Appl. Math. Lett. Vol. 4, No. 3, pp. 37-39, 1991 Printed in Great Britain. AiI rights reserved

08959659191 $3.00 + 0.00 Copyright@ 1991 Pergamon Press plc

Relationships Between the Solutions of Feigenbaum’s Equation

JOHN STEPHENSON AND YONG WANG

Department of Mathematics, University of Saskatchewan

(Received September 1990)

Abstract. In [1] we gave a numerical algorithm to solve Feigenbaum’s Equation which is g(z) = -ag(g(-z/a)), g(0) = 1, where g(z) is an even function and P is a constant. With this algorithm

we discovered many solutions in the form of 1 + c:, gi(r2”)i, where 71 = 1,2,. . . . We aIso found that the corresponding constant cx can be positive or-negative.

M. Campanino, H. Epstein and D. Ruehe [2] suggested that if g(r) is a solution of Feigenbaum’s

Equation on [-1, 11, then $(I) = (g( 6-

jr])) 2 is also a solution. We generalized Campanino, Epstein and Ruehe’s suggestion and discovere analytic relationships between the solutions. In the following theorems we will establish these relationships.

1. ANALYTIC RELATIONSHIPS BETWEEN THE SOLUTIONS OF FEIGENBAUM’S EQUATION

We use the notation {g(z), CY} to denote a solution of Feigenbaum’s Equation when g(z) satisfies Feigenbaum’s Equation and Q is the corresponding constant.

THEOREM 1. If { g(z), CY } is a solution of Feigenbaum’s Equation on [-1, 11, g(z) is an even function and g(0) = 1, then { &(c), -(-a

) n } is also a solution of Feigenbaum’s Equation for

any positive integer n with &(z) = gn(lcl’ “) having the same properties as g(z).

PROOF. To prove { $,(z), -(-a)” } is a solution of Feigenbaum’s Equation for any positive integer n, we must verify that

ha(z) = -[-(--~>“I 543 (in ([_(-z)“])) . We start with

-H-4nl lCIn (b (,_(-~)J) = (-aY ti” (h (&J) = C-Q’)” $J” (f (y$)) , = (--ay $2, (gn (+J) )

= (-&)ngn (lg (G) I)

= (-a)“g” (g (d$e)) ,

from the definition of tin(z)

since g(z) is an even function

since g(r) is an even function

37

38 J. STEPHENSON, Y. WANG

So { $,, , -(-a)” } is a solution of Feigenbaum’s

since { g(z), (Y } is a solution of Feigenbaum’s Equation

Equation, where n can be any positive integer.

I

THEOREM 2. If { g(z), Q } is a solution of Feigenbaum’s Equation on [-1, 11, g(z) is an even

function and g(0) = 1, then { G,(2), -_I@ } is also a solution of Feigenbaum’s Equation with

&l(r) = Ig(l4 >I 1 l/n n raving the same properties as g(z), where n can be any positive integer.

THEOREM 3. If { g(z), cx } is a solution of Feigenbaum’s Equation on [-1, 11, g(z) is an even

function and g(0) = 1, then { &(c), -]cx]“* } is also a solution of Feigenbaum’s Equation with $J,(z) = ]g(z”)]‘/” having the same properties as g(z), where n can be any positive integer.

THEOREM 4. If { g(e), CY } is a solution of Feigenbaum’s Equation on [-1, 11, g(z) is an even function and g(0) = 1, then { $~,,,~(e), -]c~yl~/~ } is also a solution of Feigenbaum’s Equation

with lClmn(z) = Ig(l4”/“)l m/n having the same properties as g(z), where m and n can be any

positive integer.

THEOREM 5. If { g(z), (Y } is a solution of Feigenbaum’s Equation on [-1, 11, g(z) is an even function and g(0) = 1, then { g,(z), -_Iol’ } is also a solution of Feigenbaum’s Equation with &(2) = ~g(~z~l/‘)~’ having th e same properties as g(z), where P is a real number > 0.

The proof of the above four theorems is quite similar to the proof of the Theorem 1.

2. NUMERICAL EXPERIMENT

With our numerical algorithm, we have found many solutions to Feigenbaum’s Equation. TWO

of these numerical solutions are { 1 + C gi z4i, ~1 } and { 1 + C Gi z2i, QZ, } , where gi, Gi, i = 1, 2 ,“‘> 10 and (~1, a2 are reported in Table 1.

Table 1. Coefficients of Feigenbaum’s Solutions { 1 + CSiz4’T 01 ) ad

{ 1 + C Gizziy CYZ ).

1 I -1.83410 79070 09410 I -3.66821 58140 18820 1.29622 26191 37174 x~O-~ 3.11901 73664 28453 x10-l

-6.20146 23283 84941 x 1O-2 -3.75394 76018 04479 x 1O-2 1.76482 14169 90452 x 1O-2 1.93502 99025 08961 x 1O-3

-2.81139 41151 26232 x~O-~ 9.51922 71514 45271 x 1O-5

3.38987 62669 37184 5.76255 23018 56619 x10-l

-1.26798 41100 33081 1.60489 95151 38390 x10-l 2.68674 32609 16483 x10-l

-1.00525 52625 96896 x10-l -3.18348 17255 68503 x~O-~ 2.62183 59709 65275 ~10-~

10 4.35491 31077 48550 x 1o-4 8.76308 90012 13328 x10-’

a1 1.69030 29714 05244 ff2 [ -2.85712 41351 41400

In order to demonstrate an application of Theorem 1, we calculated (1 + ~~=, gi z~“)~ and compared the resulting coefficients of ;c2i with Gi, i = 1, 2,. . . , 10, and also compared -(&I)~ with (~2. These numbers coincided to within the numerical precision of the computed coefficients. When we applied the transformation g2(]s]‘/2) to the classical solution to Feigenbaum’s Equation,

given in [l] where g(z) = 1 + C bi z2i and cr = 2.50290.. ., we obtained a solution of the form

{ 1 + cc; l4, a1 }. We used our numerical algorithm with 2 replaced by (21 and obtained the solution given in Table 2. Although this function has all the desired properties, it is not differentiable at 2 = 0. This is but one example of an infinite number of such solutions.

Solutions of Feigenbaum’s Equation 39

Table 2. Coefficients of Feinenbaum’s Solution

I il ci

1 -3.05526 59940 72602 2.54329 29632 08720

-2.66826 95924 53583 x10-l -7.76615 21268 04727 x~O-~ 1.65386 96833 28000 x~O-~ -2.25001 9129135483 x10-' -2.53662 93036 60474 x10-' 2.97187 61505 75294 x~O-~ 5.90207 39422 52519 x10-' -4.15184 4511867239 x10-'

-6.26454 78312 17037

REFERENCES

1. J. Stephenson and Y. Wang, Numerical Solution of Feigenbaum’s Equation, Applied Mathematics Notes 15, (to appear).

2. M. Campanino, H. Epstein and D. Ruelle, On Feigenbaum’s Functional Equation, Topology 21 (2), 125-129, (1982).

University of Saskatchewan, Saskatoon, S7N OWO Canada.