Universal

by Mitchell J. Feigenbaum

Behavior in

here exist in nature processes that

or

Universal numbers, T can be described as complex are

8 =

4.6692016...

and

α = 2.502907875...,

determine quantitatively

the transition from

smooth to turbulent or

erratic behavior

for a large class of

nonlinear systems.

chaotic and processes that are simple or orderly. Technology attempts to create devices of the simple variety: an idea is to be implemented, and various parts executing orderly motions. are assembled. For example, cars, airplanes, radios, and clocks are all constructed from a variety of elementary parts each of which, ideally, implements one ordered aspect of the device. Technology also tries to control or minimize the impact of seemingly disordered processes, such as the complex weather patterns of the atmosphere, the myriad whorls of turmoil in a turbulent

fluid, the erratic noise in an electronic

signal, and other such phenomena. It is

the complex phenomena that interest us

here.

When a signal is noisy, its behavior from moment to moment is irregular and has no simple pattern of prediction. However, if we analyze a sufficiently long record of the signal, we may find that signal amplitudes occur within narrow ranges a definite fraction of the time. Analysis of another record of the signal may reveal the same fraction. In this case, the noise can be given a statistical description. This means that while it is impossible to say what amplitude will appear next in succession, it is possible to estimate the probability or likelihood that the signal will attain some specified range of values. Indeed, for the last hundred years disorderly processes have been taken to be statistical (one has

given up asking for a precise causal prediction), so that the goal of a description is to determine what the probabilities are, and from this information to determine various behaviors of interest-for example, how air turbulence modifies the drag on an airplane.

We know that perfectly definite causal and simple rules can have statistical (or random) behaviors. Thus, modern computers possess "random number generators" that provide the statistical ingredient in a simulation of an erratic. process. However, this generator does nothing more than shift the decimal point in a rational number whose repeating block is suitably long. Accordingly, it is possible to predict what the nth generated number will be. Yet, in a list of successive generated numbers there is such a seeming lack of order that all statistical tests will confer upon the numbers a pedigree of randomness. Technically, the term "pseudorandom" is used to indicate this nature. One now may ask whether the various complex processes of nature themselves might not be merely pseudorandom, with the full import of randomness, which is untestable, a historic but misleading concept. Indeed our purpose here is to explore this possibility. What will prove altogether remarkable is that some very simple schemes to produce erratic numbers behave identically to some of the erratic aspects of natural phenomena. More specifically, there is now cogent evidence that the problem of how a fluid changes over from smooth to turbulent

flow can be solved through its relation to the simple scheme described in this article. Other natural problems that can be treated in the same way are the behavior of a population from generation to generation and the noisiness of a large variety of mechanical, electrical, and chemical oscillators. Also, there is now evidence that various Hamiltonian systems those subscribing to classical mechanics, such as the solar system

can come under this discipline. The feature common to these phenomena is that, as some external parameter (temperature, for example) is varied, the behavior of the system changes from simple to erratic. More precisely, for some range of parameter values, the system exhibits an orderly periodic behavior; that is, the system's behavior reproduces itself every period of time T. Beyond this range, the behavior fails to reproduce itself after T seconds; it almost does so, but in fact it requires two intervals of T to repeat itself. That is, the period has doubled to 2T. This new periodicity remains over some range of parameter values until another critical parameter value is reached after which the behavior almost reproduces itself after 2T, but in fact, it now requires 4T for reproduction. This process of successive period doubling recurs continually (with the range of parameter values for which the period is 2"T becoming successively smaller as n increases) until, at a certain value of the parameter, it has doubled ad infinitum, so that the behavior is no longer periodic. Period doubling is then a characteristic route for a system to follow as it changes over from simple periodic to complex aperiodic motion. All the phenomena mentioned above exhibit period doubling. In the limit of aperiodic behavior, there is a unique and hence universal solution common to all systems undergoing period doubling. This fact implies remarkable consequences. For a given system, if we

Thus, this definite number must appear as a natural rate in oscillators, populations, fluids, and all systems exhibiting a period-doubling route to turbulence! In fact, most measurable properties of any such system in this aperiodic limit now can be determined, in a way that essentially bypasses the details of the equations governing each specific system because the theory of this behavior is universal over such details. That is, so long as a system possesses certain qualitative properties that enable it to undergo this route to complexity, its quantitative properties are determined. (This result is analogous to the results of the modern theory of critical phenomena, where a few qualitative properties of the system undergoing a phase transition, notably the dimensionality, determine universal critical ex

f(x1)

Xn+1 = f(xn)

(4)

Analogous to Eq. (4), a starting coordinate pair (xo-Yo) is used in Eq. (14) to determine the next coordinate (x,,y,). Equation (14) is reapplied to determine (x2,y2) and so on. For some initial points, all iterates lie along a definite elliptic curve, whereas for others the iterates are distributed "randomly" over a certain region. It should be obvious that no explicit formula will account for the vastly rich behavior shown in the figure. That is, while the iteration scheme of Eq. (14) is trivial to specify, its nth iterate as a function of (x,y) is unavailable. Put differently, applying the simplest of nonlinear iteration schemes to itself sufficiently many times can create vastly complex behavior. Yet, precisely because the same operation is reapplied, it is conceivable that only a select few selfconsistent patterns might emerge where the consistency is determined by the key notion of iteration and not by the particular function performing the iterates. These self-consistent patterns do occur in the limit of infinite period doubling and in a well-defined intricate organization that can be determined a priori amidst the immense complexity depicted in the cover figure.

The maximum value of f(x) in Eq. (15) is attained at x = 1⁄2 and is equal to λ. Also, for λ > 0 and x in the interval (0,1), f(x) is always positive. Thus, if λ is anywhere in the range [0,1], then any iterate of any x in (0,1) is also always in (0,1). Accordingly, in all that follows we shall consider only values of x and λ lying between 0 and 1. By Eq. (16) for 0 ≤ λ, only x = 0 is within range, < 4, x* whereas for ≤λ≤ 1, both fixed points are within the range. For example, if we set λ = 1⁄2 and we start at the fixed point x = xo 1⁄2 (that is, we set x1 = 1⁄2), then x1 =

Figure 1 depicts this process for λ = The two fixed points are circled, and the first several iterates of an arbitrarily chosen point x, are shown. What should be obvious is that if we start from any xo in (0,1) (x = 0 and x = 1 excluded), upon continued iteration x, will converge to the fixed point at x = 2. No matter how close xo is to the fixed point at x = 0, the iterates diverge away from it. Such fixed point is termed unstable. Alternatively, for almost all xo near enough to x = 1⁄2 [in this case, all x, in (0,1)], the iterates converge towards x = 2. Such a fixed point is termed stable or is referred to as an attractor of period 1.

Now, if we don't care about the transient behavior of the iterates of xo, but only about some regular behavior that will emerge eventually, then knowledge of the stable fixed point at x = 1⁄2 satisfies our concern for the eventual behavior of the iterates. In this restricted sense of eventual behavior, the existence of an attractor determines the solution independently of the initial condition x。 provided that x, is within the basin of attraction of the attractor; that is, that it is attracted. The attractor satisfies Eq. (16), which is explicitly independent of xo. This condition is the basic theme of universal behavior: if an attractor exists, the eventual behavior is independent of the starting point.

Fig. 1. Iterates of x, at λ= 0.5.

What makes x = 0 unstable, but x = 1⁄2 stable? The reader should be able to convince himself that x = O is unstable because the slope of f(x) at x = 0 is greater than 1. Indeed, if x* is a fixed point of f and the derivative of f at x*, f'(x*), is smaller than 1 in absolute value, then x is stable. If If'(x) is greater than 1, then x is unstable. Also, only stable fixed points can account for the eventual behavior of the iterates of an arbitrary point.

We now must ask, "For what values of λ are the fixed points attracting?" By Eq. (15), f'(x): 42(12x) so that f'(0) = 42

and

f'(x)=2-42.

=

(18)

(19)

For 0 << 4, only x* = 0 is stable. At λ = 4, x = 0 and f'(x) = 1. For <λ < 4, x* is unstable and x is stable, while at 24, f'(x) = -1 and x* also has become unstable. Thus, for 0 < λ < 4, the eventual behavior is known.

Period 2 from the Fixed Point

What happens to the system when λ is in the range < λ < 1, where there are no attracting fixed points? We will see that as increases slightly beyond λ = 4, f undergoes period doubling. That is, instead of having a stable cycle of period 1 corresponding to one fixed point, the system has a stable cycle of period 2; that is, the cycle contains two points. Since these two points are fixed points of the function f2 (f applied twice) and since stability is determined by the slope of a function at its fixed points, we must now focus on f2. First, we examine a graph of f2 at λ just below 4. Figures 2a and b show f and f2, respectively, at λ = 0.7.

To understand Fig. 2b, observe first that, since f is symmetric about its maximum at x = 2, f2 is also symmetric about x = 2. Also, f2 must have a fixed point whenever f does because the second iterate of a fixed point is still that same point. The main ingredient that determines the period-doubling behavior of f as λ increases is the relationship of the slope of f2 to the slope of f. This relationship is a consequence of the chain rule. By definition

Fig. 2.λ= 0.7. x* is the stable fixed point. The extrema of f2 are located in (a) by constructing the inverse iterates of x = 0.5.