CIRCLE MAPS IN THE COMPLEX PLANE

FRACTALS IN PHYSICS L. Pietronero, E. Tosatti (editors) © Elsevier Science Publishers Β. V., 1986

439

CIRCLE M A P S IN T H E C O M P L E X PLANE

Predrag CVITANOVIC Institute of Theoretical Physics, Chalmers University of Technology, S-412 96, Goteborg, Sweden Mogens H. JENSEN, Leo P. K A D A N O F F The James Franck and Enrico Fermi Institutes, University of Chicago, Chicago, Illinois 60637, U S A and Itamar PROCACCIA Department of Chemical Physics, The Weizmann Institute of Science, Rehovot 76100, Israel Circle maps of polynomial, exponential, and rational polynomial types are studied numerically in the complex plane. The golden mean universality for real circle maps does not extend into the complex plane.

i_i_=NI£QDycTiON

For the circle maps the unstable manifold equation is

The

discovery

sality in

of the

period-doubling

search

other

dimensional dynamical

low and

period

the

for

[33,(43,(53). One the

success

the

difisomorphisms

a

scalingβ systems

of

such

ideas

transitions

to

a

in

had

some

chaos

for

(circle

maps).

variety of physical

we refer the reader to refs. discussion

of

refs.

problems

have

the circle

model

in

of

their

[63,C73 physical

applications. scaling

for the critical circle maps was discovered in a of mappings with the golden mean

numberC83,C93,C103.

An elegant

such asymptotically universal is

afforded

(

2

Ρ

by

by

the

winding

formulation

are

made

unstable

equationsC113,[123,[133,C143,[33.

manifold

=

0

9

1 +ρ / δ

(

α

(see

for

9

1 +1 / δ +ρ / δ 2

solutions

difficult

problems.

To best

successful manifold

example refs.

of

(

of

Ζ

/ )θ

equations

2

,

this equation

by

subtle

convergence

our

knowledge, only two

numerical solutions of the are

unstable

extantC163.

These

convergence problems, as well as the interest in understanding

other

universalities

with the circle

mapsC153,

has

investigate the

structure

of

circle

maps.

Such

associated

motivated us to the complexified

investigations

yielded new insights into

have

universal

scaling lawe[133,[143, C173,Ι1Θ3, as well as much beautiful

mathematics

(see for

example

refs.

C193, [203, [213, [223, [233, [243 >.

of

self-similarities

)

However, numerical

previously

The first example of a universal

study

9

in

(the

observations

reviewed

class

on

Haps of this type

for

are

universality

are

systems:

universal

experimental

doublings

which

given

[123,[153):

one-dimensional iterations CI], [23 has

prompted a

theory

univer­

The reader is [15] and maps.

In

referred

to

refs.

[63,

[253 for an introduction to the circle this note we concentrate only on some

general properties of complexified circle maps.

P. Cvitanovic et al.

440

i.i.THE.CUBIC^IRCLE.MAP

The boundary

of

the

large

(parameter values for which As the first example

of

a

complexified

circle nap, consider the critical cubic map zn + l

= Q +

»

+ 1 / 2 ) mod 1

R < ze

(a cubic circle map is

critical

if

has

a

This

discontinuous

all

y * 0.

This

fixed

point

Idz'/dzl = 1. The ζ

cubic inflection point z'=z"=0).

for

critical point converge to a stable fixed point)

which the

periodic along the real axis, and at ζ = ± 1 / 2 *iy

map

component

is given by the cardioid of parameter values for

(2.1) it

central

the iterates of the

is

is the

and

therefore

is

marginally

map is monotone

cannot exhibit

along the real axis;

however,

stable,

for ζ real

period

doubling

bifurcations

to

period 2 occur on the imaginary axis, generating the

pair

of

2-cycle "hearts".

One

of

these

crudest example of the polynomial approximations to circle maps of kind solutions

of

the

used

unstable

in

the

manifold

numerical equation

(1.1). The parameter critical

Mandelbrot

set

(the

set

of

values for which the iterates of point

tend

62ly.&omial cubic map

to is

infinity) plotted

in

for

all the a

fig. 2.1.

FIGURE 2.2 An enlargement of the "heart" from fig. 2.1.

hearts is plotted in fig. 2.2. a

stable

rational

value

boundary.

of

dz'/dz

the

Mandelbrot

ζ

c+z2*.

set

every

along

the a

set

"hearts", analogous to

for

Just as in the

each component is centered cycle, and one expects

phase

at

This gives rise to

composed of self-similar

FIGURE 2.1 The Mandelbrot set for the critical polynomial cubic mapping ζ -» c • 4 z 3.

A bifurcation to

periodic orbit takes place

cardioid

cycle-2

the

quadratic

quadratic

map

case C141,

on a superstable m/n

universal

scaling

for the infinite sequences of bifurcations.

laws

Circle maps in the complex plane

The 5ΪΣ9=§

shape of the Mandelbrot set for

»aps

will

be

described

in

the

the

next

section.

the

four The

rays

of smaller disks

higher

iterates

of

bgsin_of_attrgction

(the set of

all

initial values of ζ whose iterates do not escape to infinity) of the cubic

circle

superstable fixed point for map

(2.1)

is

plotted

in

fig. 2.3

at ±.60°,

(2.1)

similarly the other sequences of

The

the

to

±120°.

441

disks

onto the real axis.

(The

image visible abrupt

truncation close to Im ζ = .95 is an artifact of the Re ζ modularity condition in (2.1).)

in 3_z._THE_STANDARD_CIRCLE_MAP

fig. 2.3.

As the second example of

a

complexified

circle map we take the sine map zn + l

= Ω + nz

(k=l for attraction the

the

~ 7π

s

i ( 2n^ n )

critical

.

case).

mod 1

(3.1)

The

basins of

have the same basic structure as

previous

example;

point basin of attraction

the

superstable

in

fixed

is given in fig. 3.1,

and the superstable 3-cycle basin of

attraction

is given in fig. 3.2 as typical examples.

FIGURE 2.3 The upper right quadrant of the basin of attraction of the superstable fixed point for the cubic map (2.1) ( Λ « 0 ) . The remaining quadrants are obtained by reflection about the χ and y axes.

The

general

attraction

shape

of

basins

for the complex circle maps is

to understand.

radius

Izl = 1/2.

disk.

The

easy

Idz'/dzI * 1 on the circle of This

defines

periodicity

the

condition

mod 1) generates an infinity

structure

central

(Re(z*l/2)

of such disks, one

integer value of Re z.

self-similar

of

The map (2.1) has a boundary of

marginal stability

for every

the

The remaining

arises

from

this

periodicity; it is generated by all preimages of the

Izl - 1/2

maps

initial

disks along the real points

with

βχρ(±ίπ/3) onto the real

phase

axis.

axis.

FIGURE 3.1 The basin of attraction for the superstable fixed point of the sine map (3.1).

ζ

βχρ(±2ίπ/3),

This gives rise

The complexified sine map is

an

example

of an exponential map[263: with substitutions u = exp(2niz) (3.1) becomes

c = exp(2niQ)

442

P. Cvitanovic et al.

extends in the complex plane into a "hearts'* set of

fig. 2.1.

By

crossing

from

the

central

component into one of the hearts, one drives the (here k = l ) . (2.1),

or

Unlike the the

discuss in the maps

polynomial maps of type

rational maps next

which

section,

the

we

shall

exponential

have basins of attraction which extend

to

infinity and are dense everywhere in the complex plane. y

For example, the image of ζ * ± 1 / 4 + iy,

large,

lies

close

to

the

real

fig. 3.2, and in the same way any ray

axis of

in

disks

which is a preimage of the real axis extends

to

infinity.

mapping

through

a

period

n-tupling

changing its winding number

(for

without

example,

the

1/2 winding number period-doubles to 2 / 4 ) . Hence the

Mandelbrot

constructed

set

from

first, the map has complex

for

two a

a

circle

basic cubic

map

building

is

blocks:

inflection, so the

period n-tuplings are characterized

by

the self-similar hearts of fig. 2.1; second, the map

is

periodic, and that gives

rise

to

the

infinite sequences of copies of the basic hearts set along the real axis, and along the

rays

in

the complex plane. Similarly,

the basins of attraction

the critical circle maps the basin

corresponding of

are

building

attraction

for

constructed blocks:

the

for from

first,

critical

a

cubic

FIGURE 3.2 The basin of attraction for the superstable 1/3 cycle of the sine mop (3.1).

For the same reasons the for the

Mandelbrot

set

sine map (3.1), plotted in fig. 3.3, is

dense everywhere over the entire complex plane. The

general structure of the

Mandelbrot

sets for the critical circle maps is illustrated by fig. 3.3 . Along the real mode-locking interval for

(the "devil's staircase"C63). cubic

inflection,

each

axis

there

is

a

every rational number As

the map has a

mode-locking

interval

FIGURE 3.3 The Mandelbrot set for the * 3 P u < i3 i >e l?wer complex halfplane is obtained by the reflection about the real axis. The black region corresponds to values of parameter for which the critical point does not iterate away to infinity. The different shades of aray are an indication of the escape velocity; the longer the number of iterations needed to reach a cutoff, the lighter the shadina. The large "hearts" set on the ends of the interval corresponds to the fixed point, the hearts set in the middle corresponds to the orbits with the winding number 1/2, and so on.

Circle maps in the complex plane

443

polynomial map; second, from the infinity of its

series expansion of the exponentials

preimages

generated

It

condition.

This is illustrated in fig. 3.4

the

euperstable

winding

by

the

period-doubled

periodicity

cycle

number 0/1. The triangular

of

attraction

for

the

periodicity

period-2 of

polynomial

cubic

along the real axis,

and

critical

circle

map

(3.2).

in

with

a

cubic

u'

* c(l

- r 3/ 2

r

= 2niz

- r 5/ 4

+ r 6/ 8

+...) (4.2)

basins

the set is generated by

a

inflection point at u*l:

with

arrangement

of the central 4 disks is typical of the

maps; the remainder

for

is

The

Mandelbrot

set for this map

is

given

in

fig. 4.1.

its

preimages in the complex plane.

FIGURE 4.1 The Mandelbrot set for the rational map (4.1). The fixed point hearts set is centered on c*l, the 1/2 winding on c » - l . The remaining infinity of other mode lockings lie on the unit circle.

•κFIGURE 3.4 The basin of attraction for the euperstable period-doubled cycle of (3.1) with winding number 0/1. The central part of the ~li\*A % £ £ ta r *e ic o n is the same as for the period-2 polynomial cubic maps. The remainder of , 5 ? . s t# ? Γ· replicas generated by the periodicity along the real axis, and the preimages In the complex plane.

4 . A RATIONAL POLYNOMIAL MAP

As the third

example

of a circle map we

take the rational polynomial map u'

= cu(3-ku)/(3-k/u)

(k=l here). by the

The

lowest

form order

l

of this map is motivated truncation

of

the power

(

4

e

FIGURE 4.2 The basin of attraction of the rational polynomial map (4.1) for the parameter value corresponding to the golden-mean winding number.

P. Cvitanovic et al.

444

The

rational

naps

differ

fro*

exponential maps in one important aspect; Mandelbrot sets and their basins are finite rational

of

the their

attraction

in extent, because for large lul the maps

behave

like

preimages of the real axis (more precisely, as (4.1) circle map,

the

polynomials. of

the

is

preimages

an of

The

circle

map

exponentiated t u1=1 unit

the

circle) are themselves closed

loops.

illustrated by the

attraction of the

map (4.1) for the

basin

of

golden-mean

This

winding

is

number,

fig. 4.2.

The

unstable

manifold

equation

(1.1)

states that in the neighborhood of the parameter value corresponding to

the golden-mean winding,

the

is

Mandelbrot

rescaling

by

set

self-similar

the Shenker'e

universal

number δ . We have investigated this by blowing up the

scaling

numerically

neighborhood of the parameter

value corresponding to the golden mean and

comparing

the

corresponding Fibonacci

sizes of

to

the

numbers.

Mandelbrot

under

FIGURE 5.1 The Mandelbrot set fig. 3.3 f or the sine map (3.1) in the neighborhood of the parameter value corresponding to the golden mean winding. The largest hearts set corresponds to the 55/89 Fibonacci numbers ratio. While the successive Fibonacci ratios' hearts sets do scale by Shenker'e universal scaling, their exteriors contain more and more "hair".

the

successive

In this

winding,

hearts

sets

ratios

neighborhood

set does indeed scale with the

universal factor as in

the real case.

of the same

However,

the golden mean universality does not generalize to the complex plane in several important ways. The from

first

the

ions C27],C24]; "hairier" This can

non-universality is

theory the

is be

of longer

the

by

comparing

iterat­

cycle,

the exterior of the seen

familiar

polynomial

hearts

the set.

fig. 2.1, the

fixed point hearts set, with fig. 5.1. The other complex

circle

exteriors of

non-universal maps

hearts

feature

is

the

fact

sets

for

different

remain unmistakably distinguishable,

of the

that

the maps

regardless

of the degree of magnification of the the golden

FIGURE 5.2 The Mandelbrot set fig. 4.1 for the rational polynomial map (4.1) in the neighborhood of the parameter value corresponding to the golden mean winding. The largest hearts set corresponds to the 21/34 Fibonacci numbers ratio. While the successive Fibonacci ratios' hearts sets do scale by Shenker'e universal scaling, their exteriors are different from those for the exponential maps, such as fig. 5.1.

Circle maps in the complex plane

mean winding neighborhood.

This

can be seen by

comparing fig. 5.1 with fig. 5.2. While all the external raye for

the

sine sap go to infinity,

the exteriors of

hearts

polynomial

are

map

preimages these

of

loop

the

sets

decorated

unit

type

of

by

circle.

decorations

scale

universal Shenkerβ number as the hearts

for the rational looplike

Numerically, by

the

the

same

interiors

of

sets, so we can always determine the the

starting

approximation

unstable manifold, regardless of the magnification.

We

conclude

that

manifold equation (1.1) has no

to

the

degree

of

the unstable

unique

analytic

thanks

J.H.

many

to

J.H.

inspiring

Hubbard

and

Hubbard and B.

discussions. the

Department

Mathematics, Cornell University, to the Mathematics

VAX,

kind

with

assistance

iteration

programs,

DE-AC02-83-ER13044. support

by

of

support

the

of

for the access

the

two-dimensional

which

most

were generated.

grateful to H.J. Feigenbaum for the at the Laboratory

P.C.

and B. Wittner for the

with

graphics in this paper

acknowledges

4. Bai-Lin Hao, ed.. Singapore, 1984).

of

the

P.C. is

hospitality

Solid

State Physics, and

from

DOE

I.P. Minerva

Chaos

(World

in

chaos

Scientific,

5. P. Collet and J.-P. Eckmann, Iterated maps on Interval as dynamical systems (Blrkhauser, Boston, 1980). 6. M.H. Jensen, P. Bak and T. A30, 1960 (1984). 7. T. Bohr, M.H. Jensen and P. A30, 1970 (1984). 8. S.J. Shenker, Physica reprinted in ref. C33.

Bohr, Bak,

5D,

Phys

Rev.

Phys

Rev.

405

(1982),

9. M.J. Feigenbaum, L.P. Kadanoff and S.J. Shenker, Physica 5D, 370 (1982), reprinted in ref. C43. 10. S. Ostlund, D. Rand, J. Sethna Siggia, Physica D8, 303 (1983).

12. H. (1982).

ACKNOWLEDGEMENTS

Branner for

3. P. Cvitanovicf, ed.. Universality (Hilger, Bristol, 1984).

and E.D.

11. H. Daido, Phys. Lett. 83A, 246 (1961); 86A, 259 (1981).

continuation into the complex plane.

We are grateful

445

contract

no.

acknowledges

partial

Foundation,

Munich,

Daido,

Prog.

Theor.

Phys.

67, 1698

13. A.I. Golberg, Ya.G. Sinai and K.H. Usp. Mat. Nauk 38, 159 (1983).

Khanin,

14. P. Cvitanovic and J. Myrheim, Nordita preprint 84/5 (1984), submitted to Comm. Math. Phys. (Feb. 1984). 15. P. Cvitanovic, M.H. Jensen, L.P. and I. Procaccia, Phys. Rev. Lett. (1985).

Kadanoff 55, 343

16. G. Gunarathne (unpublished); H.J. Feigenbaum (unpublished). 17. N.S. Manton and H. Nauenberg, Commun. Hath. Phys. 89, 555 (1983). 18. H. (1983).

Widom,

Commun.

Hath.

Phys.

92, 121

19. A. Douadv and J.H. Hubbard, C.R. Acad. Sci. Paris 294, 123 (1982). 20. P. Fatou, Bull. Soc. Hath. France 47, 161 (1919); ibid. 48, 33 and 208 (1920).

Germany. M.H.J, and L.K. work has been supported

21. G. Julia, 47(1918)

in part by the U.S. Office of Naval Research and

22. B.B. Mandelbrot, Ann. N.Y. Acad. Sci. 357, 249 (1980).

by the Materials Research Laboratory.

J.

Hath.

Puree

et

Appl.

4,

23. B.B. Mandelbrot, The Fractal Geometry of Nature, (Freeman, San FrancTsc57"I982T. " ~ * ~ 24. John Hilnor, IHES preprint (Hay 1985). 25. P. Cvitanovic, B. Shraiman and B. Soderberg, Phys. Scripta (to appear).

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(1978),

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669

26. Similar maps have been studied by R. Devaney (unpublished) and J.H. Hubbard (private communication). 27. A. Douady communication); conjecture).

and J.H. Hubbard (private B.B. Handelbrot (unpublished