Pram~r~a, Vol. 24, Nos I & 2, January & February 1985, pp. 31-37. © Printed in India.

On the Feigenbaum-Cvitanovi~ equation in the theory of chaotic behaviour

V I R E N D R A S I N G H Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India.

Abstract. We propose an analytic perturbative approach for the determination of the Feigenbaum-Cvitanovi~ function and the universal parameter a occurring in the Feigenbaum scenario of period doubling for approach to chaotic behaviour. We apply the method to the case Z = 2 where Z is the order of the unique local maximum of the nonlinear map. Our third order approximation gives ~,=2.5000 as compared to "exact" numerical value a = 2.5029 . . . . We also obtain a reasonably accurate value of the Feigenbaum-Cvitanovi¢5 function.

Keywords. Feigenbaum-Civitanovi~ equation; chaotic behaviour; analytic perturbative approach.

PACS No. 05.45; 02-30; 02.90

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

The discovery by M Feigenbaum of universal quantitative behaviour, in the maps o f an unit interval onto itself, has generated enormous excitement as it opens up the possibility of quantitatively understanding turbulence and other chaotic natural phenomenon. In Feigenbaum scenario a system follows period doubling route to chaos (Feigenbaum 1978, 1979, 1980).

In this theory asymptotically, at each period doubling the separation between any two corresponding adjacent elements of a periodic at t ractor is scaled by a constant ratio ~. This leads to the existence of a function g(x) which reproduces itself under the mapping except for the relevant scaling. This function satisfies the equation,

- • o t g ( x / ~ ) ) = g ( x ) , ( 1 )

and is normalised by

0(0) = 1. (2)

We shall refer to this equation and g(x) as Feigenbaum-Cvitanovi6 equat ion and function respectively (Feigenbaum 1978). Fo r a generalisation to period n-tuplings o f this equat ion we refer to Cvitanovi~-Myrheim (1983). Equat ion (1) together with normalisat ion (2) determines both the functional fo rm of g(x) and ~. We shall be interested in the solution which has a local max imum at x = 0 o f the order Z. Such a solution is unique (Feigenbaum 1980, p. 15; Collet et al 1980).

The universal function g(x) and ct do, o f course, depend on the order o f local max imum i.e. if the t ransformation T considered is

T : x --, x ' = 2 ( 1 - a l x l Z ) ,

31

32 Virendra Sinffh

then they depend on Z. For Z = 2, which is the most interesting case, we have

0t = 2"502907875. . .

The function g(x) has also been determined numerically for this case (Feigenbaum 1979).

Another important universal constant 6, which determines the rate of period- doublings, is given by

6 = 4 .6692016 . . .

for Z = 2. A consideration of asymptotic period-doubling leads to the equation

- ~[h(g(x/~)) + g'(g(x/~))h (x/~)] = ,~h(x) > 1 (3)

for a determination o f the universal function h(x) and universal constant & Again numerical determination of h(x) and ~ has been carried out (Feigenbaum 1979).

In view of the importance o f these functions it would be desirable to have an analytical approach for their determination as opposed to one involving pure numerical computation. We propose an analytical perturbative scheme towards this purpose in the present paper. It is a systematic procedure and can be improved successively. Among other attempts to determine ~, but not the #(x), by analytic approximations, we may mention those of Helleman (1983) and Hu and Mao (1982) among others.

2. Basic equations and method

Replacing x by ~.x we get

g(~x) + ~,g[g(x)] = 0,

g(0) = 1.

Let g(x) = 1 + p(x),

where Z i s Further let

p ( x ) = Y. p~y~, n = l

y = Ixt z p , =~ 0,

the order o f local maximum of the transformation.

p(ax) = ~ C . [p (x) ] ' . n = l

Using (6) and (9) in (4) we obtain

1 + y. c.[p(x)]"+~,g[1 +p(x)] = 0. a = l

Equating the coefficients o f [p(x)]" to zero in (10) we obtain

1 + aO0) = 0,

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

Feigenbaum-Cvitanovik equation 33

and n!C, + ~tf~")(l) = 0 (n = 1, 2 . . . . ). (12)

On the other hand, if we expand both sides of (9) in powers of y and equate the coefficients we obtain (n = 1, 2 . . . . )

where fl = I~1 z We display first

Pl~ =

p2/~ 2 =

p3fl 3 --_ p4# 4 =

We note that since px 4: O,

G = I ~ I Z It is also useful to note that

(rZ)! f~t)(1) = l ! ( rZ - 1)! Pr

r = l

for l = 1, 2, 3 . . . . . and

f (1) = 1 + ~ p,. r = l

E C l p m t P m ~ " " " Pint t~ml + m z + . , . +m~,. I, m l , ra 2 . . . . . r a I >1 1

few of these equations.

Clpt,

CIp2 +C2p 2,

CIP3 +C2(2plP2)+C3Pal,

C1 p, + C2 (2pl Pa + P22) + C3 (3p2p2) + C,p~,

If we combine (11), (13), (19) and (20), and further define

p.~"= s.l~t z n = 1, 2 , . . .

We obtain the basic equations to be solved

_ + l + l ~ l z _s,=0 O~ r = l Oct

and (rZ)! S, Sm, Sm~ . . . Sm,

S n + ~ l ! (rZ- l ) ! ~t "-x "- ~ ( . - i ) - = 0

for n>~l .

The summation in (23) is over the set U given by

r>>-l,l>~l, ml ~>l, m2 ~>1 . . . . . ml~>l

and m l + m 2 + . . . +mt=n .

(13)

(14)

(15)

(16)

(17)

(18)

(19)

(20)

(21)

(22)

(23)

P - ~ 3

34 Virendra Sinoh

We now note that (23) for n = 1 leads to

~rZ~ S, Sl SI+ ,>IE \ l ] ~ - ; = r - ~ = 0

and since $1 ¢ 0 we get

~ i = O. (24)

In fact using this equation we can simplify the remaining equations (23) for n/> 2 to read

X mt ~>1 . . . . . mt~>l S m j S r a 2 . . • S m ~ ( ~ m l + m z + . . . + r a i n

I ~ l z ( n - l ) = 0

for n = 2, 3 . . . . (25)

Equations (22) and (24) and (25) together constitute a system o f coupled nonlinear equations to determine u and $1, $2 . . . . We regard (22) as the eigenvalue equation for while (24) and (25) are to be used to determine $1, $2 . . . . in terms of ~.

These equations are, however, highly nonlinear and coupled. One needs a systematic procedure for their solution. We now notice that (24)-(25) allow us to conclude that, as

--* oo, we have a solution for which Sn tends to constants S,, o i.e.

S,(~) ~ S, .o .

In particular Sn's admit, as a --, oo, an expansion in the inverse powers o f 0q of the form, for Z a nonrational number,

Sn(~) = Z Sn;p:q , . 4 = o . , . 5 . . . . I~l " ÷ p '

where S,. p. a are numerical constants. When Z is integral or a rational number simpler expansions can be written down.

We propose to use such ~ -+ oo expansions and use them to S, (a) in terms of • by using (24) and (25). These expansions for S~(a) when substituted in (22) would give us the eigenvalue equation for the determination of ~,.

It is known that as Z ~ 1, ~ does tend to infinity. We thus expect this procedare to clearly work when Z ~ 1 (Collet et a11980; Derrida et a11978, 1979). We will see in the next section, where we apply it to the physically most interesting case i.e. Z = 2 that the method is still quite effective.

3. Application o f the method for Z = 2

We shall now apply our method for the Z = 2 case. In this case we can use the simpler expansion given by

Feigenbaum-Cvitanovi~ equation 35

S~(~) = Y~ S . , . (26) re=O, I, 2 .... o~m

Using (24) we obtain, as • --*

St (~t) ~ - 1/2. (27)

Similarly f r o m (5) and using (27) we ob ta in the fol lowing leading behaviours , as at -* oo

1 1 1 S2 ~ ~ , S3 -~ 1 - ~ ' S4 ~ - 1280t etc. (28)

Proceed ing fur ther and equa t ing powers o f Qt on b o t h sides o f (24) and (25) we ob ta in

0 = $1,1 + 2S2, o

0 = $2, l + 3S2, oSl, 1 + 6(St, o)ZS2, o,

O ---- St, 2 + 2S2, t + 3Sa, o,

O = $2, 2 - - $ 2 , 0 + (S t , 0 )2 [$1 , 2 + 6S2, t + 15S3, o]

+ 2S,, o Sl, t [St , t + 6S2, o] + [2St, o St, 2 + S2, t ] St, o,

0 = St, s + 2S2, 2 + 3Ss. 1 + 4S,, o, etc. (29)

C o m b i n i n g the in fo rma t ion con ta ined in (28) with tha t con ta ined in (29) we ob ta in

$ 1 , o = - 1 / 2 ;

Sl, t = -- 1/4, $2. o = I /8 ;

$ I , 2 = 0, S2, 1 = 0, S3 ,0 = 0;

$1,3 = - 1/4, $2, 2 = 1/32, $3, 1 = 1/16, $4, o = 0; etc. (30)

Us ing these coefficients we ob ta in the app rox ima t ion

# ( x ) = 1 - 2 + ~ + ~ - a 2 + x + x '~

+ ( l ~ - ~ + . . . ) x 6 + . . . (31)

where all the coefficients o f the powers o f x 2 have been kep t to the o rde r (1/~2). The eigenvalue equa t ion is

J(~) = 0, (32)

1 ~ Sn where J (~t) - - + 1 + ~t ~ and we have

(X n = l

~t 7 1 5 J(0t) ~ -- ~ + g 4 ot 32r~ 2 t- 0(1/~t3). (33)

T o this o rder in a p p r o x i m a t i o n we obta in f rom (32) using (33)

0t = 5/2 = 2.5000 (34)

to be c o m p a r e d with the result o f the "exact" numerical c o m p u t a t i o n 0t = 2.5029 . . . .

36 Virendra Singh

Using the approximate value 0t --- 5/2 we get f rom the expression (31),

O(x) ~, 1 - 1 " 5 4 0 0 x 2 + 0"1300x 4 + 0"0100x 6 + . . . . (35)

We also quote, for comparison, the result of the "exact" numerical computat ion (Feigenbaum 1979)

O(x) -~ 1 - 1.5276x 2 + 0" 1048x 4 + 0.0267x 6 - 0-0035x 8 + . . . (36)

where we have kept only the four figures after the decimal sign.

4. An exact solution

An exact solution of (4) is given by

O(x) = (1 + ]xlZ) '/z, (37) 1

ct = -- 21/---- 7. (38)

This solution is, however, o f no interest for the Feigenbaum scenario for the onset o f the chaotic behaviour since O(x) has a local minimum rather than local maximum at x = 0. It however implies that the eigenvalue equation for ~t should have a root given by (38). Fo r Z -- 2 we should thus have a root at ~t = - l / x / 2 ~ - 0 . 7 0 7 . . . for the exact eigenvalue equation. Our approximate eigenvalue equat ion given by (32) and (33) i.e.

~t 7 1 5 O = - ~ + ~ + (39)

0t 320t 2

has a negative root at at -- - ( ~ i 7 + 3)/8 ~ - 0.8904 which presumably a reflection of the root ~t ~ - 0 - 7 0 7 . . . for the exact equation.

5. Concluding remarks

We thus see that the present method produces reasonably accurate results. Since the method is a systematic one the accuracy can be improved by taking higher order approximations.

The real utility of the method would be to further allow us to calculate the Z-dependance o f the universal parameter ct. Moreover whenever one needs g(x) as an input, such as for example in the equations for h(x), it is useful to have such analytic approximat ions to g(x) as we have obtained here. These and similar investigations will be reported later.

References

Collet P, Eckmann J P and Lanford O E II1 1980 Commun. Math. Phys. 76 211 Cvitanovi~ P and Myrhe im J 1983 Phys. Lett. A94 329 Derrida B, Gervois A and Pomeau Y 1978 Ann. Inst. Henri Poincare 29 305 Derrida B Gervois A and Pomeau Y 1979 J. Phys. AI2 269 Fe igenbaum M J 1978 d. Star. Phys. 19 25

Feioenbaum-Ct, itanovik equation 37

Feigenbaum M J 1979 J. Star. Phys. 21 669 Feigenbaum M J 1980 Los Alamos Science 1 4 Helleman R H G 1983 in Long-time prediction in dynamics (eds) C H Horton Jr, L E Reichl and A G Szcbehely

(New York: John Wiley) p. 95 Hu B and Mao J M 1982 Phys. Rev. A25 3259