*Tel.: #34-958-243360; fax: #34-958-243384.E-mail address: julyan@hp1.uib.es (J.H.E. Cartwright)

Computers & Graphics 23 (1999) 607}612

Chaos & Graphics

Newton maps: fractals from Newton's method for the circle map

Julyan H.E. Cartwright*

Instituto Andaluz de Ciencias de la Tierra, CSIC-Universidad de Granada, E-18071 Granada, Spain

Abstract

To understand why two interacting oscillators synchronize with each other, or lock together and resonate at somerational frequency ratio, dynamical-systems theory shows that one should study circle maps and their periodicorbits. One can easily explore the structure of these periodic orbits using Newton maps, derived from Newton'smethod for "nding the roots of an equation. I present here some interesting and beautiful examples of fractalsencountered in Newton maps while investigating the periodic orbits of circle maps. ( 1999 Elsevier Science Ltd. Allrights reserved.

Keywords: Circle maps; Coupled oscillators; Newton-Raphson method; Periodic orbits; Resonance

1. Introduction

In 1665, Christiaan Huygens made a serendipitousdiscovery. Working on a military project to developbetter time-keeping at sea, he had two pendulum clocksin his laboratory * very expensive pieces of hardwaredestined for Dutch warships * hanging from a poleslung across the backs of two chairs. He noted [1] thatover a period of half an hour or so, the two pendulumstended to lock into step with each other.

He called his discovery the sympathy of clocks. Today,it is known as locking, resonance, or synchronization inoscillators, and is an enormously important part of dy-namical-systems theory that has applications to physics,chemistry, biology, engineering, in fact to any placewhere there is something that oscillates. The theory ofdynamical systems allows one to investigate synchroni-zation by showing that the behaviour of complex oscillat-ing systems like pendulum clocks is described by simpleequations called circle maps.

The sine circle map

hm`1

"hm#X!

k

2psin 2ph

m, mod 1 (1)

is a one-dimensional discrete mapping that describeshow an oscillator of natural frequency X behaveswhen coupled to another of frequency one by a couplingof strength k. When k"0 the oscillator runs uncoupledat frequency X, but when k'0 it can lock into aperiodic orbit: a resonance with some rational ratiop/q to the driving frequency. To measure the averagerotation per iteration of the map * the frequencyof the oscillator * it is useful to de"ne the rotationnumber [2]

o(h)" lim

m?=

hm!h

02pm

. (2)

Keeping k constant, we can iterate the circle map forvalues of X between 0 and 1, and plot rotation numbero against X. Luckily, it is not necessary to iterate toin"nity; the "rst few iterations can be discarded to re-move transient e!ects, and a few hundred iterations arethen su$cient to give an accurate value for o. This plothas been termed the devil's staircase; the stairs are lock-ings: periodic orbits with di!erent rational rotation num-bers that show up as #at steps as in Fig. 1. When k(1the map is called subcritical, and intervals on which therotation number is constant and rational, where there isa periodic orbit of a particular period, punctuate inter-vals of increasing rotation number, whereas in supercriti-cal circle maps (k'1) the periodic orbits overlap. But in

0097-8493/99/$ - see front matter ( 1999 Elsevier Science Ltd. All rights reserved.PII: S 0 0 9 7 - 8 4 9 3 ( 9 9 ) 0 0 0 7 8 - 3

Fig. 1. The devil's quarry in the sine circle map for 0(X(1 and 0(k(2. The periodic orbits, the steps in the quarry, are sizedaccording to the Farey rule (3). The devil's staircase is a section front to back (k constant) through the quarry. The steps are often calledArnold tongues after the mathematician V.I. Arnold.

a critical circle map at k"1, the staircase is complete; atevery value of X there is a periodic orbit, and the rotationnumber increases in a staircase fashion with steps at eachrational rotation number and risers in between. Thestaircase has an in"nity of steps at all scales so thatsomeone climbing the staircase step by step would neverreach the top * a devilish construction indeed! Thedevil's staircase becomes the devil's quarry illustrated inFig. 1 when we look at the picture in terms of thenonlinearity k as well as the forcing frequency X, so thatwe have rotation number o represented by height ina three-dimensional quarry. The size of the step decreasesas the period of the associated cycle increases. Betweenany two steps p/q and p@/q@ associated with rationalrotation numbers, the largest intermediate step is givenby the rational number having the smallest denominatorin that interval. This ordering of periodic orbits is pro-

vided by the Farey tree [3}6] constructed using theFarey mediant

p

q=

p@

q@"

p#p@

q#q@. (3)

To obtain the Farey tree with this rule we start withthe two ends of the unit interval written as 0

1and 1

1, to

produce the ordering at the "rst level 12, at the second

level 13and 2

3, at the third level 1

4, 25, 35, and 3

4, and so on. We

can see this ordering of periodic orbits re#ected in thesizes of the steps in Fig. 1.

There are many other fascinating phenomena in thecircle map, for example, chaos is found in the supercriti-cal circle map as iterates wander between the overlappingresonances; this is seen in the devil's quarry of Fig. 1 asthe jagged rock face for k'1 [7]. Here, though, I shallconcentrate on the periodic orbits.

608 J.H.E. Cartwright / Computers & Graphics 23 (1999) 607}612

2. Newton maps

To study the Farey ordering of the periodic orbits weshould to be able to locate with great precision wherethey begin and end. To do this we can use the methodNewton invented for "nding the roots (zeroes) of a func-tion g(x)

xm`1

"xm!(J~1(x

m)) g(x

m), (4)

where J is the Jacobian matrix of g(x). We want to "ndperiodic orbits p/q of the sine circle map

f(h)"h#X!

k

2psin 2ph, mod 1, (5)

where f q(h)"h. We wish to locate the boundary points ofthese periodic orbits where

Lf q(h)

Lh"c"1. (6)

If we want to look at other aspects of the periodicorbits, we could also "nd the superstable points for c"0,or the period doubling points at which c"!1, so wewill do the analysis for general c. Using Newton's methodwe wish to have

f q(h)!h!p"0,Lf q(h)

Lh!c"0. (7)

We let

g"Cf q(h)!h!p,Lf q(h)

Lh!cD, (8)

and use Newton's method to "nd the roots of g.The Newton maps that are produced quickly become

mathematically complex as the period of the orbit in-creases. To give a simple example 1 shall look "rst at theperiod-one (i.e., q"1) or "xed points in the circle map, atwhich X!k/(2p) sin 2ph"0 mod 1. In this case we have

g"CX!

k

2psin 2ph!p, 1!k cos 2ph!cD. (9)

We can locate the period-one orbits either along linesof "xed k, varying X, or along lines of "xed X, varying k.I shall analyse the Newton maps produced by each of thetwo cases in turn.

3. The period-one [h, X] Newton map

Let us look at the Newton map of period-one ("xed)points in the sine circle map using h and X as variables(x"[h, X]). The Jacobian is

J"C!k cos 2ph 1

2pk sin 2ph 0D, (10)

which from (4) gives us the Newton map

hm#1"h

m#

1

2p Acot 2phm#

c!1

kcosec 2nh

mB,X

m`1"p#

k

2n Acosec 2nhm#

c!1

kcot 2nh

mB. (11)

Despite appearances, this is a one-dimensional map,since h

m`1and X

m`1depend only on h

m, and not on X

m.

The "xed points [hfp

, Xfp

] of the Newton map are at

Cn#a1

2p, p#

1

2pJk2!(1!c)2D, n3Z (12)

and

Cn#a2

2p, p!

1

2pJk2!(1!c)2D, n3Z, (13)

where a1

and a2

are the two values of arccos((1!c)/k) inthe range [0, 2p). Of course if DkD(D1!cD then there areno "xed points.

If, as is often the case, we are interested only in "ndingX values, whatever the value of h, we may lump together

all the "xed points with Xfp"p#1/(2p)Jk2!(1!c)2

and all the others with Xfp"p!1/(2p)Jk2!(1!c)2.

In this way we are left with only two "xed points. Thisprocedure is equivalent to putting the map on the unitcircle, that is looking at the "xed points of the map

hm`1

"hm#

1

2p Acot 2phm#

c!1

kcosec 2ph

mB,mod1. (14)

By the virtue of its in"nitely many branches, illustrated inFig. 2(a), this map has the fractal structure shown in Fig.2(b) of the boundaries of the basins of attraction of the"xed points. At the edges of the large contiguous regionsaround the two "xed points, computer investigation re-veals structure at "ner and "ner scales.

4. The period-one [h, k] Newton map

Now let us look at the Newton map produced by usingh and k as variables (x"[h, k]). The Jacobian is

J"C!k cos 2ph !1/(2p) sin 2ph

2pk sin 2ph !cos 2ph D (15)

giving us the Newton map

hm`1

"hm#1/(2pk

m)(2p(X!p) cos 2ph

m

#(c!1) sin 2phm),

km`1

"2n(X!p) sin 2phm!(c!1) cos 2nh

m. (16)

This map, in contrast to the previous one, is two-dimensional, but since h

m`1"f (h

m, k

m) and k

m`1"

J.H.E. Cartwright / Computers & Graphics 23 (1999) 607}612 609

Fig. 2. (a) Two trajectories in the Newton map hm`1

"hm#1/(2p) cot 2ph

mmod 1, that is (14) with c"1, showing sensitive depend-

ence on initial conditions. The trajectory started at 0.063 ends up in one basin of attraction; that starting at 0.064 "nishes in the otherbasin. (b) Fractal structure in the basins of attraction of the two "xed points.

f (hm), a set of points with the same h will map to a set with

the same k. We can write the map as

hm`1

"hm#R/(2pk

m) sin (2ph

m#a),

km`1

"R cos(2phm#a), (17)

where

R"J(c!1)#[2p(X!p)]2 (18)

and

a"arctan2p(X!p)

c!1. (19)

Let us call the two values of this arctangent in therange [0, 2p) a

1and a

2. It is now easy to see that the map

is bounded in k with maximum Dkm`1

D"R"Dk.!9

D.It is interesting to note that if we set

/"2phm#a, x"2ph

m`1#a, y"1!k

m`1/k

mand

b"R/km

we obtain

x"/#b sin /,

y"1#b cos / (20)

which are the parametric equations for a cycloid, thecurve traced out by a point on a wheel being rolled alonga #at surface. A curtate cycloid is obtained when DbD(1,an ordinary cycloid has DbD"1 and a prolate cycloid isgiven when DbD'1. We have b"R/k

m, so lines of con-

stant km

in the map iterate to di!erent cycloids withDk

mD"R"k

.!9being the changeover point; see Fig. 3(a).

The map has "xed points [hfp

, kfp

] at

[n!a1/(2p), R], [n!a

2/(2p), !R], n3Z. (21)

Note that points with any k except the singular valuek"0 with the same h-value as a "xed point map straight

into it. That is

[hfp#n/2, k], kO0, n3Z, (22)

iterates to [hfp

, kfp

]. Thus there exists an in"nity ofpreimages of each "xed point. The map also has a singu-lar line at k"0. By the same reasoning as above, thereexists an in"nity of preimages of points on the singularline.

[hfp#n/2#1/4, k], kO0, n3Z, (23)

iterates to [h, 0], somewhere on the singular line and

[hfp#n/2#1/4, $R/((n#1/2)p)], n3Z (24)

iterates to [hfp

, 0], on the singular line below a "xedpoint. All this interesting behaviour may be observed inFig. 3(b), showing the basins of attraction of the "xedpoints.

5. Periods two and higher

The mappings are no longer one-dimensional, andtheir structures are even more complex than those de-scribed above. In the (h, X) Newton map for periods twoand higher there are two "xed points, corresponding tothe two sides of the Arnold tongues in Fig. 1, and colour-ing their basins of attraction di!erently in an image,together with a third colour for points that do not con-verge to either solution in a certain number of iterations,shows the complexity that exists, as I illustrate in Fig. 4.This image from my doctoral thesis [8], was used for thecover of Volume 5 of the journal Nonlinearity.

6. Further ideas

I have shown some of the beautiful dynamical struc-tures that can be found in Newton maps of the circle

610 J.H.E. Cartwright / Computers & Graphics 23 (1999) 607}612

Fig. 3. (a) Cycloids produced by iterating once three lines of constant k in the map. That for k(k.!9

gives a curtate cycloid, like a pointwithin the circumference of a rolling wheel; that for k"k

.!9gives an ordinary cycloid, as a point on the circumference of the wheel; and

for k'k.!9

there is a prolate cycloid, like a point outside the rolling circumference* a #ange on a train wheel, for instance. (b) Themap (16) with parameters X"0.5, p"0, c"1. The ordinate is 0(h(1, and the abscissa !5(k(5. The basins of attraction of the"xed points [h

fp, k

fp] at [1/4, p] and [3/4, !p] are depicted in shades of crimson and turquoise; the darker tones represent more rapid

approach to the "xed points.

J.H.E. Cartwright / Computers & Graphics 23 (1999) 607}612 611

Fig. 4. Basins of attraction for a Newton map "nding the edges of the 1/2 tongue in a subcritical k"1/2 sine circle map; the boundariesof the plot are !0.5(h(1.5 for the ordinate, and !1(X(2.0 for the abscissa. Red and blue represent the basins of the two "xedpoints that arise from "nding the two edges of the tongues. The deeper the colour, the faster is the approach to the basin from that point.The yellow points are where the trajectory failed to fall into either basin within 20 iterations.

map. The idea can be carried further by iterating theprocess: one can make Newton maps of Newton maps ofthe circle map, Newton maps of Newton maps of 2, andso on recursively. But that story will have to wait foranother time.

References

[1] C. Huygens. Extrait d'une lettre escrite de La Haye, le 26fevrier 1665. Journal des Scavans, (11 (16 March)), 1665. Seethe correction published in the following issue: Observationa faire sur le dernier article de precedent journal, ou il estparleH de la concordance de deux pendules suspendueK sa trois ou quatre pieds l'une de l'autre. Journal des Scavans,(12 (23 March)), 1665. Huygens' notebook is reprinted inOeuvres Completes de Christiaan Huygens, volume 17,page 185, SocieteH Hollandaise des Sciences, 1888}1950. Agood historical perspective is provided by J. G. Yoder. Un-rolling Time*Christiaan Huygens and the mathematizationof nature. Cambridge: Cambridge University Press, 1988.

[2] Arrowsmith DK, Place CM. An introduction to dynamicalsystems. Cambridge: Cambridge University Press, 1990.

[3] GonzaH lez DL, Piro O. Chaos in a nonlinear driven oscil-lator with exact solution. Physical Review Letters1983;50:870}2.

[4] Aronson DG, Chory MA, Hall GR, McGehee RP. Bifurca-tions from an invariant circle for two-parameter families ofmaps of the plane: A computer-assisted study. Communica-tions in Mathematical Physics 1983;83:303}54.

[5] CvitanovicH P, Shraiman B, SoK derberg B. Scaling laws formode lockings in circle maps. Physica Scripta1985;32:263}70.

[6] Hao B-L. Elementary symbolic dynamics and chaos indissipative systems. Singapore: World Scienti"c, 1989.

[7] Arrowsmith DK, Cartwright JHE, Lansbury AN, PlaceCM. The Bogdanov map: bifurcations, mode locking, andchaos in a dissipative system. International Journal of Bi-furcation and Chaos 1993;3:803}42.

[8] Cartwright J.H.E, Chaos in dissipative systems: bifurcationsand basins. Ph.D. thesis, Queen Mary and West"eld Col-lege, University of London, 1992.

612 J.H.E. Cartwright / Computers & Graphics 23 (1999) 607}612