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3. Driven Systems, Cantor Sets and Strange Attractors

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1988 Phys. Scr. 1988 34

(http://iopscience.iop.org/1402-4896/1988/T20/003)

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Physica Scripta. Vol. 20, 34-46, 1988.

CHAPTER 3

Driven Systems, Cantor Sets and Strange Attractors

3.1. Introduction: The Lorenz Model and the Tent Map

In Ch. 1 we described the results of numerical computation on the Henon-Heiles system which indicated motions rang- ing from integrable through near-integrable and perhaps beyond via a sequence of bifurcations as the energy E was increased. That bifurcations occured can be inferred from the increase in the number of elliptic and hyperbolic fixed points appearing in the Poincare section as the energy was increased. We have shown how Liapunov exponents and Bernoulli shifts describe the motion of a completely unstable area- preserving map, the bakers’ transformation, and it is known that the unstable motion near a homoclinic point can be mapped symbolically onto a Bernoulli shift at the crudest level of description, that of symbolic dynamics [7]. We turn now to chaotic behavior in driven systems.

It has been known since the time of Maxwell, Boltzmann and Gibbs in physics, and since the time of Poincare in mathematics, that chaotic motions must occur in Hamil- tonian systems. Boltzmann’s argument in favour of statistical irreversibility, although not correct in every detail, was essentially correct in spite of the objections of the mathema- ticians [2]. Modern chaos theory has given us simple discrete models of statistical mechanical behavior which show explic- itly that time reversibility and Poincare recurrence do not prevent the irreversible approach to statistical equilibrium, and has also taught us that we do not always need the thermodynamic limit for the latter: two degrees of freedom in a non-integrable system can be enough. The question arises whether some unsolved problems of complex open systems can be understood by few-degree of freedom deterministic dynamics. Recent experiments indicate that some features of the early stage of the transition to turbulence in certain hydrodynamic systems can be understood, at least qualita- tively, via low-degree-of-freedom dissipative dynamics. Some of those experiments are discussed in Ch. 4.

The mathematical discovery of chaotic motion in a driven dissipative systm goes back to E. Lorenz (cf. Lorenz (1963) in Cvitanovic [3]), who mentioned the question of predictability of the earth’s weather system as a motivation for his work. Since equations describing the earth’s atmosphere must be solved under boundary conditions corresponding to heating by the sun and simultaneous radiation of energy into outer space, it was useful to consider a simpler problem, the flow of a fluid between two flat plates in a gravitational field, the one heated from below and the other cooled from above, so that the fluid is driven by a constunt temperature difference between the two plates. Lorenz noted that for a non-linear system, constant forcing doesn’t always imply constant response, indeed far from it. But even this system was too difficult to solve. To obtain a more transparent model, the Navier-Stokes equations were expanded in a discrete set of modes and truncated arbitrarily to three coupled ordinary

Physrca Scripta T20

differential equations, representing three modes of oscillation [4]. These equations are given below and are now known as the “Lorenz model”: 5, 6

i = -ox + cry

p = - x z + rx - y (3.1)

i = XY - bz

Since

(3.2)

is generally non-zero, the system is non-conservative. If we let b > 0 and [r > 0, then V * V < 0 and phase space volumes contract according to

Jfl(t) = e-‘‘+1+b)r6fl(0). (3.3)

Since 6Cl(t) + 0 as t -+ CO, one might be tempted to conclude that initial conditions initially uncertain by an amount -6R(0)’3 yield easily predictable motion as t + 00. This conclusion is wrong: 6 f l ( t ) -+ 0 requires only that the attrac- tor should have a dimension less than 3, e.g., contraction onto a 2 dimensional subspace would do the job. However, chaos in two dimensions is ruled out by the Poincare- Bendixson theorem, but we can expect chaotic motion if the orbit tends toward a “strange attractor”. A simple model is discussed in the next section. First, let us turn to Lorenz’s discovery of dissipative chaotic motion and his attempt to characterize it. It is worthwhile to bear in mind that the attractor is the closure of the set of points visited asymptotic- ally by the orbit. It is the closure, not the orbit itself, that can have a non-integral fractal dimension. The orbit itself is always a one-dimensional curve, for any finite time t. Fractals and their dimensionalities are defined below in Section 3.2. First, obtain and classify all equilibria (fixed points);

-o(x - y ) = 0

--xz + rx - y = 0

XY - bz = 0

yielding

x = y if a f O

.x(-z + r - 1) = 0

z = x2/b > 0 if b > 0.

(3.4)

(3.5)

If x = 0, then (0, 0, 0) is a fixed point. If r < 1, this is the only fixed point. If r > 1, there are two additional fixed points

(3.6) (4-J v/b(r - l ) , Y - 1)

Driven Systems, Cantor Sets and Strange Attractors 35

step toward qualitative understanding was to ignore the details of this map and to replace it analytically by a much simpler map, the tent map

with a = 2. We can permit a to be any integer g 2 and will arrive at the same conclusion. Since the Liapunov exponent is lb = In a. we have orbital instability whenever a > I . The tent map dynamics can be easily understood in base-a arith- metic (a = 2, 3, 4, . . .) via the expansions A g 3 1 Numerically computed discrete map for the Lorenz model

Exercise: (a) Show that the point (0, 0, 0) is stable if 0 < r < 1. (b) if 1 < r < r2 show that (0, 0, 0) is unstable but that the two other fixed points are stable. Show also that r, = a(a + b + I)/(cr - b - 1). (c) Show that when r > r , , there are no stable fixed points.

Case (a) corresponds qualitatively in the fluid-mechanical system to pure heat conduction (diffusion) with no convec- tion. Case (b) corresponds to steady heat transfer via convec- tion. Case (c) opens up the possibility of chaos, if the motion is bounded and there are no stable fixed points, limit cycles, or invariant tori. There, the orbit is continually repelled by the unstable fixed points and cannot settle down to rest: it is condemned to wander about forever inside a bounded region because, as Lorenz proved, the motion is bounded. Qualita- tively, this seems essential for chaos. Actually, there is a finite 6r > 0 such that for r, - 6r < r < r 2 , chaotic motion occurs while both fixed points are stable. However, in this regime each stable fixed point is enclosed by an unstable limit cycle [5] so that the orbits continually are scattered by the two interior regions of finite size that are bounded by unstable limit cycles. Hence, the two fixed points are “effectively” unstable, preserving our simple qualitative view of the pre- requisite for chaos: bounded motion plus the absence of stable fixed points, stable limit cycles and invariant tori upon which stable quasi-periodic motion can occur.

Lorenz studied the above model numerically on a com- puter, and gave a partial description of the problem of numerical error on a finite state machine, noting the eventual imposition of periodic orbits by the machine even when none exist for the system. He discovered orbital instability in the form of sensitivity with respect to small changes in initial conditions, and in order to characterize the resulting chaos he numerically constructed a return may by plotting successive maxima of the variable z ( t ) , which we denote by 2 , . His next

4n+L

’ i

X

x, = ci(n)/al, E , @ ) = 0, 1, 2, . . . , a - 1. (3.9) (3.7) 1 = 1

Fig. 3.2. The tent map.

In base a arithmetic, the map is exactly equivalent to a “shift”; e.g., for a = 2, we have

(3.10)

which is also an example of a simple cellular automaton. Initial conditions of the form

N

xo = &,(O)/a’ 1 = 1

(3.1 1)

with N finite, truncate to zero in finite time whereas non- truncating rationals yield unstable periodic orbits. Irrational initial conditions yield chaotic non-periodic orbits. These irrational orbits yield statistical mechanical behavior by producing a uniform invariant probability density [4, 61 for almost all irrational initial conditions.

Exercise: Derive the master equation for the tent map,

(3.12)

and determine the condition on the parameter a such that p,(x) + 1 as n + CO, given some initial non-equilibrium density po(x).

For the tent map, the “route to chaos” is simple: when la1 < 1, all solutions are attracted toward the stable fixed point z = 0 whereas for la/ > 1 there are no stable fixed points, yet the motion is self-confined to the unit interval [0, 11. In analogy with the Landau theory of critical phenom- ena, we can characterize this transition to chaos by a critical exponent b, where

A / a - a,Ifl*sgn (a - a,) (3.13)

and,? < Ofora < a,whilel, > Ofora > a,. SinceA = In a a n d a , = l , A = l n ( l + a - l ) = a - l a s a + l s o t h a t p = 1. Among the details that are lost when we replace the Lorenz map by a tent map are the following: (a) the sudden change from stable critical points to “all orbits are unstable” predicted by the tent map is wrong, and (b) the “windows of stability” found numerically for the Lorenz model [7] have been lost. One problem is that the tent map is too smooth: the Lorenz map is expected to be fractal. The tent map exhibits mixing rather than strange attractor behavior and the tent map’s attractor has fractal dimension equal to 1 if a > 1 and xo is irrational, but is otherwise zero. This can be seen by applying the method of Section 3.2 to the tent map.

Physica Scripta T20

36 Joseph L. McCauley

In order to obtain a simple model of a strange attractor with a non-integral Hausdorff dimension, we return to the bakers' transformation in the next section. Even though the transformation is itself smooth, it gives rise to an attractor that is a Cantor set.

Computed orbits for the Lorenz model in the chaotic regime are shown in Lichtenberg and Lieberman [8] and in Schuster [4]. Lorenz proved that the motion is bounded, and the computed orbits behave as follows: an orbit unwinds from an unstable spiral, switches to the domain of the other unstable equilibrium point and then unwinds about it in like fashion. Since the motion is bounded, but there are no stable equilibria or limit cycles, this motion continues forever. Qual- itatively, the idea of symbolic dynamics seems useful for understanding the chaos encountered here: we can assign a bit E , = 0 every time that the trajectory makes a complete rotation about one unstable fixed point, a bit E, = 1 for one rotation about the other, and in this sense the hopping from one unstable to one fixed point to the other is characterized by a binary, or "head-tails'' sequence, an example of sym- bolic dynamics. For non-periodic motion we expect a non- periodic sequence of bits, corresponding to an irrational number.

3.2. The dissipative bakers' transformation and fractal

The attractor for the Bernoulli shift with irrational .yo is the unit interval, with dimension 1 . The attractor for the bakers' transformation is the unit square, with dimension 2. We will show that by adding "dissipation" to the bakers' transfor- mation, we obtain orbits which are attracted to a set that is "strange' in the sense that i t has a non-integral Hausdorff dimension, also called here the fractal dimension. The attracting set also turns out to be self-similiar, an example of a scaling fractal.

The dissipative bakers' transformation is given (with a > 0) by

.T,,+~ = 2x,,, mod 1

dimension of a strange attractor

ay" > 0 6 x, < 1/2 r ay, + 112, 1/2 6 X" 6 1 Y n + i =

The transformation is dissipative for a < 1,'2 since

(3.14)

(3.15)

The dimension of the attractor is d = d, + d, where d, = 1 is due to the Bernoulli shift .x,,-~ = 2x,, mod 1 when so is irrational (d, = 0 if xo is rational). To understand d, let us observe how the unit square is mapped by the bakers' trans-

'13 x - 0

Y Y

0 ' 1

I

--t--+x 0 1

n-t 50 n=i n.- 2

Fig. 3.4. Dissipative bakers' transformation (continued).

formation whenever 0 < 2a < 1:

The region 0 6 xo 6 112, 0 < yo 6 1 is mapped

onto 0 6 xi 6 I , 0 6 yi 6 a < 112 while

the region 1/2 6 xo 6 1, 0 6 yo 6 1 is mapped

onto 0 6 x0 6 1, 1/2 6 y , 6 1/2 + a:

where we have assumed a < 112 ( a is slightly less than 112 in Fig. 3.3). Note that the total area shrinks by a fctor J = 2a per iteration. For the second iteration one gets Fig. 3.4, where the numbers show which block at n = 1 is mapped into a corresponding block for n = 2. Now, with a given length- scale E , we ask how many rulers of length E are required to cover the attracting set in they-direction. For n = 0, we need No = 1 ruler of length = 1. For n = 1, we need N, = 2 rulers each of length El = a. For n = 2, we need N, = 4 rulers, each of length E~ = a*, . . . , so that at the nth iteration we will need N, = 2" rulers, each of length E,, = a" to cover the resulting set of points.

L, = N,E, (3.16)

is the total length of the attracting set in the y-direction and we can see that the systematic contraction of area defines a scaling law

N(E) = E?, (3.17)

or

. In N(E) d, = lim ~

e+o In ] / E '

Since for E = a" we find N = 2", we have

(3.18)

In 2" In 2 d,, = lim = - < 1 since a < 11'2. (3.19)

i l + x In a - IIn al

Since d, < 1, L, = N, = E d~ + 0 as E -+ 0 so that the attractor occupies no length in the y-direction. This par- ticular attractor exhibits scaling behavior (self-similarity) for every length scale E = a" and so is called a scaling fractal [lo].

I E - 1 , N = 1 0 1

Y Y

n= o n= 1

Fig. 3.3. Dissipative baker's transformation, iterating a block of initial con- ditions in the left half of the units square.

Phyhicu Scriptu T20

E = 1/2", N = 2"

Fig. 3.5. Diagram for computing the fractional dimension of the unit inter- val, or of the rationals therein.

Driven Systems, Cantor Sets and Strange Attractors 31

A similar ruler-covering argument shows that the x-dimension of the attractor is unity for an ergodic orbit, so that

In N n In 2 In I / & n In 2

= 1, d x = - - - -

0 I

A 2L

113 %I3 1 as is shown in Fig. 3.5.

The resulting point set along the y-axis is uncountable because 2“ has the cardinality of the continuum, i.e., of the irrationals, but is nowhere dense. This strange behavior is caused by the two Liapunov exponents which can be obtained as follows. With

0

A 0 1/4 ’ % 1

Fig. 3.7. The Koch curve.

(3.20) More generally, fractal geometry [lo] has been used to describe not only certain strange attractors and Cantor sets, but also some irregular shapes such as the coastline of Norway. As an idealization of a coast line, the Koch curve is an example of a continuous curve that, as n + a, is nowhere

so that the x-errors are stretched by a factor of 2 while the y-errors are compressed by a factor of a < 1/2;

differentiable and is also a scaling fractal. The Koch curve is based upon the above Cantor set and has the dimensionality 2 In 2/ln 3, and is shown in Fig. 3.7. Here, we see that E = 113“ and N = 22” after n iterations, yielding

SX, = ( f ;) SX,, (3.21)

for IlSX, 1 1 sufficiently small SO that dx, = en’.6Xo yields I , = In 2 > 0, while Sy, = ~ ~ ’ ~ d y , yields I,, = In a < 0.

(3.23) 2n In 2 In 2 Note that A, + i2 = In 2 + In a < 0, in agreement with d = - = 2 -.

1 J I < 1. The resulting fractal structure makes all the bakers’ n In 3 In 3 dough disappear as n + a, so that there is nothing left to d 2 1 follows from the fact that we have a curve, while bake, asymptotically; the strange attractor is a “sink” for the d < 2 is demanded by the fact that the set occupies dough. no finite area in the plane. Clearly, we have again a Cantor

The above fractal structure is qualitatively characteristic set on the x-axis, and on every finite segment ofthe curve that of Cantor sets [l 11, of which the “middle-thirds” set is the lies off the x-axis as well, due to self-similarity of the curve on best known. In order to construct it, we begin with the unit every length scale. interval and pull out the open middle third, leaving behind Notice that for the dissipative bakers’ transformation, we the end points 112 and 2/3 (i.e., remove the open set (1/3, 2/31 have 2, = In 2, iy = In a and so the fractal dimension of the from [0, 11 as is shown in Fig. 3.6. attractor,

Next remove the (open) middle thirds of the two remaining (closed) segments, leaving behind 4 closed sets, each of length d =

E = 1/3” is the length scale used to measure the covering of can also be written the resulting set after n iterations, then N = 2” “rulers” with

d = 1 + - . 4 (3.25) length 1/3“ are required, yielding the fractal dimension

(3.24) I In 2

/In a/ 119, and continue the operation ad infinitum. Clearly, if

IS I (3‘22) This sort of result leads to the Kaplan-Yorke conjecture

t 4 This dimension characterizes the best-known Cantor set, which consists of end points 0, 113, 213, 1, 119, 2/9, 719, 8/9, d = j + - . . . and all limit points of sequences of the end points. I)”, + 1 I

(3.26)

which can be arrived at by an intuitive sort of scaling argu- ment [9], and where the exponents are presumed ordered so that I,, > ,I2 > . . . > Idand j is the largest integer for which - 1-1 €-1 /3 ,h , -2 2, > 0. (3.27)

0 ‘13 213 1 Note that for the conservative baker’s transformation when a = 1/2, we obtain d = 2, which is correct. However, it is not well understood as to when the Kaplan-Yorke conjecture holds and when it should fail.

Having introduced a nowhere dense, uncountable set of zero measure, the “middle thirds” Cantor set, we will now show how to construct a Cantor set with finite measure [l 11.

b € 5 1 , # = I 0 7

J

> = 1

C I C I - - E = 1 / 3 L , N = 2 2 o 119 Zk ‘I3 r/ , ’?h !4 7 .

5 113: N = Zp

Fig. 3.6. The middle-thirds Cantor set.

Physica Scripta T20

38 Joseph L. McCauley

3 I

0 1

Fig. 3.8. A finite-measure Cantor set.

Begin with the unit interval and remove the open middle section with length 4 2 .

Here, E = 1/2 - 4 4 is the length of a ruler needed to cover the Cantor set, and N = 2 such rulers are needed. The end points 0, 1/2 - 44, 1/2 + 44 , 1 belong to the Cantor set (as will all such end points, in what follows). Now, remove the open middle “a”’,’’ of the two remaining closed sets. The first end point beyond 0 point has the coordinate 1/2 (1/2 - 4 4 ) - 416, so this is E , the length of the required ruler, and we need N = 22 such rulers to cover the set. In general, N = 2” rulers, each of length

c = q 1 2“ - 2 , = I l i2.1 (3.28)

will be required in order to cover the point set. This yields a Hausdorff dimension

In N n In 2 d, = lim - = lim

n - ~ ; In I / & 1 n I n 2 - - l n 1 - E- n+J) [ n i ,:, 31 n In 2 n In 2

- I -

Since dF = 1 , the length of this Cantor set is finite,

L = lim NE = 1 - n - X

” 1 “c, = 1 - 2 . I = ,

(3.29)

(3.30)

Yet, there is space between every pair of end-points in the set. As with the middle-thirds set the set consists of all end points and limits of sequences of end points. This example shows that the idea of a non-integral fractal dimension is not always an adequate measure of the “jaggedness” of a point set.

Jan Froyland has pointed out that the Koch curve con- structed by “lifting while not adding to the roof’ has Haus- dorff dimension equal to 1 . Yet, the resulting curve, exhibited in Fig. 3.9, is certainly no less jagged than the original Koch curve.

3.3. Hopf bifurcations and the Ruelle-Takens-Newhouse

The route to chaos via differential equations and discrete maps that exhibit an infinite sequence of subharmonic bifurcations (“period doubling”) is by now well-known and has been thoroughly discussed in standard books on chaos theory [3,4,8]. We will discuss in this section a route to chaos that occurs via two or three Hopf bifurcations [stable node -+ stable limit cycle -+ stable 2-torus + chaos]. This route was first discovered in abstract mathematics by Ruelle. Takens and Newhouse [3]. The Ruelle-Taken-Newhouse work was not until several years after its discovery given much weight in physics, primarily due to the lack of a simple, transparent model of that route to chaos.

We begin by illustrating the Hopf bifurcation [ 121, where a fixed point becomes unstable in favour of the birth of a stable limit cycle. Consider the system

route to chaos

i = p r - ? = r (P - r> e = - 1 (3.31)

in the (xl , x2) phase plane where 2 = .xf + x; and tan 0 = x 2 / x , . When p < 0 the spiral is unstable, but the limit cycle at r = p > 0 is then stable. In Cartesian coor- dinates we have

xl = r cos 8

X I = i cos 8 - r8 sin H

i, = r ( p - r ) cos 0 + r sin 8

= x1(p - \ x: + x:, + .Y2

x2 = r sin 0

.i2 = i sin O + r B cos O

= x2(p \ x: + x:, - x ,

Linearizing about (0, 0) we obtain

(3.32)

61, = p6x, + 6x, 6X2 = p6x2 - 6x,

or

,i = p k i

(3.33)

i Z I = (p - )“)I + 1 = 0

(3.34)

(3.35)

Fig. 3.9. The Koch-Froyland curve

Physicu Soipru T20

Fig. 3.10. .4 Hopf bifurcation

Driven Systems, Cantor Sets and Strange Attractors 39

’ I Fig. 3.11. Hopf bifurcation diagram.

Here, we have an example of a Hopf bifurcation, where a set of complex conjugate eigenvalues crosses the imaginary axis, yielding a transition from a stable spiral at r = 0 to a stable limit cycle at r = p. (cf. Fig. 3.10). The bifurcation diagram is shown in Fig. 3.11, where for p > 0, stable asymptotic solutions lie on the surface of the cone in a plane transverse to the p-axis.

The inverse Hopf bifurcation is illustrated by

5tP-W e 3 --f 4ttrarSOTj

perhrps b ; + h %a- i n 4 w l d

>table f , r t d p m n t

s table p a n - 3 S W e i ; m . i ? e r ; od ,L cycle ( d z 1 ) ( d =2)

(AZO)

Fig. 3.13. Transition to chaos via bifurcations to instability of quasi-period- ic orbit.

i = r(p + r )

e = - 1

a “strange attractor” as a bounded point set in phase space toward all orbits within some finite basin of attraction are attracted. There is a Liapunov exponent 2, < 0 “off the attractor”, but “on the attractor” there is sensitivity with respect to small changes in initial conditions: there is at least one Liapounov exponent I , > 0, for any two such neigh- bouring orbits. Finally, 2, = 0 for any two points on the same orbit [8]. To be precise, one should think of the attractor as the closure of the asymptotic motion, so that a system is typically near, but not on an attractor. For a more careful discussion of strange attractors, see Eckmann in “Cvitanovic” [31.

(3.36) 3.4. Dissipative circle maps: Introduction

where an unstable spiral gives rise to a stable spiral bounded by an unstable limit cycle as p decreases through 0 from the right as is indicated in Fig. 3.12. The transition to chaos in the Lorenz model is via an inverse Hopf bifurcation [ 5 ] .

Some evidence against the route to fluid turbulence that was suggested by L. Landau was first reported in an early experiment by Gollub and Swinney [3], who studied the motion of a normal liquid confined between two concentric rotating cylinders (Taylor-Couette system). Gollub and Swinney used a light scattering technique to analyze the radial component of fluid velocity and found that the power spectrum was somewhat consistent with the Ruelle-Takens route to chaos, with destruction of all discrete spectral peaks plus the appearance of broadband noise in the power spec- trum after apparently 3 bifurcations. Interest in this route to chaos has increased in recent years, following the explosion of activity in the field of nonlinear dynamics that was generated by Feigenbaum’s discovery of universality in the period- doubling route to chaos.

The Poincare-Bendixson theorem rules out chaotic motion on a two dimensional torus since in that case the trajectories can be described by a system of two coupled ordinary differential equations. Any chaotic motion must occur on an attractor with dimension d greater than two (cf. Fig. 3.13). For present purposes, it is enough to think of

We begin with a topic that leads directly to the route to chaos via quasi-periodicity, namely the equation of the damped, driven pendulum in the form of the Josephson-Junction equation [ 141

r8 + r(38 + y sin 9 = A + K cos cot (3.37)

in which there are two competing “bare” frequencies a and w. With xI = 9, x2 = 8, and x3 = wt we have

XI = x,

1 i, = - - [Bx, + y sin x, ] + [ A + K c o s x,]/a (3.38)

ci

x3 = 0,

namely, a three dimensional phase space, which is the mini- mum dimension required for chaos in solutions of ordinary differential equations. Rather than work directly in the phase space of the differential equations, we will consider the stroboscopic map

(3.39)

where n = 0, 1, 2 , . . . corresponds to times 0, T, 2T, 3T, . . . where T = 2n/w is the period of the external force. As we showed in Section 1.7, this choice of T gurantees that the functions GI and G2 do not change upon iteration of the map; also these functions can be constructed as power series in nT directly from the differential equation, at least at high fre- quencies (see the Appendix). The Jacobian J of the map is simply the Jacobian

(3.40)

of the first-order system

8( t ) = B(t) 1

(3.41)

I d(t) = - 1. ( f i ~ ( t > + y sin e(t>> + (K sin wt + ~ ) / a ! C L ,

Fig. 3.12. An inverse Hopf bifurcation. (3.42)

Physica Scripta R O

40 Joseph L. McCauley

evaluated at any time t , but with T = 2 7 1 . j ~ . Since

we have

J ( t + T, t ) = J(r)e t P z J r = J(t)e 2nlizw (3.44)

so that areas in the two-dimensional ( O n , e,) phase space are contracted at the constant rate

(3.45) h = e 2nliwa

where f l > 0 (case of ordinary friction). One can ask what is the dimension of the point set that the areas contract onto, i.e., what is the nature of the attractor? If 8,, relaxes rapidly relative to 0, so that

0 , 2 g(Q,,), (3.46)

as n + x, then we are led directly to the study of “circle maps”

(3.47)

wheref(d,, + 271) = f ( 8 , ) + 271. To prove that this dimen- sional reduction actually occurs and thatfis a smooth func- tion is a non-trivial exercise, and no general such proof exists [ I 51, although it is reasonable within the non-chaotic regime to assume that the attractors have dimensions which are integers 0 or 1. The study of this problem is mathematically similar in some respects to the study of the KAM problem for area-preserving maps: the question whether there exists a smooth map of the circle f onto itself is analogous to the question whether there exist smooth, invariant tori in KAM theory. Arnol’d shows that so long a f i s smooth and invert- ible, then no chaotic motion can occur [16, 171. This agrees qualitatively with out understanding of the Bernoulli shifts and other one dimensional maps where non-invertibility is essential for chaos. Non-trivial scaling behavior is known to occur a t the values of the control parameters wherefbecomes non-invertible [3], and by studying the growth in width of the Arnol’d tongues as the perturbation strength increases, we expect that this corresponds to the point where the wedges of parametric resonance begin to overlap [14]. (For area preserving maps, resonance overlap has been used to describe the breakup of smooth, invariant K.4M tori [8, 181, and the appearance of chaos.) It has been found numerically [I41 that the return map of the Josephson Junction equations can no longer be regarded as a smooth circle map once one enters the chaotic regime. If the chaotic Josephson attractor has a frac- tal dimension 1 < dF < 2 then we expect that the full two dimensional map will be required in order to describe the chaotic motion, whereas in the non-chaotic region, we know that the possible attractors will be either points or smooth one-dimensional curves (stable limit-cycles).

A model that has received much attention to the literature is the sine-circle map

f ( 0 ) = 0 + 2nR - K sin 0 (3.48)

which reduces at zero-perturbation-strength ( K = 0) to the simplest circle map, the linear map

O,,,] = 0, + 2nR. (3.49)

We encountered this return map in Ch. 1 when we discussed the dynamics of stable Hamiltonian systems in terms of tori

P/zy.ircu Sc ripru T20

in phase space. When R is rational, the motion is periodic while irrational R corresponds to quasiperiodic motion that is ergodic but non-mixing. In what follows, we take K 2 0. As we have seen in Ch. 1, this map can be used to represent one dimensional motion on the surface of a two dimensional torus for a two degree of freedom integrable system. There, R is the ratio of two frequencies. Since we began with a two-degree of freedom dissipative system and have assumed contraction onto a one-dimensional orbit asymptotically, i t will be useful in what follows to think of Cl as the ratio of two competing frequencies in the original system. In practice, frequency-locking is observed in such systems: there is a tendency for the “effective” or “renormalized” frequency called the winding number W (yet to be defined) to lock in at a rational value, reflecting the tendency of two nonlinear oscillators to synchronize. The winding number W is defined as follows:

If we consider the most general map satisfying our circle map condition

f (s , + 271) = 27r + f ( e , ) (3.50)

then we can also write

en+] = 8, + 27151 + Kh(0,)

h(B, + 271) = h(8,J mod 27r.

(3.51)

(3.52)

From this it follows that it is not 0 but the winding number W.

(3.53)

that is the correct measure of periodicity, or lack of same. e.g. if O,, , , = 8, + 271Q then 8, = Bo + 271nR and we retrieve W = R, as expected. Wcan be interpreted quite generally as the average number of rotations through 2n per iteration, since ( f ( 8 , ) - e,) is the total distance travelled along the circle in n iterations. The mathematicians [ 16, 171 have shown that if W is irrational then the map On+, = f ( 0 , ) is topologi- cally equivalent to a rotation through some angle c,$, when- ever f is invertible:

AI+, = c,$?! + 2nW. (3.54)

with sufficient smoothness conditions on A this transforma- tion is analytically possible [17]. Because of this “universal behavior” in the regular regime, whether we assume thatfis a sine-circle map or something else doesn’t matter so long as the map is invertible. The equivalence of circle maps to rotations was first considered by Poincare in this study of differential equations.

The results from mathematics are quite strong: it is known, under assumptions that correspond to a restriction to regular, or stable motion, that W “locks in” at every possible rational frequency P/Q for a finite range of 0 whenever K is finite and that W(K, R) is continuous and monotonically increasing with R. This frequency-locking behavior produces wedges of parametric resonance called Arnol’d tongues in the (R, K ) plane, which we also should expect for small K from our analysis in Ch. 1. From that analysis, we should expect the growth of an Arnol’d tongue from every rational point on the R axis as K increases from zero, as is indicated in Fig. 3.14.

The region of finite width in 0, for fixed K, where W

Driven Systems, Cantor Sets and Strange Attractors 41

K

Fig. 3.15. The sine-circle map with (Kl < 1 (invertible). Fig. 3.14. Arnol’d tongues.

assumes a given rational number P/Q is just the width of the tongue for a given K-value. It seems difficult geometrically, at first sight, to understand why for aribitrarily small but finite K we do immediately run into the problem of resonance overlap since, (a) the rationals are dense, and (b) a tongue grows from every rational value of R. Geometrically, this poses a difficult conceptual problem, but an easier one ana- lytically: resonance overlap need not occur for small enough K simply because the rationals are countable. Although it does not correspond to the right widths for the tongues, the ~/2”-covering that we used in Ch. 2 to illustrate that coun- tability implies that the rationals have measure zero can be used to make the point. Since the rationals can be labeled by the integers, let the nth rational be covered by a “ruler” of width E/T. The total length of all such rulers, laid end to end is

30

E / 2 ” = E ,=I

(3.55)

so that the total width associated with all rationals can take up as little space as we like: it follows from this that Arnol’d tongues can grow from all rationals without having to over- lap. Whether or not any two or more tongues overlap is a problem is dynamics. In what follows we will indicate why the tongue widths should vanish rapidly as K -+ 0. Although this is quantitatively different from the above covering, the reason why there need be no resonance overlap for small enough K is qualitatively the same.

In order to go further, we will specialize to a definite model, the sine-circle map

f(6,) = 6, + R - K sin 2716,/2n (3.56)

where also we have made the change of variables 6, -+ 2n0,, and R -+ 271R so that the range of 6, is now 0 < 0, < 1. This model can be obtained as follows: if we begin with the sine- standard map, an area preserving map that models a two degree of freedom conservative system [8], and add terms that make it dissipative [14], and if the 8, motion relaxes rapidly relative to the 6, motion, the above circle map follows. Whether the sine-circle map actually can be derived analyti- cally in the same limit as an approximation for the Josephson junction equation is difficult to prove except in the limit cr 4 I where the differential equation suffers the loss of its highest derivative (thereby reducing the phase space to one dimension) in the spirit of boundary-layer theory [15].

Since in one-dimension non-invertibility offis essential for chaos, the map

f(Q) = Q + R + K sin 2z6/2n (3.57)

has the possibility for chaotic behavior only if K > 1, as can be seen from the following observations. Since

f ’ ( 6 ) = 1 - K COS 2716, (3.58)

’L 0 e Fig. 3.16. The sine-circle map with IKl = 1

f ( 6 ) is monotonically increasing for K < 1. Sincef( 1) = f ( 0 ) mod 1, f ( 6 ) is single-valued, hence invertible (cf. Fig. 3.15). At K = 1,f’(6) = 0 at 6 = 0 (= 1 mod l), as is indicated in Fig. 3.16. When K > 1 f(0) has a maximum and minima (mod 1) at

1 1 fj = - -cos-’- 271 K (3.59)

and

1 1 1 6 = -cos- - 271 K

(3.60)

respectively.

non-invertible with a 3-valued inverse (cf. Fig. 3.17). Hence, there are two regions of finite width 6 where f is

3.5. Mode-locking, devil’s staircases and complementary

One can use examples of circle maps to illustrate how the motion locks into any given rational winding number in a finite width AR of the R axis, thus forming the resonant wedges called Arnol’d tongues in the (R, K ) plane. For the cosine circle map x,,, = X, + R + K COS ZZX,, (3.61)

if we set the winding number

Cantor sets

(3.62)

equal to a definite rational number P/Q, we can solve algebraically for the range of SZ over which this condition holds, for fixed K E (0, I). Since W = R gives the average rotation through 271 when K = 0, we will solve for R as expansion in powers of K as K -+ 0.

The idea is to fix Wand K and to ask for the range of R

Fig. 3.17. The sine-circle map with IKI > 1 (non-invertible).

Physicu Scripta R O

42 Joseph L. McCauley

K K

O1

1 w= 1/2

Fig. 3.20. .4rnol’d tongue with W = 112 grows from R = 112. Fig. 3.18. ‘4rnol’d tongue growing from R = 0 , with W = 0.

(corresponding to different choices .yo) over which such a yielding the tongue boundaries - - periodic orbit exists. If W = 0, we have fixed point

.yo = .yo + C2 + K COS 27tx0 (3.63) AR(K) Y k n K 2 / 2 (3.74)

growing symmetrically from the point R(0) = 112 as is or shown in Fig. 3.20.

C2 + K COS 2nx0 = 0,

yielding the wedge boundaries

R(E) = k ]Kl

since - I < cos 2nx, < 1 (cf. Fig. 3.18). For W = 1 , we solve

.T~ = .yo + R + K COS 2 ~ x ,

with

(3.64) Exercise: find the shape of the tongues to lowest order in K when (a) R = 1/4 and 3/4, (b) R = 113 and 2/3.

(3.65) We can therefore expect the following results when

.y,+i = x,~ + + Kh(X,),

/z(.x,,+,) = h(x,) mod 1 (3.75)

(3.66) unders suitable differentiability conditions on h. For a resonant wedge, W is rational,

W = P/Q (3.76)

and this corresponds to cycles of the form SI = .Yo + 1. (3.67)

Setting C2 = 1 + AR, we again obtain AQ(K) = +lKl (3.68) .YQ = .yo + Q w = .yo + P. (3.77)

One then solves The wedge is illustrated in Fig. 3.19. W = 112 corresponds to x2 = -yo + 2(1/2) = xo + I , so that with xQ = .yo + Q C2 + F(x0, R, K ) = .YO + P (3.78) x2 = )io + R + K COS 2nx0 + C2 where F is obtained from functional composition, and

+ K cos 2n(xo + Q + K cos 2 m 0 ) (3.69)

we again can set !2 = 112 + A 0 for a given .yo to obtain

2Ai2 + KCOS 27i,yO + K COS 2n(x0 + !2 + KCOS 2 7 1 ~ ~ ) = 0.

(3.70)

Since cos 271c(x0 + 112) = - cos 2nxo and sin 271(x0 + 1,2) = - sin 271x0, we can use

COS 271 (xo + 112 + Afl + K cos 3n.xo)

= COS 271 (xo + 112) COS 2n(AR + K COS 2nx,)

- sin 2n(xo + 1/21 sin 2n(AC2 + K cos 3nxo)

(3.71)

to obtain

2AQ - ( K sin 2nxo) (271) (An + K cos 2nx0) 2 0 (3.72)

C2 = Pi0 + A i l , in order to obtain the shape of the resonant wedge for a cycle as K --+ 0.

In order to understand how quasi-periodic orbits can define a Cantor set of finite measure, yielding a “devil’s staircase” of resonant wedges, let us make the following assumption. Without proof, suppose that for fixed lK 1 4 1 all Q-cycles have roughly the same tongue widths - K Q , where there are at most Q - 1 distinct cycles with winding numbers W = PIQ, P = 1, 2, , . . , Q - 1 . We can then form a Cantor set by systematically removing the tongues of width Q z K”, n = 1, 2, 3, . . . one by one. Then, consider the intersection of the remaining set with a line corresponding to K = constant in the (R, K ) plane. With Q - 1 distinct wedges of - maximum width Ke, the total length I??, removed from the interval 0 < R < 1 would be on the order of

r niT - 2K + 2 (Q - + - K2 K)2 - K (3.79)

Q = 2

to lowest order in AC2 + K COS 2nx0, SO that to lowest order in K we have

(3.73) 2AR - K2n sin 4nx0 = 0

where various numerical factors have been ignored. This defines a Cantor set, so that the remaining tongues are removed from a Cantor set, producing yet another Cantor set. The argument given here is far from rigorous, but reflects the essence of the truth.

That m, --+ 0 as K --+ 0 corresponds to the fact that quasi- periodic orbits have unit measure in 0 < R < 1 when K = 0. For K > 0, the quasiperiodic orbits in 0 < C2 < 1 occupy a Cantor set with measure - 1 - mT - 1 - K. The plot of winding number vs. 0, for small K yields a comp- lementary point-set called the devils’s staircase (cf. Fig. 3.21). The devilish aspect follows from the fact that W has a finite

K

0

Fig. 3.1Y. Amol’d tongue growing from Cl = 1 , with W = I .

Physica Scripiu 710

Driven Systems, Cantor Sets and Strange Attractors 43

W

Fig. 3.21. A few steps in the incomplete devils staircase

width AR at every rational value and is a non-decreasing function of R. An explicit numerical representation of such a function, called a Cantor-function, can be constructed for thee middle-thirds Cantor set via binary arithmetic [ 111. That staircase has measure 1, hence is called “complete.” Since mT < 1 when K > 0, we have an “incomplete” evil’s stair- case. Bohr, Bak and Jensen [I41 have suggested that as K + 1 , mT + 1 - , yielding a complete devil’s staircase, and resonance overlap presumably occurs when K > 1. They have obtained numerical evidence for the scaling law

1 - mT - (1 - K ) 5 as K + I - , (3.80)

and have extracted the fractal dimension D = 1 - B / v - 0.87 for the complementary zero measure Cantor set to which quasiperiodic orbits are confined at K = 1. The critical exponent v is given by the crossover from regular behavior with D = 1 at K < 1 to critical behavior with D - 0.87 at K = 1. The evidence for the behavior suggested by Bak, Bohr and Jensen is numerical rather than analytic. To obtain an analytic description of criticality ( K + I - ) , it is necessary to use an analog of Feigenbaum’s renormalization group treatment of period doubling for quadratic maps [3].

3.6. Scaling and renormalization group equations for an

So far, we have the following picture of dynamical behavior for 0 < K < 1. When Wis rational, the motion corresponds to stable cycles (stability follows from /z < 0) in a finite range of R in 0 < R < 1, yielding the so-called Arnol’d tongues describing the frequency-locking regime. Here, the map is invertible and chaos cannot occur. However, stable quasi- periodic motion corresponding to irrational W occurs on a Cantor set of finite measure in R when K < 1. It is of interest to investigate the dynamics of the transition as K + 1- for irrational winding numbers, since those orbits are expected to become chaotic at K > 1 (cf. Section 3.3).

According to the mathematicians, for smooth maps such as, e.g.,

f(x) = x + R - ( K sin 27rx)/2n, (3.81)

which are invertible for K < 1, the motion is topologically equivalent to the pure rotation

# n + l = T(4n) = 4 n + W (3.82)

where the winding number

irrational winding number

f”(x0) - xo W = lim n-m n

(3.83)

is irrational andf”(x,) = f(f(f . . . f(xo)) . . .). This equiv- alence of f to a pure rotation can be expressed as an analytic transformation from x to 4 i f f is sufficiently well-behaved

[16, 171. The idea is to construct a transformation 4 = H(x) so that the function R = HfH-’ that defines x,+~ = R(4,) obeys R(4) = # + 2xW. Hence, the “universal” analytic equivalence of invertible circle maps to pure rotations. W is a non-decreasing function of R and locks into every rational frequency ratio in a finite range of R as R increases. The following method of analysis, due to Shenker [3, 191, is based upon Kadanoff’s treatment of the KAM problem for two- degree of freedom conservative systems [20]. Since one can- not determine from the condition

f”(xo> - xo W = lim (3.84) n-w n

which value of R, for a given xo, generates a given irrational value of W, we will use a continued fraction expansion of W to describe the approach to irrationality of W via a sequence of stable cycles of increasingly higher order. Because its continued fraction development yields a particularly simple form for the renormalization-group equations, the so-called Golden mean winding number

f i - 1 w = - 2

(3.85)

has been chosen for study [19, 211. The continued fraction representation [22] follows from the properties of the Fibo- nacci sequence given by Fo = 0, F, = 1,

Fn+, = F, + FnPl n 2 1. (3.86)

If we define the winding number

K = Fn/Fn+l* (3.87)

then using the recursion relation for Fn+ I , we obtain the finite continued fraction

w, = l / ( l + l /( l + l /( l + . . . + 1)) . . .) (3.88)

If we now ask for the limit

W = lim W, n-3o

(3.89)

we obtain on the one hand the infinite continued fraction (corresponding to irrational W ) W = l / ( l + l /( l + l / ( l + . . . ))) and on the other hand,

(3.90)

W = I / ( ] + W ) (3.91)

or W(l + W ) = 1, whose positive solution is given by

6- 1 W = - 2

(3.92)

If we choose a point x , , ~ belonging to the Fm+, cycle with winding number W, = F,/F,+, , we have

(3.93)

wheref‘”’ denotesf(xFm+,), the F;‘+l iterate off, and if we choose coordinates so that 0 belongs to the same cycle then

Fm f‘”’(0) w m = - - - - Fm+ I F m + I

(3.94)

Physica Scripta T20

44 Joseph L. McCauley

as f ’miI(0) = F,, . So, if we fix K E [0, I ] , we can calculate the particular rational bare frequency R,, that yields the par- ticular F,,,+l cycle.

Exercise: Prove that W, converge geometrically if lKI < 1

Solution: We begin with W, = Fn/FnTI. The idea is to show that the convergence is geometric, wl = W, - cd-”, with 6-1 = - @-* . F irst, note that

(3.95)

so that

FnFn+i .+ - W 2 as n + o (3.96) K t l - w, - w, - e-1

- - 42, I F n +?

which shows that

(3.97)

Some motivation for the study of continued fractions given asymptotically by an infinite string of 1’s can be given: Arnol’d [ 161 has pointed out that irrational winding numbers that can be accurately approximated by rationals exhibit continuous but nonuniform invariant densities, p ( x ) # 1,

it follows that d,, is the distance to the point on the cycle nearest to S = 0 (mod 1): For example iff(x) = x + W, then f ( 0 ) = W , and FFn(0) = Fn W, = F:/e+, . It follows that

dit = F,2IFn+I - F n - I = Fn(FnIFn+I - F n - I / F n )

= Fn(K - K-1) (3.102)

and we can understand from a few special cases how this works. If n = 2, W2 = F2/F, = 112, which is a 2-cycle and we obtain d2 = - 1/2, which is half a rotation from x = 0 mod 1. Ifn = 3, W, = F3/F, = 1/3 and d3 = 1/3 which is 1/3 of a full rotation (mod 1) beyond x = 0.

The main assumption that is made is that sZ,(K) -+ n ( K ) as n + m for fixed K where n ( K ) yields @ = (8 - 1)/2 as winding number for the m a p 5 The scaling follows from the geometrical rate of convergence of the winding numbers to W.

corresponding to “ghosts” of nonexistent cycles, due to the fact that low-order rationals WO, 6, etc. are accurate approximations to the irrational winding no. W. Among the irrationals that can be generated by continued fractions, those ending in strings of 1’s are the hardest to accurately approximate by rationals, so that for these we expect to obtain p ( x ) = 1. Among computable quasi-periodic orbits, these orbits have the winding numbers that have the slowest convergence rate in a continued fraction approximation.

When / K / = 1 (criticality), Shenker [I91 uses a numerical method to find instead that d,/d,+, + - 1.2857 . . . = - W-’ with x = 0.5267 . . . .

We can now use the recursion properties of the Fibonacci sequence along with the circle map condition f ( Q + 1) = 1 + f(Q) to set up the recursion formulas that are required for the renormalization group analysis of criticality. From f ( S + 1) = 1 + f ( S ) we have also that

f ( 0 + 2) = f ( Q + 1) + 1 = f ( Q ) + 2 Exercise: Show that

(3.104)

Solution: This is more or less guaranteed by the geometry of the devil’s staircase (plot W vs. Cl).

where F, is any integer, in particular, F, is a Fibonacci number. On the circle it follows also that

Hence, we have !2,(K) - Cl,(K) - c6-“ with 6-I = - W 2 for a given IK( < 1. This is one of the scaling laws in the subcritical case. Shenker [19] assumed that

6-1 ‘c -w.l (3.99)

and found numerically that 2’ z 2.1644 for sine-circle map when lK/ = 1, or 6 N -2.8336 . . .

for lKl < 1 also follows from the properties of the Fibonacci sequence, where

The second scaling relation d,/d,+l + - - as n -+ -- F,+l F, times times

d, = f’”(0) - F,-] (3.100) andf i s fixed at the frequency R, corresponding to winding number W, = F,/Fn+l (i.e., an E,+,-cycle) where also f’+’ (0) - Fn = 0. If we write down the d, for the case where

refers to an F,+>-cycle with winding number W,,, = F,+,,I Fn+2. We can then rewrite the latter expression as follows:

f””’(0) = f ( f . . . f f * . f (0) - F,- I - 43 f i s the pure rotation with winding number W,, --

F,,,, - 1 F, times . f ( ~ = e + w,, (3.101) times

Physrca Scripra T20

Driven Systems, Cantor Sets and Strange Attractors 45

= f . . . f ( f . ' . f ( O - F n + l ) ) - F" U-

= F,+, F,, times times

= f ' " " ( f ' " (Q) - E,-') - 6,

= f'"' ( f " " (Q) ) .

f'"' 1) (0) = f ' " - 1' ( f W (0)).

f;, (x) = z"f(n' ( a - ' 1 x) .

= f 5 + l ( f ( " - 1 ) ( O > ) - F,

By a similar procedure we can show that

To go further, we define

Fn

(3.107)

(3.108)

(3.109)

This form is motivated by Shenker's scaling assumption"

f ' " ' (x) 2. P f ( a " x ) (3.110)

where z = f i- ' is the "scaling length" andf is "universal". These assumptions were motivated by Feigenbaum's treat- ment of scaling near criticality for one dimensional maps with quadratic maxima [3, 41. An analytic development of the scaling theory can be found in Ostlund, Rand, Sethna and Siggia [23]. The goal here is to derive a circle map version of the universal renormalization group equation that was written down by Cvitanovic for the Feigenbaum period- doubling sequence, and which follows from n = m limit of a certain recursion relation.

We begin withf(")(cc-"x) = ccc"f,(.x) which can be rewrit- ten as f '"'(x) = z-"f(a"x),

f '"+ "(x) = f'"- " (f'"'(.)).

along with

Then

f ' " + I ' ( X ) = z - " - l ~ f ; l + , ( X n + l x ) (3.111)

and

f ' " - l ' ( z ) = r-"-'f,-I(C('I+lZ). (3.112)

Set z = f ( " ' ( x ) = a-'lf,(a"x) to obtain

z -"- y2 + ( zn + I x) = a - ' I+ 'fn - (a"- a"f, (a" x))

with z"+'x + x or x -, z-"-'x, we have

fn+l(X) = ~" - I (z - l fn (~- Ix ) ) . (3.1 14)

We can also usef("'l'(x) = f ( " ' ( f ( " - ' ' ( x ) ) to obtain

f n + I ( X > = .f,(.f-I(.-'x>>. (3.115)

These are analogs of the renormalization group equations of critical phenomena. The assumption of scaling corresponds to the assumption that the n = CO limit of these equations makes sense: ,f(x) = a2T(z-l f (a-1x)) (3.1 16)

(3.113)

or

f ( x ) = xf(af(a-2x)). (3.117)

Notice that f(x) = - 1 + . x is an exact solution, e.g.,

- 1 + x = 2 - 1 + a- ' x ) ) = z2(- 1 - a-I + a-2x)

- - - - 2 - z + x (3.118)

. . . - 1 = - 1 = - z 2 - z

-1 * $ a2 4- z -- 1 = 0, t( = 2

Hence, we obtain "Golden mean universality" with "classi- cal" exponents x = 2, y = 1 when IKI < 1. The behavior is universal in the sense that it holds for all invertible circle maps f that are sufficiently well-behaved.

For the case lKI = 1, no such trivial solution is known, but the case has been treated in the literature [23, 241. How universal is "Golden mean universality"? Shenker found that one obtains more or less the same critical exponents x 2: 0.527 . . . and y N 2.164 . . . by following the con- tinued fraction route to ($ - 1)/2 with the map

x,+, = x, + i2' - K (sin 271x,, + . 2 sin (6nxn))/2n.

(3.1 19) However, the original map with winding number w = 1 (3.120)

2 + - l 1 2 + G . . .

yields different critical exponents [19]

x 0.523. . . y 2. 2.174 . . . (3.121)

Hence, a more useful development of the theory of universal critical exponents for circle maps requires a less restricted viewpoint than the one developed above for only one winding number. Cvitanovic et al. have used the properties of Farey fractions and the Farey tree to provide one such generaliz- ation [25,26], and other generalizations have been construc- ted as well [27].

Here, our chapter ends, where the hard work begins, namely, the nature of the motion for K > 1 . At K 2 1, the quasiperiodic orbits occupy a zero-measure Cantor set in (K, Cl)-space, and are presumably chaotic, in agreement with the Ruelle-Takens-Newhouse scenario. However, the stable periodic orbits persist for some distance above the line K = 1 and go chaotic via a period-doubling sequence [27]. Unfor- tunately, circle maps have little to do with the Josephson junction map beyond criticality, and that system must then be analyzed by other means. That is to say, the universal behav- ior predicted for circle maps seems to work for the Josephson Junction problem at IKI 5 1 , but the latter system is not describeable by a one-dimensional circle map when IKJ > 1 [14, 271.

References to Chapter 3

1. Lanford, 0. E. 11, in "Chaotic Behavior in Deterministic Systems (Edited by Ioos et al, North-Holland (1983).

Physicu Scripta R O

46 Joseph L. McCauley

-I -. 3. 4. 5 . 6.

7. 8.

9. IO. 1 1 .

12.

13. 14.

15. 16. 17.

Brush, S. G . , Kinetic Theory. Vol. 11, Pergamon Pr. (1966). Cvitanovic. P., Universality in Chaos, Adam Hilger (1984). Schuster, H. G.. Deterministic Chaos, Physik-Verlag (1984). Sparrow. C., The Lorenz Equations, Springer-Verlag (1982). Grossman, S., in Nonequilibrium Cooperative Phenomena in Physics and Related Fields (Edited by M. G. Velarde). Plenum Pr. (1983). Friiyland, J . and Alfsen, K . H., Phys. Rev. A29, (1984). Lichtenberg, A. J. and Lieberman. M. A., Regular and Stochastic Motion. Springer-Verlag (1983). Farmer, J.. Ott, E. and Yorke, J . A.: Physica 7D. 153 (1983). Mandelbrot, B., The Fractal Geometry of Nature, Freeman (1982). Gelbaum, B. R. and Olmsted, J . H., Counterexamples in Analysis, Holden-Day (1964). Guckenheimer, J. and Holmes, P., Nonlinear Oscillations. Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag (1983). Physica Scripta T9 (1985). Bohr. T. Bak. P. and Jensen, M. H. , Physica Scripta T9, 50 (1985); Phys. Rev. A30, 1960 (1984); Phys. Rev. A30, 1970 (1984). Azbel, M. Y. and Bak, P., Phys. Ref. B30, 3722 (1984). Arnol’d, V. I. Trans. Am. Math. Soc. 46. 213 (1965). Arnol’d. V. I.. Geometric Theory of ordinary Differential Equations,

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Springer-Verlag (1 983). Zaslavskii, G . M. , andchirikov. B. V.. Sov. Phqs. Usp. 14. 549 (1972). Shenker, S., Physica 5D, 405 (1982). Kadanoff, L. P., Phys. Rev. Lett. 47, 1641 (1981). Green, J. M., J. Math. Phys. 20. 1183 (1979). Schroeder, Number Theory in Science and Communication, Springer- Verlag, (1 985). ostlund, S., Rand, D., Sethna. J. and Siggia, E.. Physica SD. 303 (1 983). Feigenbaum. M., Kadanoff. L. P. and Shenker, S.. Physica 5D. 370 ( 1982), Cvitanovic, P., in Instabilities and dynamics of Lasers and Nonlinear Optics (Edited by R. N . Boyd et al.), Cambridge University Press (1985). Cvitanovic, P.. Shraiman, B., and Soderberg, Bo, NORDITA preprint- 85/21 “Scaling Laws for Mode Locking in Circle Maps” (1985). Alstrom, P.. Christiansen, B., Hyldgaard, P., Levinson, M . T.. and Rasmussen, R.. “Scaling relations at the critical line for the sine map and the damped driven pendulum”, to be published in Phys. Rev. A (1986).