e g i o n w i ll n o w b e b r i e f ly d is c u ss e d .
V A . R e g i o n I : ]bJ < 1
I n t h is r e g io n t h e P o i n c a r 6 m a p is a c o n t i n u o u s , m o n o t o n i c a n d d i f fe r e n t ia b l e m a p
o f t h e u n i t c i r c le o n t o i ts e lf . T h e q u a l i t a ti v e p r o p e r t i e s o f th e d y n a m i c s c a n b e
c o m p l e t e l y d e s c r i b e d b y t h e r o t a t i o n n u m b e r p . I f p i s r a t i o n a l t h e n t h e r e is
Phase Locking, Period Doubling Bifurcations and Chaos 13
s t a b l e p h a s e l o c k i n g ( F i g s . 3 a , 3 c , 3 d ) w h e r e a s i f p is i r r a t i o n a l , q u a s i p e r i o d i c
d y n a m i c s r e s u l t ( F ig . 3 b ) . I n q u a s i p e r i o d i c d y n a m i c s , t h e r e is a c o n t i n u a l s h i ft o f t h e
f i r i n g t i m e w i t h r e s p e c t t o t h e s t i m u l u s .
I n r e g i o n I , t h e r o t a t i o n n u m b e r i s a c o n t i n u o u s f u n c t i o n o f "c a n d b w h i c h is
p i ec e w i se c o n s t a n t o v e r t h e s e t o f r a t io n a l s . T h e m e a s u r e o f t h e s e t o n w h i c h t h e
r o t a t i o n n u m b e r is i r r a ti o n a l i s p o s i ti v e ( H e r m a n , 19 77 ). T h u s , t h e d y n a m i c s f o u n d
i n re g i o n I a r e to p o l o g i c a l l y e q u i v a le n t t o t h e d y n a m i c s f o u n d i n o t h e r m o d e l s o f
p h a s e l o c k i n g in t h e li m i t o f " s m a l l " p e r t u r b a t i o n s ( A r n o l d , 1 9 65 ; K n i g h t , 1 97 2 ;
G l a s s a n d M a c k e y , 1 9 7 9 ; K e e n e r e t a l. , 1 98 1). T h e s e r e f e re n c e s s h o u l d b e c o n s u l t e d
f o r a f u r t h e r d i s c u s s i o n o f th e d y n a m i c s i n r e g i o n I .
Fig. 9. The stable fixed points as a function of b for v = 0.35. Onlythe first two p eriod doubling bifurcations and the period 3 orbitare show n. Lying in the blank space between the p eriod 4 orbitand the period 3 orbit are orbits of all other periods (May , 1976;Guck enheim er, 1977). For ease of p resentation, q5 is not givenmodulo 1
V B . R e g i o n H : P e r i o d D o u b l i n 9 B i f u r c a t i o n s w i t h Jb[ > 1
I n r e g i o n I I , t h e r e a r e o r b i t s o f p e r i o d 2 i ( i a n o n - n e g a t i v e i n t e g e r) . T h e r e a r e t w o
d i f fe r e n t ty p e s o f b o u n d a r i e s b e t w e e n a d j a c e n t p h a s e l o c k i n g z o n e s : b o u n d a r i e s
d u e t o p e r i o d d o u b l i n g b i f u r c a ti o n s , a n d b o u n d a r i e s d u e t o c h a n g e s i n t h e r o t a t i o n
n u m b e r . T h e a n a l y t ic c o m p u t a t i o n o f b o u n d a r i e s i n r e g i o n I I i s d i sc u s se d i n d e ta i l
i n A p p e n d i x B .
O n a p e r io d d o u b l i n g b o u n d a r y , t h e r e is a p e r i o d d o u b l i n g b i f u r c a t i o n o f t h e
a s s o c ia t e d P o i n c a r6 m a p w i t h o u t a n y c h a n g e in r o t a t i o n n u m b e r . F o r e x a m p l e ,
s u c h p e r i o d d o u b l i n g b i f u r c a t io n s a r e r e s p o n s ib l e f o r t h e b o r d e r s b e t w e e n t h e 1 : 0
a n d 2 : 0 z o n e s a n d t h e 1 : 1 a n d 2 : 2 z o n e s o f F i g . 2 a , a n d b e t w e e n t h e 2 : 1 a n d 4 : 2
z o n e s a n d t h e 2 : 2 a n d 4 : 4 z o n e s o f F i g . 2 b . I f q~* i s a f i x e d p o i n t o f t h e P o i n c a r 6m a p T N, t h e n a p e r i o d d o u b l i n g ( o r " p i t c h f o r k " ) b i f u r c a t i o n t o a n o r b i t o f p e r io d
2 N w i l l a r i s e w h e n
0TN = - 1 . (22)
I n F i g . 9 w e s h o w t h e s t a b l e f i x e d p o i n t s f o r ~ = 0 . 35 a s a f u n c t i o n o f b. F o r
- 1 . 8 < b < - 1 .0 t h e r e i s t h e f a m i l i a r c a s c a d i n g s e q u e n c e o f fi x e d p o i n t s
c u l m i n a t i n g i n th e e m e r g e n c e o f a n o r b i t o f p e r i o d 3 , a n a l o g o u s t o t h e s e q u e n c es
f o u n d i n s tu d i es o f th e i t e r a ti o n o f o n e p a r a m e t e r f a m i l ie s o f i n t e rv a l m a p s w i t h a
s in g le m a x i m u m ( M a y , 1 9 76 ; G u c k e n h e i m e r , 1 97 7).
D e l i c a t e l y i n t e r l a c e d w i t h t h e s e p e r i o d d o u b l i n g b o u n d a r i e s a r e b o u n d a r i e s
a c r o s s w h i c h t h e r o t a t i o n n u m b e r c h a n g e s , b u t a c r o s s w h i c h t h e n u m b e r o f f ix e d
p o i n ts o n t h e P o i n c a r6 m a p r e m a i n s c o n s ta n t . F o r e x a m p l e , s uc h b o u n d a r i e s o c c u r
b e t w e e n t h e 1 : 0 a n d 1 9 1 zon es (F i g . 2a ) , t h e 2 : 1 an d 2" 2 zon es (F i g . 2a , 2b ) , t he
4" 2 and 4 : 3 zone s (F i g . 2b ) , a nd t he 4 : 3 and 4 : 4 zon es (F i g . 2b ) . I f ~b* i s a f i xed
p o i n t o f t h e m a p T N f o r b < - 1 , t h e n s u c h a b o u n d a r y o c c u r s w h e n
q~* = 0. (23 )
N ot e t ha t in F i g . 9, ~b* = 0 ( = 1 , m o d 1 ) f o r b -~ - 1 .220, ~ = 0 . 35 . Th i s po i n t li e s
o n t h e b o u n d a r y o f t h e 4 " 0 a n d 4 1 p h a s e l o c k i n g re g i o n s.
T h e t w o d i s t in c t t y p e s o f b o u n d a r i e s i n t e r s e c t a t s i n g u l a r p o i n t s a t w h i c h ( 22 )a n d ( 2 3 ) a r e s i m u l t a n e o u s l y s a t i s f i e d . T h e s i n g u l a r p o i n t s r e p r e s e n t p o i n t s o n t h e
c o m m o n b o u n d a r i e s o f f iv e d if f e r e n t p h a s e l o c k i n g z o n e s . T h e p o i n t [b[ = 2 ,
= 0 . 5 , w h i c h i s o n t h e b o u n d a r i e s o f th e 1 " 0 , 1 : 1 , 2 : 2 , 2 : 1 , a n d 2 " 0 p h a s e l o c k i n g
r e g i o n s ( F i g . 2 a ) r e p r e s e n t s s u c h a s i n g u l a r p o i n t . A p p l y i n g ( 2 2 ) a n d ( 2 3) e n a b l e s u s
t o co m pu t e t he s i ngu l a r p o i n t a t Ib[ = 2 , z = 0 . 5 a s w e l l a s a pa i r o f s i ngu l a r po i n t s
loc ate d a t Ibh = 51/2 - 1 , z -~ 0 .5 +__ 0 .14 (Fig . 2b , A pp en di x B ) . In a dd i t io n to the se
s i n g u l a r p o i n t s , n u m e r i c a l s t u d i e s s h o w t h a t t h e r e a r e f o u r a d d i t i o n a l s i n g u l a r
po i n t s a t w h i ch pe r i od 8 o rb i t s a r i s e l oc a t ed a t t he va l ues Ibh ~ 1 . 17 , r ~ - 0 . 5 _ 0 . 16
and [b[ -~ 1.06, ~ -~ 0.5 + 0.10.
O n t h e b a s is o f th e s e re s u lt s, w e p r o p o s e t h a t t h e t o p o l o g y o f t h e d i f fe r e n t p h a s el o c k i n g z o n e s a n d s i n g u la r p o i n t s i n r e g io n I I c a n b e d e p i c t e d u s i n g t h e g r a p h i c a l
r e p r e s e n t a t i o n s h o w n i n F ig . 1 0. T h i s g r a p h is t h e g e o m e t r ic d u a l o f t h e p h a s e
l o c k i n g z o n e s i n th e p e r i o d d o u b l i n g r e g i o n o f F ig . 2 . T h u s v e r ti c e s o f t h e g r a p h
r e p r e s e n t t h e p h a s e l o c k i n g z o n e s , w h i le t h e e d g e s o f t h e g r a p h r e p r e s e n t t h e
b o u n d a r i e s b e t w e e n p h a s e l o c k i n g z o n e s. N o t e t h a t e a c h r o w o f th e g r a p h i s r e l a t e d
t o t h e r o w s a b o v e i t b y p e r i o d d o u b l i n g b i f u r c a t i o n s . W i t h i n e a c h r o w t h e r o t a t i o n
n u m b e r i n c r e a s es a s o n e p r o g r e s s e s f r o m l e f t t o r i g h t. N u m e r i c a l s t u d i es h a v e n o t
b e e n p e r f o r m e d t o c h e c k t h e v a l i d i ty o f th e e x t e n s i o n o f t h e s c h e m e i n F i g . 1 0 t o
orb i t s o f p er i od 2% i >~ 4 .
V C . R e g i o n I I I : S t i p p l e d A r e a s P l u s 3 : 1 a n d 3 " 2 Z o n e s in F ig . 8
I n r e g i o n I I I , t h e r e i s a n o v e r l a p b e t w e e n t h e r e g i o n s w h e r e t h e P o i n c a r 6 m a p
r e d u c e s t o a n i n t e r v a l m a p w i t h a s i ng le m a x i m u m ( t h e d i a g o n a l l y s t r ip e d r e g i o n s o f
Phase Locking, Period Do ubling Bifurcations and Chaos 15
F i g . 8 ) a n d t h e r e g i o n s w h e r e s t a b l e p e r i o d 3 o r b i t s e x i st ( c o r r e s p o n d i n g t o 3 : 1 a n d
3 : 2 p h a s e l o c k i n g ) . A c o n s e q u e n c e o f t h i s f a c t is t h a t " c h a o s " e x i s t s a t v a l u e s o f
(z, b ) wh ere such a pe r iod 3 o rb i t ex i s ts , in tha t the re a re in i t i a l phase s s t a r t ing f ro m
w h i c h i t e r a t i o n o f t h e P o i n c a r 6 m a p w i ll p r o d u c e o r b i ts o f a r b i t r a ri l y l ar g e p e r i o d a s
we l l a s ape r iod ic o rb i t s (ga rko vsk i i , 1964 ; g te f an , 1975; L i an d Yo rke , 1975).
H o w e v e r , t h e s e o r b i t s a r e u n s t a b l e , a n d t h e s o l e a t t r a c t i n g o r b i t i s t h e o n e o f p e r i o d
3 . T h u s , i n a p h y s i c a l o r b i o l o g i c a l s y s t e m , o n e w o u l d n o t o b s e r v e a n y o f t h e i n f i n i t y
o f u n s t a b l e o r b i ts , b u t o n l y t h e a t t r a c t i n g o n e o f p e r io d 3 .
W e e x p e c t t h a t t h e a r e a o f (z , b ) s p ac e l y i n g b e t w e e n t h e p e r i o d d o u b l i n g z o n e o f
r e g i o n I I a n d t h e p e r i o d 3 o r b i ts ( i.e . t h e s t i p p l e d a r e a o f F i g . 8 ) w i l l c o n t a i n N : M
p h a s e l o c k i n g z o n e s w i t h a r b i t r a r y N , a s w e l l a s p o i n t s w h e r e a p e r i o d i c p a t t e r n s
e x i st ( M e t r o p o l i s e t a l ., 1 9 73 ; M a y , 1 9 76 ; G u c k e n h e i m e r , 1 9 77 ; H o p p e n s t e a d t ,
1 98 1). F o r m o s t v a l u e s o f (~ , b ) in t h i s r e g i o n , w e f i n d c o m p l i c a t e d d y n a m i c s w i t h a ni r r e g u l a r a p p e a r a n c e s u c h a s t h a t s h o w n i n F i g . 4 d ( b - - 1 .1 3, z = 0.6 5). T h i s
p a t t e r n s h o w s n o s i g n o f p e r i o d i c i t y a f t e r 1 0 00 i t e r a t io n s . N o t e t h a t t h e i r r e g u l a r i t y
is m a n i f e s t b y a s e e m i n g l y r a n d o m d r o p p i n g o r s k i p p i n g o f b e a ts o f t h e d r i v e n
o s c i l la t o r , e v e n t h o u g h t h e i n t e r v a l s b e t w e e n s t i m u l i a n d f i r in g t im e s a r e r e s t r i c t e d
t o a c o m p a r a t i v e l y n a r r o w r a n g e .
T h e t e r m " c h a o t i c d y n a m i c s " is a p p r o p r i a t e t o d e s cr ib e t h e e x tr e m e l y c o m p l e x
b e h a v i o u r d i s p l a y e d b y th e m o d e l i n t h e s t i p p le d a r e a o f F ig . 8 . A s d i s c u s s e d a b o v e ,
t h e p a t t e r n g e n e r a t e d a t a p a r t i c u l a r v a l u e o f (z , b ) in t h i s a r e a i s i n g e n e r a l v e r y
c o m p l i c a t e d a n d m a y e v e n b e a p e r i o d i c. I n a d d i t i o n , s i nc e q u i t e s m a l l c h a n g e s in
o r b a w a y f r o m t h a t p a r t i c u l a r v a l u e o f (z , b ) w il l c a u s e a c h a n g e i n t h e p a t t e r no b s e r v ed , t h e e f f e c t o f f l u c t u a t i o n s o r " n o i s e " i n h e r e n t t o a n y e x p e r i m e n t a l s y s t e m
w i l l b e t o d e s t r o y p a t t e r n s i n z o n e s w i t h h i g h o r d e r p h a s e l o c k i n g r a t i o s ( C r u t c h f i e l d
a n d H u b e r m a n , 1 98 0; G l a s s e t a l. , 1 98 0; G u t t m a n e t a l ., 1 9 80 ). T h u s , h i g h e r o r d e r
z o n e s t h a t c o v e r s m a l l e n o u g h a r e a s o f (z , b ) p a r a m e t e r s p a c e w i ll n o t b e o b s e r v e d i n
t h e l a b o r a t o r y . I n s t e a d , t h e y w i ll b e r e p la c e d b y z o n e s t h a t a r e n o t p h a s e l o c k e d a n d
w h i c h d i s p la y i r r e g u l ar d y n a m i c s .
O r b i t s i n r e g i o n I I I ( s u c h a s t h e p e r i o d 3 o r b i t ) c a n a r i s e f r o m t a n g e n t
b i f u r c a t i o n s ( se e A p p e n d i x B ). T h e d e t a i l e d t o p o l o g y o f t h e p h a s e l o c k i n g z o n e s in
t h i s r e g i o n i s n o t w e l l u n d e r s t o o d .
V I . D i s c u s s i o n
W e a r e i n t e r e st e d i n m e c h a n i s m s f o r t h e g e n e si s o f c a r d i a c a r r h y t h m i a s , a n d h a v e
b e e n c o n d u c t i n g e x p e r i m e n t s i n w h i c h s p o n t a n e o u s l y b e a t i n g a g g re g a t es o f c a r d i a c
c e ll s a r e p e r i o d i c a l l y s t i m u l a t e d w i t h p u ls e s o f c u r r e n t ( G u e v a r a e t al ., 1 98 1) . O u r
d e v e l o p m e n t o f t h e s im p l e m a t h e m a t i c a l m o d e l i n th i s p a p e r w a s m o t i v a t e d b y a
d e s i r e t o o b t a i n i n s i g h t i n t o t h e d y n a m i c s o f a n o s c i l l a t o r i n r e s p o n s e t o p e r i o d i c
s t i m u l a t io n . W e h a v e f o u n d t h a t t h e e x p e r i m e n t a l p r e p a r a t i o n b e h a v e s i n a m a n n e r
t h a t is s i m i l a r t o t h e b e h a v i o r o f t h e m o d e l . I n p a r t i c u l a r , t h e e x p e r i m e n t a l l yd e t e r m i n e d P o i n c ar 6 m a p s c a n s o m e t i m e s b e r e d u c e d t o i n t e r v a l m a p s c o n t a i n i n g a
s in g le m a x i m u m . W h e n i t er a t e d , th e s e m a p s s h o w e v i d en c e o f p e r i o d d o u b l i n g
b i f u r c a t io n s a n d c h a o t i c d y n a m i c s . F u r t h e r m o r e , in r e s p o n s e t o p e r io d i c s t i m u -
l a t i o n , p a t t e r n s d u e t o p e r i o d d o u b l i n g b i f u r c a t i o n s ( s u c h a s 2 : 2 a n d 4 : 4 p h a s e
locking) are experimentally found, as well as irregular patterns which may be due to
underlying "chaotic dynamics".
The experimental observation of these phenomena in cardiac tissue, coupled
with the predictions of a rather general mathematical model, leads us to believe that
the results of this study have direct implication for the analysis of both normal and
pathological cardiac rhythms. Under normal physiological conditions, the primary
pacemaking site in the heart is located in the sinoatrial (SA) node. The cardiac
impulse originates in the SA node, spreads to the atrial musculature, proceeds to the
atrioventricular (AV) node, and then passes through the bundle of His, the bundle
branches, and the Purkinje network to the ventricular muscle (Mandel, 1980).
Thus, firing of the SA node leads to an almost immediate contraction of the atria,
followed after a small delay by contraction of the ventricles.
There are at least two ways in which to interpret the delay between activation ofthe atria and the ventricles. The traditional and most widely held view is that the
delay is due to slow conduction of the cardiac impulse when passing through the AV
node (Tsien and Siegelbaum, 1978).
One alternate view is based on the hypothesis that the specialized electrical
conduction system of the mammalian heart contains two or more autonomous
oscillators (van der Pol and van der Mark, 1928). Normally, one oscillator (situated
in the SA node) is the driving oscillator that entrains another (situated in the region
of the AV node) in a 1:1 fashion. Recent work has provided simultaneous
anatomical and electrophysiological evidence that the site of the driven (subsidiary
or latent) oscillator is at the border of the AV node and the bundle of His (James etal., 1979). Since the driven oscillator has an intrinsic frequency that is lower than
that of the SA node (Nadeau and James, 1966; Urthaler et al., 1973 ; Urthaler et al.,
1974), there will be a phase shift between the two oscillators when synchronization
in a 1 : 1 pattern occurs (Fig. 3a, 4a). This phase shift would appear as a delay
between the contraction of the atria and the contract ion of the ventricles. Indeed, if
the intrinsic frequency of the SA node is decreased sufficiently by pharmacological
intervention, the phase shift reverses its sign, and contraction of the ventricles
precedes contraction of the atria (Nadeau and James, 1966; Roberge et al., 1968).
Modeling the heart as a system of two or more coupled nonlinear oscillators was
pioneered by van der Pol and van der Mark (1928) using an electrical circuit. In theintervening 50 years, several other investigators have extended their work using
electrical and electronic analogues and computer simulations (Grant, 1956;
Roberge et al., 1968; Sideris and Moulopoulos, 1977; Katholi et al., 1977) as well as
physiological experimentation (Nadeau and James, 1966; Roberge et al., 1968;
Roberge and Nadeau, 1969; Urthaler et al., 1973; Urthaler et al., 1974). The
modeling work shows that the disturbances of atrioventricular conduction (AV
blocks) seen in the electrocardiogram can be simulated by changing either of the
intrinsic frequencies of the two oscillators or by altering the degree of inter-
oscillator coupling. The physiological work also demonstrates that pharmacologi-
cal manipulat ion of the intrinsic frequencies of the oscillators or of the level of blockat the AV node can produce dysrhythmias when 1 : 1 synchronization is lost. Prior
modelling largely relied on electrical analogues. However, there has not been a
theoretical investigation and an analysis of the topology of the phase locking zones
Phase Locking, Period Doubling Bifurcations and Chaos 17
I n e l e c t r o c a r d i o g r a p h y , e l e c tr i c a l e v e n t s a s s o c ia t e d w i t h t h e c o n t r a c t i o n o f t h e
a t r i a ( P w a v e ) a n d v e n t r i c l e ( Q R S c o m p l e x ) c a n b e o b s e r v e d b y r e c o r d i n g e l e c tr i ca l
p o t e n t i a ls o n t h e s u r f a ce o f t h e b o d y . T h e c o u p l i n g p a t t e r n s b e t w e e n t h e s t im u l u s
a n d o s c i l l a to r o b s e r v e d i n F i g s . 3 a n d 4 a r e s i m i l ar t o c l in i c a l ly o b s e r v e d p a t t e r n s o f
A V b l o c k . I n p a r t i c u l a r , w e m a k e t h e f o l l o w i n g i d e n t i fi c a t io n s :
i) N o r m a l s i nu s r h y t h m o r fi rs t d e g re e A V b l o c k - F i g . 3 a, F ig . 4 a .
i i) 1 : 1 A V c o n d u c t i o n w i t h a l t e r n a t i n g P R i n t e r v a l s - F i g . 4 b ( 2 : 2 b l o ck ) .
i i i ) S e c o n d d e g r e e A V b l o c k w i t h W e n c k e b a c h p e r i o d i c i t y - F i g . 3 c ( 4 : 3
b l o c k ) , F i g . 3 d ( 3 : 2 b l o c k ) .
i v) S e c o n d d e g r ee A V b l o c k w i t h a t y p i c a l W e n c k e b a c h p e r i o d i c i t y - F i g . 4 e
(4 : 3 b loc k) .
v ) S e c o n d d e g r e e A V b l o c k w i t h v a r i a b l e c o n d u c t i o n o r t h i r d d e g r e e A V
b l o c k w i th a c c r o c h a g e - F ig . 4 d.v i) H i g h g r a d e se c o n d d e g re e A V b l o c k - F i g . 4 g ( 2 : 1 b lo c k) .
v i i) S e c o n d d e g r e e A V b l o c k (2 :1 b l o c k w i t h a l t e r n a t i n g P R i n t e r v a l s ) - F i g . 4 f
( 4 : 2 b l o c k ) .
v iii) T h i r d d e g r ee o r c o m p l e te A V b l o c k - F i g . 3 b.
T h e c u r r e n t a n a l y s i s p r e d i c t s t h e ex i s te n c e o n p u r e l y t h e o r e t i c a l g r o u n d s o f s u c h
p a t t e r n s a s 2 : 2 a n d 4 : 2 p h a s e l o ck i n g , w h e r e a s p r io r t h e o r e t ic a l w o r k d i d n o t
i d e n t i f y t h es e p a t t e r n s . A s t r i k i n g d e s c r i p t i o n o f a 2 : 2 p a t t e r n i s i n a n e a r l y c a n i n e
s t u d y b y L e w i s a n d M a t h i s o n ( 1 91 0) o f t h e r e s ul ts o f a s p h y x i a o n t h e h e a r t b e a t :
" A t o r a b o u t t h e t im e w h e n t h e h e a r t a p p e a r s t o w a v e r b e t w e e n a c o n d i t i o n o f 2 : 1h e a r t b l o c k a n d r e g u l a r s e q u e n t ia l c o n t r a c t i o n a c c o m p a n i e d b y p r o l o n g a t i o n o f t h e
P - R i n t e r v a l , i t n o t i n f r e q u e n t l y h a p p e n s t h a t p a s s i n g i n t o t h e l a t t e r s t a te i t e x h i b i ts
a r e g u l a r a l t e r n a t i o n o f t h e P - R i n t e r v a l s " ( se e F i g . 5 o f L e w i s a n d M a t h i s o n ( 1 91 0)
a n d c o m p a r e w i t h F i g. 4 b o f th i s p a p e r ). A r e c en t r ev i ew ( W a t a n a b e a n d D r e i f u s,
1 98 0) h a s e m p h a s i z e d t h a t t h e m e c h a n i s m s f o r g e n e r a t i n g p a t te r n s w i t h a l t e r n a t i n g
P R i n t er v a ls ( 2 : 2 r h y t h m s ) a r e n o t w e ll u n d e r s t o o d a n d a r e c o n t r o v e rs i a l. N o t e
t h a t f o r 51/2 - 1 < b < 2 , a s t h e f r e q u e n c y o f t h e s t i m u l a t i o n is i n c r e a s e d o n e p as s e s
i n t u r n f r o m 1 : 1 t o 2 : 2 t o 2 : 1 p h a s e l o c k i n g ( F i g . 2 ). T h u s , w e h y p o t h e s i z e t h a t t h e
p a t t e r n w i t h a l t e r n a t i n g P R i n te r v a ls e x p e r i m e n t a l l y o b s e r v e d c o r r e s p o n d s t o a 2 : 2
p h a s e l o c k i n g p a t t e r n a n d a r i se s a s a c o n s e q u e n c e o f a p e r i o d d o u b l i n g b i f u r c a t i o n .T h e 4 : 2 p a t t e r n s c l i n ic a l l y o b s e r v e d ( S e ge r s, 1 95 1) m a y b e d u e t o a s e c o n d p e r i o d
d o u b l i n g b i f u r c a t i o n ( F ig . 4 f ) .
A n o t h e r f e a t u r e o f o u r a n a l y s i s i s t h a t c h a o t i c d y n a m i c s c a n a r i se a s a
c o n s e q u e n c e o f p e r i o d i c i n p u t s t o a l i m i t cy c l e o s c i ll a t o r. A r e v ie w o f c li n i c a ll y
o b s e r v e d d y s r h y t h m i a s r e v ea ls t h a t t h e r e a r e m a n y d y s r h y t h m i a s w h i c h h a v e a n
e x t r e m e l y i r re g u l a r a p p e a r a n c e . T o c it e a f e w ex a m p l e s o f ir r e g u l ar A V b l o c k :
i) " A d v a n c e d s e c o n d d e g re e b l o ck w i t h f lu c t u a t i o n in th e d e p t h o f p e n e t r a t i o n
i n to t h e A V n o d e " - F ig . 1 6 - 1 7 i n W a t a n a b e a n d D r e if u s (1 98 0).
ii) " I n t e r m i t t e n t 2 : 1 A - V b l o c k a n d a n a r e a o f 3 : 2 M o b i t z t y p e I I A - V
blo ck " - F ig . 291 in Ch un g (1971) .i ii ) "W en ck eb ac h A-V b loc k o f va r y in g degree (6 : 5 , 5 : 4 , 4 : 3 , 3 : 2 )" - F ig . 295
in Chung (1971) .
iv ) " A t r i a l f lu t t e r w i t h v a r y i n g s e c o n d d e g re e A V b l o c k a n d p h a s i c a b e r r a n t
v e n t r i c u l a r c o n d u c t i o n " - C a s e 8 , p . 3 4 9 i n S c h a m r o t h ( 1 97 1) .
v ) " Pa r t ia l A V h e a r t b l o c k " - F i g . 1 -1 6E in B el le t ( 19 71 ).
These examples serve to i l lust ra te the fac t tha t c l in ica l ly observed cardiac
dysrh y thm ias a re f requen t ly qu i t e i r regu la r. On the bas i s o f our ana lys is , we
prop ose tha t one source o f i r r egu la r i ty in e l ec t roca rd iograph ic pa t t e rns o f AV
block may we l l be chao t i c dynamics re su l t ing f rom in t e rac t ion be tween d i f fe ren t
au ton om ou s pac em aker s ite s in the hea r t (F ig . 4d). Tes t ing o f t h is hypo the s i s
requ i re s co l l ec t ion and ana lys i s o f l ong e l ec t roca rd iograph ic records , a s we l l a s a
b e t t e r u n d e r s ta n d i n g o f t h e d i ff e re n t wa y s in wh i c h c h a o t ic d y n a m i c s m a y b e c o m e
mani fe s t i n ma themat i ca l mode l s .
Ou r i d e n ti f ic a ti o n o f c ar d i a c d y s r h y t h m i a w i t h d e sy n c h r o n i z a t io n o f a u t o -
nomous osc i l l a to r s i s h igh ly specu la t ive . The reade r shou ld be aware tha t t hese
ideas run coun te r t o conven t iona l c l in i ca l t each ing , and a l so tha t a l t e rna t ivetheore t i ca l mode l s based on fa t igue and re l a t ive re f rac to r iness (Landah l and
Gri f fea th , 1971 ; Ke ener , 1981a, 1981b) display som e qual i ta t ive fea tures s im i lar to
the p rope r t i e s desc r ibed he re . H ow ever , t he de ta i l ed topo log ica l s t ruc tu re o f t he
phase lock ing p resen ted in these o the r mo de l s is com ple te ly d i f fe ren t f rom tha t o f
our m ode l (Keene r , 1980 , 1981 a , 1981 b ) . Fo r exam ple , o the r m ode l s do no t i nc lude
p h a se lo c k i n g z o n e s o f t h e f o r m 2 N: 2 M a n d t h e " c h a o s" o b se r v e d b y Ke e n e r d o e s
no t a r i se f rom a sequ ence o f pe r iod do ub l ing b i fu rca t ions . F ina l ly , a l though the
theory i s gene ra l, t he ab ov e d i scuss ion has cen te red o n dy srhy thm ias a r is ing in the
AV c o n d u c t i o n sy s te m . Fo r i n s ta n c e, i n t e r ac t io n s b e t we e n n o d a l p a c e m a k e r s a n d
ec top ic spon taneo us ly ac t ive foc i wo uld l ead to s imi l a r dynam ics and d ysrhy thmias(Moe e t a l . , 1977; Ja l i fe and Moe, 1979) .
In the ma the mat i ca l l it e ra tu re , a l thou gh i t has long been kno w n tha t a pe r iod ic
so lu t ions can be fou nd in re sponse to pe r iod ic inpu t s t o two d im ens iona l osc i l l a to r s
(Car twr igh t and L i t t l ewoo d , 1945 ; Lev inson , 1949) the topo lo gy o f t he d i f fe ren t
phase lock ing zones und e r changes in the s t imula t ion pa ram ete r s i s no t com ple te ly
un ders too d (Gu ckenhe im er , 1980 ; Lev i, 1981). The nove l con t r ibu t ion o f our w ork
is t o show tha t ov e r ce rt a in ranges o f pa ram ete r space , t he phase lock ing p rob lem
reduces to a p rob lem in the ana lys i s o f in t e rva l maps . W e show tha t t r ans i ti ons
be tween phase lock ing zones a re due to pe r iod doub l ing b i fu rca t ions , t angen t
b i f u r c a t i o n s , o r s i m p l y c h a n g e s i n r o t a t i o n n u m b e r . W e h a v e a l so p r o p o se d ascheme fo r t he topo log y o f the phas e lock ing zones (F ig . 10) in the pe r iod dou b l ing
r e gi o n. Pe r i o d d o u b l i n g b i f u r c a ti o n s a n d c h a o t i c d y n a m i c s h a v e b e e n o b se r v e d in
prev ious wo rk on s inuso ida l ly fo rced non l inea r osc i l l a to r s (To mi ta an d Ka i , 1978 ;
Huberman and Cru tchf i e ld , 1979) . In add i t ion , "chao t i c dynamics" have been
o b se r v e d f r o m t w o m u t u a l l y c o u p l e d t u n n e l d i o d e o sc i ll a to r s b u t n o e v id e n c e o f
p e r i o d d o u b l i n g b i f u r c a t io n s w a s f o u n d ( Go l l u b e t a l. , 1 9 8 0 ) . Ho we v e r , a s t u d y o f
m utua l en t ra inme nt o f two e l ec t ron ic mode l s o f ca rd iac pacem aker ce ll s appa re n t ly
d id no t show chao t i c dynamics (Ypey e t a l . , 1980) .
One o f t he ma in ways fo r s tudy ing b io log ica l osc i l l a to r s i s t o sub jec t t he
osc i l l a to r s t o pe r iod ic pu l sa t i l e i npu t s . Th i s pape r has desc r ibed the qua l i t a t iveprope r t i e s o f phase lock ing fo r a s imple ma thema t i ca l m ode l o f a two d im ens iona l
osc i ll a to r . On the bas i s o f t h is t heore t i ca l wor k w e p red ic t t ha t pe r iod d oub l ing
b i fu rca t ions l ead ing to chao t i c dynam ics a s a conseq uence o f pe r iod ic s t imula t ion
of spon taneo us ly osc i ll a ting b io log ica l sys t ems wi ll be widespread .
P h a s e L o c k i n g , P e r i o d D o u b l i n g B i f u r c a t io n s a n d C h a o s 1 9
A c k n o w l e d g m e n t s . T h i s r e s e a r c h h a s b e en s u p p o r t e d b y g r a n t s f r o m t h e N a t u r a l S c ie n c es a n d
E n g i n e er i n g R e s e a r c h C o u n ci l o f C a n a d a a n d t h e C a n a d i a n H e a r t F o u n d a t i o n . W e t h a n k M . M a c k e y
a n d C . G r a v e s f o r h e l p f u l d i s c u s s io n s , S . J a m e s f o r t y p i n g t h e m a n u s c r i p t , a n d B . G a v i n f o r d r a f t i n g t h e
f i g u r e s .
A p p e n d i x A
I n t h i s a p p e n d i x w e p r o v e P r o p o s i t i o n 1 o f S e c t io n I V B . T h e p r o o f s o f t h e o t h e r t w o p r o p o s i t i o n s f o l lo w
i n a s i m i l a r f a s h i o n .
P r o o f o f P r o p o si t io n 1 . F r o m t h e s t a t e m e n t o f th e p r o p o s i t i o n a n d t h e d e fi n it i on o f t h e P o i n c a re m a p w e
k n o w t h a t
4 , = f ( O , 1,7) + 0 .5 - 6 , ( A 1 )
A~b~ =f (~ b~ _l ,7 ) + 0 .5 - 6 - qSi 1 , (A 2)N
M = Z A~) ,. ( A 3 )
i - I
S t a r t i n g f r o m p h a s e ~b~ ~ = 1 - ~b~ ~ a n d a s s u m i n g z = 0 . 5 + ~ w e c o m p u t e t h e n e w p h a s e , y /
r = f ( 1 - q ~ - l , Y ) + 0 . 5 + 6 ( r o o d 1 ). ( A 4 )
A p p l y i n g t h e s y m m e t r y in E q . ( 6 a ) a n d s u b s t i t u t i n g f r o m E q . ( A 1 ) , w e f i n d
= 1 - q~i. (A S)
T h u s , t h e s e q u e n c e # o , ~ bl . . . . f l u - 1 w i l l f o r m a c y c l e f o r z = 0 . 5 + 6 w h e r e ~ = 1 - ~ bl. T h e p h a s e
l o c k i n g p a t t e r n i s c o m p u t e d b y s u m m i n g t h e A ~kl, w h e r e
A ~/~ = f O P ~ - l , 7 ) + 0 .5 + 6 - r ( A 6 )
O n c e a g a i n u s i n g t h e s y m m e t r y in ( 6 a ) a n d s u b s t i t u t i n g f r o m ( A 2 ), w e f i n d
9 A ~ = 1 - A~b~. (A 7)
S u m m i n g a n d a p p l yi n g ( A 3) , w e c o m p u t e
N
Z A ~ b l = X - M . ( A S )
T h e r e f o r e , t h e s e q u e n c e tP o, ~ P ~ , . . . , ~ N - 1 i s a s s o c i a t e d w i t h a n N : N - M p h a s e l o c k i n g p a t t e r n .
A p p e n d i x B
I n t h i s a p p e n d i x w e g i v e a b r i e f s u m m a r y o f t h e a n a ly t i c c o m p u t a t i o n o f t h e b o u n d a r i e s o f th e p h a s e
l o c k i n g z o n e s a n d s i n g u l a r p o i n t s i n ( ~ , b ) p a r a m e t e r s p a c e . I n t h i s a n a l y s i s w e u ti l iz e r e s u l t s f r o m l o c a l
b i f u r c a ti o n t h e o r y i n c o n j u n c t i o n w i t h a c o n s i d e r a t i o n o f t h e r o t a t i o n n u m b e r .
A s s u m e t h a t 0 * i s a p e r i o d i c p o i n t o f p e r io d n o f t h e P o i n c a r ~ m a p ( 1 3 ). T h e n , c h a n g e i n t h e s t a b i l it y
o f t h e p e r i o d i c p o i n t ( b i f u r c a t i o n ) w i ll o c c u r i f
= _ 1 . (B1 )
I f t h e d e r i v a t i v e i n ( B 1 ) i s + 1 t h e b i f u r c a t i o n i s c a l le d a s a d d l e - n o d e o r t a n g e n t b i f u r c a t i o n a n d i f t h ed e r i v a t i v e i s - 1 t h e n t h e b i f u r c a t i o n i s c a l le d a p i t c h f o r k o f f l ip b i f u r c a t i o n ( M a y , 1 9 7 6; G u c k e n h e i m e r ,
1 97 7) . I n a d d i t i o n t o s a t i s f y i n g ( B 1 ) t h e r e a r e a d d i t i o n a l n o n - d e g e n e r a c y c o n d i t i o n s ( G u c k e n h e i m e r ,
1 97 7) w h i c h w e d o n o t c o n s i d e r in t h e c o m p u t a t i o n s w h i c h f o l lo w . A t a p i t c h f o r k b i f u r c a ti o n t h e n u m b e r
o f s t a bl e f i x ed p o i n t s i s d o u b l e d b u t t h e r o t a t i o n n u m b e r i s i n g e n e r a l th e s a m e . A t a t a n g e n t b i f u r c a t i o n
t h e n u m b e r o f s t ab l e fi x ed p o i n ts c h a n g e s a n d t h e r o t a ti o n n u m b e r m a y o r m a y n o t c h a n g e .
I n a d d i t i o n t o c h a n g e s i n r o t a t i o n n u m b e r a s s o c i a t e d w i t h t a n g e n t b i f u r c a t io n s , t h e r o t a t i o n n u m b e r
o f t h e s ta b l e p h a s e l oc k e d o r b i t m a y c h a n g e w i t h p a r a m e t r i c c h a n g e s in b a n d z , ev e n t h o u g h t h e n u m b e r
a n d s t a b il it y o f t h e f i x e d p o i n t s r e m a i n u n c h a n g e d . S u c h c h a n g e s i n r o t a t i o n n u m b e r o n l y o c c u r f o r
Ibb > 1 . T o s h o w h o w t h e s e c h a n g e s a r i s e c o n s i d e r t h e c a s e i n w h i c h b > 1 , a n d a s s u m e t h a t a l o n g t h el o c u s o f p o i n t s
= o ( b ) (B2)
~b* = 0 .5 i s a s t a b le f i x e d p o i n t o f p e r i o d N . A s s u m e t h a t i n t h e n e i g h b o r h o o d o f t h e c u r v e i n ( B 2 ) th e r e
e x i s t s a s t a b l e f i x e d p o i n t qS* o f t h e m a p TN
4 " = 0 . 5 - ~ ( b , T ) , ( a 3 )
w h e r e ~ is a c o n t i n u o u s d i f f e r e n t i a b l e f u n c t i o n
e(b, g(b)) = 0 (B4)
a n d e is p o s it i v e o n o n e s i d e o f t h e l in e i n ( B 2) a n d n e g a t i v e o n t h e o t h e r s id e . A s s u m e t h a t i n t h e r e g i o n i n
w h i c h e i s n e g a t i v e t h e r e i s N : M p h a s e l o c k in g . T h e n f r o m a d i r e c t a p p l i c a t i o n o f ( 1 7 ) th e r e w i ll b e
N :M + 1 p h a s e l o c k i n g o n t h e o t h e r s id e . C o n s e q u e n t l y t h e l o c u s o f p o i n t s in ( B 2 ) w il l c o n s t i t u t e t h e
b o u n d a r y b e t w e e n t h e t w o p h a s e l o c k i n g r e g io n s . I n a s i m i l a r f a s h i o n , a n d r e c a l l i n g t h e s y m m e t r i e s i n
S e c t i o n I V , a l o c u s o f p o i n t s a l o n g w h i c h ~b* = 0 c o n s t i tu t e s a b o u n d a r y o f a p h a s e l o c k i n g r e g i o n f o r
b < - 1 .
I n t h e r e m a i n d e r o f t h is A p p e n d i x w e g i v e a b r i e f s u m m a r y o f t h e c o m p u t a t i o n o f th e b i f u r c a t io n
b o u n d a r i e s a n d s i n g u l a r p o i n t s i n (~ , b ) p a r a m e t e r s p a c e . T h e r e s u l ts a r e a p p r o p r i a t e f o r b < 0 a n d
0 <~ z ~< 0 . 5. U s i n g t h e s y m m e t r i e s i n S e c t i o n I V a p p r o p r i a t e f o r m u l a e f o r s y m m e t r i c a l l y p l a c e d
b o u n d a r i e s c a n b e c o m p u t e d . S i n c e t h e c o m p u t a t i o n s e n t a i l s im p l e a l g e b r a a n d t r i g o n o m e t r y , o n l y t h e
m a i n r e s u l t s o f th e c o m p u t a t i o n s a r e g iv e n . T h e f o r m u l a e p r e s e n t e d h a v e b e e n c h e c k e d a g a i n s t r e s u l ts
f r o m n u m e r i c a l i t e r a t i o n o f t h e P o i n c a r 6 m a p .
T h e 1 : 0 p h a s e l o c k i n g b o u n d a r y c o i n c i d e s w i t h a t a n g e n t b i f u r c a t i o n f o r ] bl < 1 , a p i t c h f o r k
b i f u r c a t i o n f o r 1 < I bl < 2 a n d a c h a n g e i n r o t a t i o n n u m b e r f o r [ b[ > 2 .
C o n s i d e r f i r s t t h e l o c u s o f t h e t a n g e n t b i f u r c a t i o n . S e t t i n g 3T /O~ = + 1 a n d u s i n g (7 ) w e f i n d
1qb = 1 - - - c o s - l ( - b ). ( B 5)
2~
N o t e t h a t t h e o n l y r e a l s o l u t i o n o f ( B 5 ) o c c u r s f o r Ibl < 1 , a n d t h u s t h e b o u n d a r y o f t h e 1 : 0 p h a s e
l o c k i n g c o i n c i d e s w i t h a t a n g e n t b i f u r c a t i o n o n l y f o r Ib l < 1 . A p p l i c a t i o n o f ( 5 ) a n d ( 1 1 ) g i v e s t h e
b o u n d a r y o f t h e 1 : 0 p h a s e l o c k i n g f o r - 1 < b < 0
z = l ~ c o s - l b - 0 .2 5 . ( B 6)27r
A p i t c h f o r k b i f u r c a t i o n i s c o m p u t e d b y s e t t i n g ~T/OO = - 1 . U s i n g ( 7 ), a p i t c h f o r k b i f u r c a t i o n o c c u r s
fo r
~b = 1 - l c o s - l ( - b 2 + 2 ~ ( B 7)
2~z 2 3b / "
N o t e t h a t t h e o n l y r e a l s o l u t i o n s o f ( B 7 ) o c c u r f o r 2 / > Ib ~> 1 . S u b s t i t u t i n g (B 7 ) i n t h e P o i n c a r 6 m a p
g i ves
1 1 [ ( b Z + 2 ~ + ( 1 - b2 / 4~1 / 2~z . . . . c o s - 1 - t a n 1 (B 8 )
2 27r 3 b J \ ~ / J "
T h i s i s t h e b o u n d a r y b e t w e e n 1 : 0 a n d 2 : 0 p h a s e l o c k i n g r e g i o n s fo r 0 < z < 89 nd - 2 < b < - 1 .
T h e b o u n d a r y b e t w e e n t h e 1 : 0 a n d 1 : 1 p h a s e l o c k i n g z o n e s f o r b < - 2 i s a s s o c i a t e d w i t h a c h a n g ei n r o t a t i o n n u m b e r . A s s u m i n g t h a t t h e r e i s a f i x e d p o i n t qS* = 0 o n t h e P o i n c a r 6 m a p ( 11 ) w e f i n d t h a t
z = 89 (B9)
T h e i n t e r s e c ti o n o f ( B 8 ) a n d ( B 9) g iv e s t h e s i n g u la r p o i n t o n t h e c o m m o n b o u n d a r i e s o f t h e I : 0 , 1 : 1 ,
_ 12 : 2 , 2 : 1 an d 2 : 0 ph ase l ock i ng r e g i ons . Th e i n t e r se c t i on o ccu r s fo r z - ~ , Ib l = 2 .
P h a s e L o c k i n g , P e r i o d D o u b l i n g B i f u r c a t i o n s a n d C h a o s 2 1
T h e b o u n d a r y b e t w e e n 2 : 0 a n d 2 : 1 p h a s e l o c k i n g o c c u r s w h e n
T2(0) = 0 . (B10)
S u b s t i t u t i n g in t h e P o i n c a r 6 m a p (1 1) w e f i n d
1 1 ( b ) 1 c o s _ l _ b ' ( B l l )z = - - - - - C O S - 1 - - =2 2~ 27r 2
N o t e t h a t t h e f i x e d p o i n t s a l o n g t h e b o u n d a r y i n ( B l l ) o c c u r f o r ~b = 0 , ~b = 0 . 5 + z .
T h e s i n g u la r p o i n t o n t h e c o m m o n b o u n d a r i e s o f t h e 2 : 0 , 2 : 1, 4 : 2 , 4 : 1 a n d 4 : 0 p h a s e l o ck i ng
r e g i o n s o c c u r s w h e n ( B 1 0 ) i s s a t i s f i e d a l o n g w i t h t h e c r i t e r i o n
63T2 _ OT c~T 4,=o.5+~
c~b c~b ,=0 c~ b = - 1, (B1 2)
w h e r e w e h a v e u s e d t h e c h a i n r u l e. U s i n g ( 7 ) w e f i n d t h e s i n g u l a r p o i n t o c c u r s f o r
b = 1 - 5 1 / 2 , r = - - c o s - 1 - ~0 .3 5 6.2~
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