Breakup of an invariant curve in a dissipative standard mapping

Volume 116, number 7

PHYSICS LETTERS A

30 June 1986

BREAKUP O F AN INVARIANT CURVE IN A D I S S I P A T I V E S T A N D A R D M A P P I N G Yoshihiro Y A M A G U C H I Center for Studies in Statistical Mechanics, The University of Texas at Austin, Austin, TX 78712, USA Received 4 March 1986; accepted for publication 28 April 1986

A critical value for the existence of an invariant curve in a dissipative standard mapping is calculated by a new method in relation to Arnold's tongue. It is shown that the golden mean invariant curve is not the last curve for the strongly dissipative parameter region. In this paper we obtain a critical value K¢ for the existence of an invariant curve in a dissipative standard mapping [1-12]. We study the following type of standard mapping: X,,+ I = J X , - (K/2~r) sin(2~rq~,),

'h,+l = q~, + X,+l + 12 (mod 1),

(1)

where J(0 ~
lim (q,, - d p o ) / n .

(2)

n~oO

The periodic motions have a rational winding number and the quasiperiodic motions have an irrational one. In the limit K = 0 the winding number is equal to 12. For K > 0, it is shifted from the bare winding number 12. When both parameters K and I2 are changed, the invariant curve expressed by X, = g(q~,) loses smoothness and the transition to chaotic motion occurs. Such transition was also observed in the forced Josephson oscillator [7,13]. The critical

value depends on the winding number. It is well known that the last invariant curve is characterized by the golden mean winding number W - - ( ¢ ~ - 1)/2 ( = [ 1 , 1 .... ] expressed by the continued fraction). In the following, we pay attention to this invariant curve. The smooth continuous golden mean invariant curve can exist for 0 < K < K¢ in the (~, X) plane. This curve changes its structure to a nondifferentiable curve at K = K c. The invariant curve is the infinite-valued function with respect to q~. The theoretical approach in terms of Birkhoff's criterion [14] was made by Mather [8] in an area preserving case, and by Bohr [9] in the dissipative case. The result by Mather and Bohr is expressed by Kc = 2(1 + J ) / ( 2 + J ) . This value is an upper bound for the existence of a smooth curve. Mackay and Percival [10] gave the critical value 6 3 / 6 4 for the case J = 1. This is very close to the numerical value K c = 0.97163... by Greene [2]. In the opposite limit J = 0, eqs. (1) reduce to the climbing sine mapping. For this case the critical value is exactly equal to 1. The reason of the breakup of smoothness is the breakdown of invertibility. For any value J (0 < J < 1), it is expected that Kc decreases from 1 to 0.9716 .... In order to calculate K~, we introduce a new method in relation to the structure of the phase-locking states. The socalled Arnold's tongues [15]. This method was partially discussed by Bohr, Bak and Jensen (BBJ) [71. Here we introduce a circle map ep,+ 1 = f ( ~ , ) .

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Volume 116, number 7

PHYSICS LETTERS A

U s i n g the circle m a p , eqs. (1) can be rewritten in the functional form

f(ep.)=fo(ep.)+J[ep.-~-ffl(ep.)],

f ( q ' , ) = f 0 ( o , ) +Jg(q',,).

(4)

T h e n the a b n o r m a l i t y of f(q~,) induces that of

g(,t,.). D i f f e r e n t i a t i n g eq. (3) with respect to q,., we get ,)/dqs.

~],

(5) where E ( q , , ) = 1 + J - K cos(2"~q,,). This is a key e q u a t i o n to d e t e r m i n e the critical value K~. Both functions f(dp,) a n d g ( ~ , ) lose their s m o o t h n e s s if the slope d f ( ~ , ) / d ~ , changes its sign from positive to negative at s o m e p o s i t i o n of q,, (Birkhoff criterion) [7]. But it is very difficult to find K c by using this criterion b e c a u s e eq. (5) is an infinite recursion e q u a t i o n for the i n v a r i a n t curve with i r r a t i o n a l w i n d i n g n u m b e r . It is best that we a p p r o a c h the golden m e a n in terms of the F i b o n a c c i series: W = F./F.+ t where F.+ 1 = F. + F~ 1, F0 = 0 a n d F 1 = 1. Since the w i n d i n g n u m b e r is r a t i o n a l ( W = p / q ) , we can rewrite eq. (5) b y using the values of the p e r i o d - q fixed p o i n t s

c I = E a - J/e.q,

E i. T h e n a t u r e of the roots of this e q u a t i o n a n d of f(4~n) d e p e n d s on the sign of function Dq defined by

(3)

where f0(q'.) = ~'. - ( K / 2 . ~ ) sin(2,~q,.) + 12, a n d f - l ( q , ) d e n o t e s the inverse circle map. T h e circle m a p is related to the invariant curve

df(ep~)/dep,=E(q~,)-J/[df(O,

30 June 1986

Oq = ( OLq-}- ~q)2 __ 4flq2/q.

(9)

N o t e that Dq is a cyclic function with respect to E,. If Dq > 0, the e q u a t i o n a d m i t s two real roots a n d then the fixed p o i n t s are nodes a n d the circle m a p is a single-valued function with respect to ¢knIn the o p p o s i t e case Dq < 0, the e q u a t i o n a d m i t s a p a i r of c o m p l e x conjugate roots a n d the fixed p o i n t s c h a n g e their structure to foci. The circle m a p b e c o m e s an infinite-valued function. The p h a s e - l o c k i n g regions are s e p a r a t e d b y the curve Dq = 0 a n d the critical value Kc,(q ) satisfying

Dq(K~,q,) = 0

(10)

can be d e t e r m i n e d . Let the m i n i m u m p o i n t of K~ q) be the critical value K(cq) of the p h a s e - l o c k ing state with W = p / q (see fig. 1). W e also find the value 12~q) to give Kc( q ) . The critical value K,. for the golden m e a n i n v a r i a n t curve is given by

K~(J, WG)= lim K~q~(J),

where we a s s u m e that the critical value K c is the

wG

¢.2 = E2 - J/¢.l . . . . .

~.q = Eq - J/¢q-1.

•

(6)

'm,'

t-'"

13

where % = df(q~i)/dO, a n d E, = E(~i). Eliminating E, ( i = 1, 2 . . . . . q - 1), the finite c o n t i n u e d fraction to d e t e r m i n e ~q is derived b y

__8! r3j

-j -Z=

_j

Eq + Eq 1 +

(7)

-J

where Z = J/C~q. T h e r i g h t - h a n d side can be form a l l y expressed as

-Z = ( OLqZ + [~q)/("[qZ ~- ¢~q),

*

f

,...,'

51 -g J~

-J

Eq 2 . . . E I _ Z

(8)

where aq, flq, yq, a n d 6q are functions of J a n d 308

(11)

q~oO

Fig. 1. Schematic diagram of the phase-locking states around the golden mean invariant curve labeled by Wc in the I2-K plane. The fraction in each region is the winding number W. The tongues defined by eq. (10) are shown by the dotted curve. The closed circles show the value g(c q) (the minimum point of K~q) defined by eq. (10)) where q implies the denominator of W = p/q. The closed square implies the critical value Kc at which the golden mean curve becomes the nondifferentiable curve.

Volume 116, number 7

PHYSICS LETTERS A

accumulation point of the minimum point of the tongue. Note that the truncation of Z on the right-hand side is meaningless even if J << 1 because the right-hand side is a real constant if Z is truncated. Here we consider two exactly solvable cases. The first one is the case with I2 = 0 ( I V = 0 / 1 or 1/1). The point (0, 0) is the period-1 fixed point and then the function D 1 is given by D 1 = (1 + J - K )2 _ 4J.

(12)

The critical value is K~ a) = (1 - v/J) 2 and ~2~a) = 0 or 1. It is easy to check that this value is a minimum of the tongue in the period-1 phaselocking state. The second one is the case with [2 = 0.5. There are two fixed points: (0, 0) and (0.5, 0). The function D 2 is D2 = EIEz(E1E 2 - 4J).

(13)

The critical value is K (2) = 1 - J and 12~2) = 0.5. These values are equal to the results by BBJ. It is impossible to find the fixed points with period-q ( q > 2) analytically. As our system is dissipative for 0 < J < 1, the mapping point rapidly approaches the attractor after many iterations. Using the information of the fixed points q~i, we can calculate aq, flq, "yq, and 6q. From eq. (7) the recursion relations are derived by

30 June 1986

Table 1

W = p/q

K(cq)

~'~(q)

8/13 13/21 21/34 34/55 55/89 FKS's value

0.97770 0.97850 0.97891 0.97881 0.978848 0.97883778

0.60982 0.61244 0.61151 0.61193 0.611723 0.61175390

(11) are justified. The critical values for several values of J are illustrated in fig. 2. In the limit J ~ 0, the critical value tends to 1. In the limit J ~ 1, it approaches 0.9716 . . . . for example, K¢ = 0.9717... at J = 0.9. Here we try to find the critical value K¢(W~) for the silver mean W~ = ¢~- - 1 ( = [2, 2 . . . . ]) by using the same method. The results are also shown in fig. 2. The critical value K c ( W c ) is certainly larger than K~(W~) in the region 0.35 _
,,'+,

,-r

"','+, ",,,,++,

"Yq+l = Eq+l'~q "Jr Oiq,

~q+l = Eq+l~q at- ~ ,

(14)

~

o. t.)

where q>~2 and a 2 = J ,

f12 = - J E ] ,

7z = - E

z

W G

.,:?., ..... . /

,,I

and 8 2 = E t E z - J.

In order to find K~(q), we use the following procedure: (i) The determination of the pattern of the phase-locking state with I V = p / q . ( i i ) T h e determination of fixed points and the estimation of E,. (iii) Using eqs. (14), the coefficients Olq, flq, yq and 8q are determined and then the function Dq is estimated. (iv) The shape of the tongue is defined and the minimum value K~ q) is determined. We shall compare our answer with the value K~ = 0.97883778 at J = 0 . 5 given by Feigenbaum, Kadanoff and Shenker [4]. Our results are listed in table 1. By increasing q using the Fibonacci series, our results for K (q) and [2~q) approach the values of ref. [4]. Then our method and the assumption

"4- . . . .

÷ ....

4-

o,,

"+ "4"-

~ooo

'

0'.20

'

o',4o

'

o'.6o

'

o'.ao

'

doo

J Fig. 2. The critical values of the invariant curve characterized by the winding number WG = (v~- - 1)/2 and ~ = ¢'2 - 1 for several values of J. The values at J = 1 are the results by Greene for WG and by Shenker and Kadanoff for W~. The critical value for W~ in the vicinity of J = 1 cannot be estimated using our method due to the overlapping of the phaselocking states.

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Volume 116, number 7

PHYSICS LETTERS A

Table 2 W

Kc

[2, 1) [2, 2, 1) [2, 2, 2, 1) [2, 2, 2, 2, 1) [2, 2)

0.9885 0.9905 0.9908 0.9908 0.9908

s i m i l a r p r o b l e m o n the t w o - d i m e n s i o n a l c u b i c m a p has b e e n d i s c u s s e d b y Y a m a g u c h i a n d M i s h i m a [16,17]. T h e a u t h o r w o u l d like to t h a n k D r s . K . T a n i k a w a a n d P. T r a n for u s e f u l discussions. H e also a p p r e c i a t e s Professor I. P r i g o g i n e ' s k i n d encouragement.

b e t w e e n 1 - W G a n d W~. T h e critical v a l u e s for such i r r a t i o n a l n u m b e r s are also larger t h a n K c ( W G ) (see table 2). T h e s e facts suggest that the g o l d e n m e a n i n v a r i a n t c u r v e is not the last curve in such a region. T h e q u e s t i o n of w h y the g o l d e n m e a n i n v a r i a n t c u r v e is n o t the last c u r v e r e m a i n s as a f u t u r e p r o b l e m . F i n a l l y we c o m m e n t o n the m u l t i v a l u e d n e s s of f(q~n)- Eq. (3) c a n be r e w r i t t e n i n the f u n c t i o n a l form

f(f(e&))

= f0 ( f ( ~ , ) )

+ J [ f ( q ' , ) - q~, - ~2].

(15) S u p p o s e that f(q~n) is a n n ( > D - v a l u e d f u n c t i o n w i t h respect to qs. T h e n the l e f t - h a n d side is a n n 2 - v a l u e d f u n c t i o n a n d the f i g h t - h a n d side is a n n - v a l u e d f u n c t i o n . As a n u m b e r s a t i s f y i n g n 2 = n, we c a n find, for e x a m p l e , 2 s°, 3 ~° . . . . . No~° (N 0 is the Aleph-null d e f i n e d b y C a n t o r ) b e c a u s e of ( 2 s ° ) 2 = 2 s ° + ~ ° = 2 s° (N O + N O = NO). A c c o r d i n g to the figures in refs. [3,4,7,13], the i n v a r i a n t curve has m a n y c u b i c ( t h r e e - v a l u e d ) kinks. I n this case the f u n c t i o n f(q~n) is the i n f i n i t e (3So)-vahied f u n c t i o n ; o t h e r w i s e f ( ~ , ) is a s i n g l e - v a l u e d f u n c tion. T h e p r o p e r t i e s of fo(~),) give the v a l u e 3. A

310

30 June 1986

References

[1] B.V. Chirikov, Phys. Rep. 52 (1979) 263. [2] J.M. Greene, J. Math. Phys. 20 (1979) 1183. [3] S.J. Shenker and L.P. Kadanoff, J. Star. Phys. 27 (1982) 631. [4] M.J. Feigenbaum, L.P. Kadanoff and SJ. Shenker, Physica D 5 (1982) 370. [5] S.J. Shenker, Physica D 5 (1982) 405. [6] M.H. Jensen. P. Bak and T. Bohr, Phys. Rev. A 30 (1984) 1960. [7] T. Bohr, P. Bak and M.H. Jensen, Phys. Rev. A 30 (1984) 1970. [8] J.N. Mather, Ergod. Th. Dynam. Sys. 4 (1984) 301. [9] T. Bohr, Phys. Lett. A 104 (1984) 441. [10] R.S. MacKay and I.C. Percival, Commun. Math. Phys. 98 (1985) 469. [11] P. Cvitanovid, M.H. Jensen, L.P. Kadanoff and I. Procaccia, Phys. Rev. Lett. 55 (1985) 343. [12] M.H. Jensen and I. Procaccia, Phys. Rev. A 32 (1985) 1225. [13] M.J. Kajanto and M.M. Saloma, Solid State Commun. 53 (1985) 99. [14] G.D. Birkhoff, Dynamical systems (American Mathematical Society, New York, 1927). [15] V.I. Arnold, Am. Math. Soc. Trans. Ser. 2, 46 (1965) 213. [16] Y. Yamaguchi and N. Mishima, Phys. Lett. A 104 (1984) 179. [17] Y. Yamaguchi and N. Mishima, Phys. Lett. A 109 (1985) 196.