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On the universality and computation of Feigenbaum's δ
Volume 84A, number 9
PHYSICS LETTERS
31 August 1981
ON THE UNNERSALITY AND COMPUTATION OF FEIGENBAUM’S ö James B. McGUIRE Physics Department, FloridaAtlantic University, Boca Raton, FL 33432, USA
and Cohn J. THOMPSON
1
School of Natural Sciences, The Institute for Advanced Study, Princeton, NJ 08540, USA Received 22 June 1981
Universal properties of maps on the unit interval with a single zero minimum and unit maximum are shown to depend on the product of the powers describing the behavior of the function at its minimum and at its maximum. A rapidly convergent algorithm for computing 6 is presented.
For functions ~ of the form shown in fig. 1 and depending on a parameter X that tunes the slope ~~(x*) of ~ at its single non-trivial fixed point x”, it is well known [1] that as 0I(x*)becomes increasingly negative, x” bifurcates to a cascade of stable 2n cycles of 0 at successive parameter values Xn, n = 1,2, and that typically the Xn have a finite accumulation point ?ç, as n tends to infinity. Feigenbaum discovered recently [2] that the rate of convergence I X~ x I n as n (1) ...
—
—
1
On leave 1980—81 from Mathematics Department, University of Melbourne, Parkville, Victoria 3052, Australia.
~
/ / / /
Fig. 1. Relevant part of the function
is governed by a parameter ~ that is apparently insensitive to the form of 0 except in the immediate neighborhood of its maximum. For example in the usual case of aquadratic maximum, Feigenbaum found that ~ 4.669 201 6 There are now elaborate theories for this universal phenomenon [2] some parts of which have been made completely rigorous [3]. Our purpose here is to report briefly on an alternative approach to this phenomenon [4] that further elucidates the nature of the universality and which also provides a rapidly convergent method for computing ~ from an associated universal function. For reasons which will become clear in a moment we prefer to deal with functions 0 defined on the unit interval as shown in fig. 2. These functions are related by a conjugacy, defined in general by ...
.
,
(2) to the relevant part of functions shown in the square of fig. 1. (The particular conjugacy g(x) = I x in this case amounts to turning the square upside down and back-to-front.) Also shown in fig. 2 is the second iterate 0(2) ç~p of 0 for the case A = 0(1)
inside the square.
451
Volume 84A, number 9
PHYSICS LETTERS
~
31 August 1981
powers
describing the behavior off near its minimum
and its maximum. To be more precise if one takes 0(x)=I1 _X~x~I~ (s,t>0), (6) the universal properties derived from its associated fixed point function will only depend on the product st, since all power law conjugates of (6) are conjugate to =
_______________________
—~
Fig. 2. Form of conjugated ~. The relevant parts of ~(~x)) are inside the two small squares.
piece of 0(2) resembles a scaled down version of 0. By rescahing to the unit square we are led naturally to define a doubling transformation T by Tçb(x) =A’O(O(Ax))
(A
=
0(1) < 1).
—
XxIst.
(7)
It is thus possible to h~efunctions with quadratic minima (or maxima in fig. 1) that do not have 6 =4.6692016.... The conventional approach to 6, eq. (1), is to investigate the local instability of a fixed point f of T. Thus, for small e one obtains from (3) and (4) To(f+ h)(x) f(x) + e%oh(x) + O(e2), (8)
1
o
Il
(3)
What one typically finds is that under the action of T, successive iterates TkO either terminate in stability or chaos except when A = A,,, where they converge to a fixed point f of Tsatisfying f(x) = a~f(f(ax)) (a = f(1) < 1). (4) Such a fixed point is, of necessity, unstable. If, however, one has a sequence of functions 0k = Tkcb that converge tof then conjugates (2) of 0 also converge to conjugates off and moreover the asymptotic form of this conjugacy depends solely on the form of the mitial conjugacyg atx = 0. In particular, ifg(x)’~xPas x -÷ 0 (p > 0) ‘~-f(x)=[f(x1~)]~’ ask 00 1’g~~~g~(x) (5) T andjllx) is also a fixed point of Twith reduction parameterf(1) = a~. In terms of T then, one can identify power law conjugacy classes of functionsf and related by eq. (5) as universality classes. This leads immediately to the observation based on (5) that universal properties depend not only on the behavior off near its (zero) minimum but also on its behavior near its maximum (atx
f
where the linear operator J2~is defined by Efoh(x) = a~[h(l)(xf’(x) —f(x)) (9) +f’(f(ax))h(ax) + h(f(ax))I. Repeated application of (8) then gives
r~o(f+eh)(x) =f(x) + eJ2~oh(x) + O(e2) ~f(x) + e6’1H(x) + 0(e2) as n where 6 > 1 is the maximum eigenvalue of
(10)
-~
.I2~(and H
£4.
depends on h(x) and the principal eigenfunction of Now if one chooses say 0(x) =f(X 0x) with a stable 2’s-cycle, and expands for large n about A,,, ( 1 in this case), e = A,~ A,, and h(x)xf’(x) in (10), the lefthand side of (10) remains bounded as n 00, and eq. (1) follows. The is a universal quantity only onfact st inthat (6) 6follows from the fact thatthat thedepends point spectrum of .2~is invariant under a power law conjugacy eq. (5) [4]. Furthermore, with a certain amount of work [4] one can express 6 in terms of its associated regular fixed point function (f’(O) finite and non-zero) obtamed for example by iterating (7). Firstly one shows from (9) that as 1 -÷ —
-~
2~on 0~’(x)
0). In fact, because of power law conjugacy, universal properties will only depend on the product of the 452
(11) n~(0)[f’(x)+f’(O)(xf’(x) —f(x))],
where
nl(x)=a-’
31 August 1981
PHYSICS LETTERS
Volume 84A, number 9
2’- 1 n (f(k)&)) c * k=O f’k”(&)
(12)
X,+,/X,
is asymptotically independent of x E: [0, l] (since a < 1) and also, as it turns out, on the functional form of no(x). (fck) =fof@-L) is the kth iterate offunder functional composition andflk)’ its fist derivative.) Setting x = 0 and no(x) = x in (12) and comparing (11) and (10) we then have the result (13) where, as a kind of continued
Using Feigenbaum’s algorithm [ 2 ] for computing f(x) for the regular case st = 2 one obtains the estimates = 4.671,4.669
22,4.669
204,4.669
201 9, (15)
from (14) for I = 1,2,3,4 compared with Feigenbaum’s value of 4.669 201 6 ... for 6. C.J.T. would like to thank the Institute Advanced Study for their kind hospitality.
for
References [l] R.M. May, Nature 261 (1976) 459.
fraction expansion,
[2] M.J. Feigenbaum, J. Stat. Phys. 19 (1978) 25; 21 (1979) 669. [3] P. Collet, J.-P. Eckmann and O.E. Lanford III, Commun.
x [fww +f'&-(o))
j
1
I
fWS”))) + -.- f
(14)
Math. Phys. 76 (1980) 211. [4] J.B. McGuire and C.J. Thompson, Asymptotic properties of sequences of iterates of non-linear transformations, to be published.
453
PHYSICS LETTERS
31 August 1981
ON THE UNNERSALITY AND COMPUTATION OF FEIGENBAUM’S ö James B. McGUIRE Physics Department, FloridaAtlantic University, Boca Raton, FL 33432, USA
and Cohn J. THOMPSON
1
School of Natural Sciences, The Institute for Advanced Study, Princeton, NJ 08540, USA Received 22 June 1981
Universal properties of maps on the unit interval with a single zero minimum and unit maximum are shown to depend on the product of the powers describing the behavior of the function at its minimum and at its maximum. A rapidly convergent algorithm for computing 6 is presented.
For functions ~ of the form shown in fig. 1 and depending on a parameter X that tunes the slope ~~(x*) of ~ at its single non-trivial fixed point x”, it is well known [1] that as 0I(x*)becomes increasingly negative, x” bifurcates to a cascade of stable 2n cycles of 0 at successive parameter values Xn, n = 1,2, and that typically the Xn have a finite accumulation point ?ç, as n tends to infinity. Feigenbaum discovered recently [2] that the rate of convergence I X~ x I n as n (1) ...
—
—
1
On leave 1980—81 from Mathematics Department, University of Melbourne, Parkville, Victoria 3052, Australia.
~
/ / / /
Fig. 1. Relevant part of the function
is governed by a parameter ~ that is apparently insensitive to the form of 0 except in the immediate neighborhood of its maximum. For example in the usual case of aquadratic maximum, Feigenbaum found that ~ 4.669 201 6 There are now elaborate theories for this universal phenomenon [2] some parts of which have been made completely rigorous [3]. Our purpose here is to report briefly on an alternative approach to this phenomenon [4] that further elucidates the nature of the universality and which also provides a rapidly convergent method for computing ~ from an associated universal function. For reasons which will become clear in a moment we prefer to deal with functions 0 defined on the unit interval as shown in fig. 2. These functions are related by a conjugacy, defined in general by ...
.
,
(2) to the relevant part of functions shown in the square of fig. 1. (The particular conjugacy g(x) = I x in this case amounts to turning the square upside down and back-to-front.) Also shown in fig. 2 is the second iterate 0(2) ç~p of 0 for the case A = 0(1)
inside the square.
451
Volume 84A, number 9
PHYSICS LETTERS
~
31 August 1981
powers
describing the behavior off near its minimum
and its maximum. To be more precise if one takes 0(x)=I1 _X~x~I~ (s,t>0), (6) the universal properties derived from its associated fixed point function will only depend on the product st, since all power law conjugates of (6) are conjugate to =
_______________________
—~
Fig. 2. Form of conjugated ~. The relevant parts of ~(~x)) are inside the two small squares.
piece of 0(2) resembles a scaled down version of 0. By rescahing to the unit square we are led naturally to define a doubling transformation T by Tçb(x) =A’O(O(Ax))
(A
=
0(1) < 1).
—
XxIst.
(7)
It is thus possible to h~efunctions with quadratic minima (or maxima in fig. 1) that do not have 6 =4.6692016.... The conventional approach to 6, eq. (1), is to investigate the local instability of a fixed point f of T. Thus, for small e one obtains from (3) and (4) To(f+ h)(x) f(x) + e%oh(x) + O(e2), (8)
1
o
Il
(3)
What one typically finds is that under the action of T, successive iterates TkO either terminate in stability or chaos except when A = A,,, where they converge to a fixed point f of Tsatisfying f(x) = a~f(f(ax)) (a = f(1) < 1). (4) Such a fixed point is, of necessity, unstable. If, however, one has a sequence of functions 0k = Tkcb that converge tof then conjugates (2) of 0 also converge to conjugates off and moreover the asymptotic form of this conjugacy depends solely on the form of the mitial conjugacyg atx = 0. In particular, ifg(x)’~xPas x -÷ 0 (p > 0) ‘~-f(x)=[f(x1~)]~’ ask 00 1’g~~~g~(x) (5) T andjllx) is also a fixed point of Twith reduction parameterf(1) = a~. In terms of T then, one can identify power law conjugacy classes of functionsf and related by eq. (5) as universality classes. This leads immediately to the observation based on (5) that universal properties depend not only on the behavior off near its (zero) minimum but also on its behavior near its maximum (atx
f
where the linear operator J2~is defined by Efoh(x) = a~[h(l)(xf’(x) —f(x)) (9) +f’(f(ax))h(ax) + h(f(ax))I. Repeated application of (8) then gives
r~o(f+eh)(x) =f(x) + eJ2~oh(x) + O(e2) ~f(x) + e6’1H(x) + 0(e2) as n where 6 > 1 is the maximum eigenvalue of
(10)
-~
.I2~(and H
£4.
depends on h(x) and the principal eigenfunction of Now if one chooses say 0(x) =f(X 0x) with a stable 2’s-cycle, and expands for large n about A,,, ( 1 in this case), e = A,~ A,, and h(x)xf’(x) in (10), the lefthand side of (10) remains bounded as n 00, and eq. (1) follows. The is a universal quantity only onfact st inthat (6) 6follows from the fact thatthat thedepends point spectrum of .2~is invariant under a power law conjugacy eq. (5) [4]. Furthermore, with a certain amount of work [4] one can express 6 in terms of its associated regular fixed point function (f’(O) finite and non-zero) obtamed for example by iterating (7). Firstly one shows from (9) that as 1 -÷ —
-~
2~on 0~’(x)
0). In fact, because of power law conjugacy, universal properties will only depend on the product of the 452
(11) n~(0)[f’(x)+f’(O)(xf’(x) —f(x))],
where
nl(x)=a-’
31 August 1981
PHYSICS LETTERS
Volume 84A, number 9
2’- 1 n (f(k)&)) c * k=O f’k”(&)
(12)
X,+,/X,
is asymptotically independent of x E: [0, l] (since a < 1) and also, as it turns out, on the functional form of no(x). (fck) =fof@-L) is the kth iterate offunder functional composition andflk)’ its fist derivative.) Setting x = 0 and no(x) = x in (12) and comparing (11) and (10) we then have the result (13) where, as a kind of continued
Using Feigenbaum’s algorithm [ 2 ] for computing f(x) for the regular case st = 2 one obtains the estimates = 4.671,4.669
22,4.669
204,4.669
201 9, (15)
from (14) for I = 1,2,3,4 compared with Feigenbaum’s value of 4.669 201 6 ... for 6. C.J.T. would like to thank the Institute Advanced Study for their kind hospitality.
for
References [l] R.M. May, Nature 261 (1976) 459.
fraction expansion,
[2] M.J. Feigenbaum, J. Stat. Phys. 19 (1978) 25; 21 (1979) 669. [3] P. Collet, J.-P. Eckmann and O.E. Lanford III, Commun.
x [fww +f'&-(o))
j
1
I
fWS”))) + -.- f
(14)
Math. Phys. 76 (1980) 211. [4] J.B. McGuire and C.J. Thompson, Asymptotic properties of sequences of iterates of non-linear transformations, to be published.
453