TA has an unstable (nonattractive) internal fixed point; its approximate coordinates are So = 0.5885696 and ao = 0.1388662. Some twenty years after the appearance of our paper, TA was examined on a Cray computer by Erica Jen. The results strongly suggested that its limit set is what is today called a strange attractor, with a fractal (noninteger) dimension of about 1.7. The term "strange attractor" was coined by Ruelle and Takens in 1971 in the course of a study of turbulence. Strange attractors are now known to arise often during iteration of the nonlinear differential or difference equations used to describe phenomena in, for example, meteorology and fluid dynamics.

Several other messes have been classified as strange attractors by present-day criteria, the main one being sensitive dependence on initial conditions. That is, a limit set is a strange attractor if any two points within the set, no matter how close, move farther and farther apart under the action of the mapping. If the limit set is bounded away from infinity (as it is here), the points cannot keep moving apart, and the criterion then is that the relative positions of the limit points become uncorrelated-a feature of chaos. Unfortunately, no numerical experiment can prove that some limit set is a strange attractor. For example, what appears to be a strange attractor may actually be a periodic limit set of very high order. To my knowledge, rigorous measures of the likelihood that a computer-generated limit set is a strange attractor have not yet been developed.

Having said that, I shall pretend that some of our cubic maps do illustrate strange attractors. How can those maps be studied further? One way is to introduce another variable 8 (0 < < 1). Letting S' = F(S, a) and a' = G(S, a) denote the defining equations of the map (cf. Eq. 11b), we write a new set of equations as follows:

Note that 8 = 1 corresponds to the original map. (If 8 = 0, Eq. 12 reduces to the identity transformation.) So long as 8 lies in the given range, the first-order fixed points are independent of this parameter. The original system may have a nonattractive fixed point; it cannot, of course, be found by iteration. If, however, the fixed point can be made attractive by decreasing & (from unity), then iteration can be used, thus avoiding some messy algebra. In fact, a sufficient decrease in 8 will-in almost all cases decrease the absolute value of both eigenvalues of the Jacobian matrix of Eq. 12 to less than unity at the fixed point, which is precisely the criterion for the attractive

THE INFINITE LIMIT SET OF
TA-A "MESS"

Fig. 8. The transformation TA (see text for defining equations) is one of our two-dimensional cubic maps that iterates to a mess. Shown in (a) is its messy limit set; its nonattractive fixed point has been superimposed on the photograph. The magnifications in (b) and (c) reveal ever greater complexities.

EFFECT OF & ON THE LIMIT SET
OF TA

Fig. 9. Our two-dimensional cubic maps can
be generalized by introducing the parameter &
as described in the text. Shown here is the
effect of varying this parameter on the messy
limit set of TA (see Fig. 8). As & is decreased
from unity, the limit points at first coalesce into
seven distinct bunches, forming what we call a
pseudo-period. (a) Then at 8≈ 0.9930, the in-
finite limit set becomes periodic (and hence fi-
nite), with an order of 7. (b) This configuration
persists over a range of 6 values, although the
coordinates of the limit points vary. (c) Then
at ≈ 0.9770, the periodic limit set changes
into a closed curve. (d) As & is decreased fur-
ther, the curve becomes smaller and smaller.
Finally, at ≈ 0.9180, the curve collapses to
a single point, the nonattractive fixed point of
the original transformation (8 =
1).

character of such a point. This is the fact that motivated the introduction of 8, but the effect of its variation turned out to be much more interesting than we expected. Decreasing & may cause a remarkable change in the appearance of a messy limit set (Fig. 9). Points may start to cohere, forming a pattern of disjoint arcs. Further decrease of 8 may lead to a periodic limit set of finite order, which persists over a range of 8 values. As 8 approaches the value at which the limit set collapses to the fixed point, the set may metamorphose into a closed curve (at least something that looks like a curve) that shrinks continuously with 8. This behavior is typical; even more complex changes have been observed in some cases (Fig. 10).

Another way to study cubic maps with messes as their limit sets is to vary the coefficients. This is done just as it was for the quadratic maps, but the results are far more dramatic. Figure 11 shows a few examples of the fascinating behavior that has been observed. Here the coefficients constitute a twenty-parameter set, so exploration of all possibilities is not feasible; the usual practice is to vary the coefficients of one or two terms at a time. Much numerical work of that type was done at the Laboratory in 1984 and 1985 on a Cray computer, and many new strange attractors turned up. The aim of this work is to find some "structural" (geometric or algebraic) principle underlying the relatively bizarre phenomena our computer screens reveal.

One-Dimensional Maps and Universality

The first part of this section is a historical note on the origins of a 1973 paper by Metropolis, Stein, and Stein. The paper dealt with a certain universal structure and hierarchy of the periodic limit sets that can arise in the iteration of one-dimensional maps; it has been cited by Mitchell Feigenbaum as a source of inspiration for his later work on the universal nature of the approach to chaos by "period doubling."

The origins of our paper lie in the work discussed above by Stan and me on cubic maps. We had found fifteen or sixteen that had the property of transforming a pair of sides of the S, a reference triangle into each other. It is clear that the "square" of such a map (the second iterate) transforms one side of the triangle into itself, and the map is therefore one-dimensional. We rewrote some of these as maps defined on the unit interval and iterated them on MANIAC II. In every case we obtained a periodic limit

The restrictions on insure that the iterates of the maps lie within the specified x interval and that the nonzero first-order fixed points of the maps are nonattractive. (Equation 14a, the "parameterized parabola," is well known in ecology as the logistic equation. It is a transform of a quadratic map studied in the early sixties by the Finnish mathematician P. Myrberg. Had we been aware of his study, considerable time would have been saved.)

Equations 14a and 14b are examples of maps of the general form

Tx(x): x' = f(x),

where f(x) is defined on the interval [0,1] and has a single maximum (at which dx'/dx = 0). For simplicity we placed the maximum at x = and at first restricted ourselves to functions symmetric about that point. This restriction does not affect the results presented in the "MSS" paper (a name due to Derrida, Gervois, and Pomeau). We also required f(x) to be strictly concave; relaxing this requirement can have drastic effects, as we learned later.

ANOTHER EXAMPLE OF THE
EFFECT OF & ON A MESSY LIMIT SET

Fig. 10. An even more striking example of the
effect of varying & on a messy limit set. (a)
The limit set for the original two-dimensional
cubic map (8 = 1) consists of three separate
pieces. Photographs (c) through (g) focus on
the changes that occur in the piece shown
in greater detail in (b); similar changes occur
in the other two pieces. As & is decreased
monotonically from unity, the limit points (c)
consolidate, (d) form a set of disjoint arcs, (e)
disperse, (f) collapse to a periodic limit set
of order 26, and (g) form a closed curve that
eventually collapses to a single point.

T

EFFECT OF COEFFICIENTS ON THE
LIMIT SET OF TA

Fig. 11. Our two-dimensional cubic maps can also be generalized by varying the coefficients. In the examples above the transformation TA was modified by varying only two coefficients. The photographs show the dramatic effect of such modifications on the messy limit set of TA (Fig. 8). The modification given in (a) changes the mess into a seven-member set of closed curves, one of which is shown in detail. The very similar modification given in (b) changes the mess into a pseudo- period of order 7, that is, into seven distinct bunches of points, three of which are shown in detail. The modification given in (c) results in a remarkably different but still messy limit set.

In addition to the parabola and the sine, we also studied two other functions satisfying the conditions given above. One, a sixth-degree polynomial, was the transform to the unit interval of one of the one-dimensional cubic maps mentioned previously; the other was a trapezoid (in the x'x plane).

For all four maps we calculated the periodic limit sets of order k that begin and end with x = . These correspond to À values that are solutions of

and are necessarily attractive because of the condition that dx'/dx|x= = 0. (This
condition guarantees what is referred to as superstability.) To characterize the limit
sets in a function-independent way, we used the minimum distinguishing information,
namely, the positions of the successive iterates relative to x =
. For this purpose
we employed the letters R and L ("right" and "left"). For example, when k = 5, all
our maps have three distinct periodic limit sets of order 5, each associated with a
different value of A. Naturally, for different functions the values are different, as
are the actual values of the iterates, but the R, L (or MSS) patterns are identical. The
three patterns for k = 5, in order of increasing A, are RL R→ R
->

- R-L-L →R , and . Omitting the initial and final 's, we may write these patterns in simplified form as RLR2, RL2R, and RL3. The identity of the MSS patterns and their ordering on A was found to hold among all of our four functions for all values of k such that 2 ≤ k ≤ 15. We immediately

noted the phenomenon that Feigenbaum later called period doubling. As an example, consider the period of order 2, the pattern of which is R. The patterns of its first two doublings are RLR (k = 4) and RLR3LR (k = 8). A simple rule relates the pattern P of a given period and that of its doubling: if P contains an odd (even) number of R's, the pattern of its doubling is PLP (PRP). Note that P must be an MSS pattern; that is, it must begin and end at the x value for which x' is maximum. (Obviously, not every R, L succession is such a pattern.)

Period doublings are, of course, ordered on increasing X. The values corresponding to two successive doublings, A1 and A2, are "contiguous" in the sense that no A between 1 and 2 corresponds to a periodic limit set beginning at 1.

Our initial work indicated that a large class of maps generates the same sequence of patterns ordered on increasing A. Later experiments on some fifty additional maps confirmed this conclusion. It is still not known exactly, however, how this "large class" (almost certainly infinite) should be defined.

One of the most interesting results presented in the MSS paper is an algorithm for generating the MSS sequence. No iterations are needed, and no functions are explicitly specified. The algorithm is purely logical; given a limiting value kmax for the period order, it produces all MSS patterns with kkmax in the canonical ordering (that is, on increasing X). An independent proof of this algorithm is given for trapezoidal maps in Louck and Metropolis 1986. Others have found new algorithms for generating the MSS sequence, but, in my opinion, none of these are substantially simpler than ours.

Since the publication of these results, many mathematicians and physicists have studied one-dimensional maps, but much more work has been done on Feigenbaum's "quantitative" universality than on the "structural" universality represented by the MSS sequence. A few years ago Bill Beyer, Dan Mauldin (of North Texas State University), and I initiated new attacks on some of the problems suggested by MSS. We also considered a few new questions. One of these has to do with maps that exhibit a multiple appearance of some MSS patterns. If a map is strictly concave, it is our conjecture that each pattern occurs for just one value of X. We found that something else can happen otherwise. Consider the "indented trapezoid" map shown in Fig. 12, which is not strictly concave. For certain ranges of the parameters b and c, the same MSS pattern corresponds to three different values. (This phenomenon implies that Feigenbaum's quantitative universality, which hinges on the occurrence of period doublings at unique A values, is not applicable to certain maps and hence is less than truly universal.)

Our multiplicity, as we called it, is more than an interesting mathematical fact. It has helped in understanding the latest results of an extensive study of the BelousovZhabotinskii reaction by H. L. Swinney and his collaborators. (The B-Z reaction, the oxidation of malonic acid by an acidic bromate solution in the presence of a cerous ion catalyst, is an oscillating chemical system, that is, a system in which the concentrations of the chemical species do not vary monotonically with time but instead oscillate, sometimes chaotically, sometimes periodically.) In 1982 Simoyi, Wolf, and Swinney had identified certain members of the MSS sequence in the periodic concentration variations of the bromide ion, one of some thirty chemical entities involved in the reaction. In addition they found that the MSS patterns observed were ordered on a parameter 7 (the residence time of the reactants in the reaction vessel, which is inversely proportional to their rate of flow through the vessel) in exactly the same manner as the patterns in the MSS sequence are ordered on X. Several years later