Problems with computer experiments [in chaos theory]

The defining characteristic of a system that exhibits chaos is that on successive steps the system samples digits which lie further and further to the right in its initial condition. But in a practical computer, only a limited number of digits can ever be stored. In Mathematica, one can choose how many digits to store (and in the pictures shown in the main text, enough digits were used to avoid the problems discussed in this note). But a low-level language such as Fortran, C or Java always stores a fixed number of digits, typically around 53, in its standard double-precision floating-point representation of numbers.

So what happens when a system one is simulating tries to sample digits in its initial conditions beyond the ones that are stored? The answer depends on the way that arithmetic is handled in the computer system one uses.

When doing high-precision arithmetic, Mathematica follows the principle that it should only ever give digits that are known to be correct on the basis of the input that was provided. This means that in simulating chaotic systems, the numbers produced will typically have progressively fewer digits: later digits cannot be known to be correct without more precise knowledge of this initial condition. (An example is NestList[Mod[2 #, 1]&, N[Pi/4, 40], 200]; Map[Precision, list] gives the number of significant digits of each element in the list.)

But most current languages and hardware systems follow a rather different approach. (For low-precision machine arithmetic, Mathematica is also forced to follow this approach.) What they do is to give a fixed number of digits as the result of every computation, whether or not all those digits are known to be correct. It is then the task of numerical analysis to establish that in a particular computation, the final results obtained are not unduly affected by digits that are not known to be correct. And in practice, for many kinds of computations, this is to a large extent the case. But whenever chaos is involved, it is inevitably not.

As an example, consider the iterated map x -> Mod[2x, 1] discussed in the main text. At each step, this map shifts all the base 2 digits in x one position to the left. But if the computer gives a fixed number of digits at each step, then additional digits must be filled in on the right. On most computers, these additional digits are always 0. And so after some number of steps, all the digits in x are 0, and thus the value of x is simply 0.

But it turns out that a typical pocket calculator gives a different result. For pocket calculators effectively represent numbers in base 10 (actually so-called binary-coded decimal) not base 2, and fill in unknown digits with 0 in base 10. (Base 10 is used so that multiplying for example 1/3 by 3 gives exactly 1 rather than the more confusing result 0.9999... obtained with base 2.)

Pictures (a) and (c) below show simulations of the shift map on a typical computer, while pictures (b) and (d) show corresponding simulations on a pocket calculator. (Starting with initial condition x the digit sequence at step n is essentially

IntegerDigits[Mod[2^{n} Floor[2^{53} x], 2^{53}], 2, 53]

on the computer, and

Flatten[IntegerDigits[IntegerDigits[Mod[2^{n} Floor[10^{12} x], 10^{12}], 10, 12], 2, 4]]

on the calculator. In both cases the limited number of digits implies behavior that ultimately repeats—but only long after the other effects we discuss have occurred.)

For the first several steps, the results as shown at the top of each corresponding picture agree. But as soon as the effect of sampling beyond the digits explicitly stored in the initial condition becomes important, the results are completely different. The computer gives simply 0, but the pocket calculator yields apparently random sequences—which turn out to be analogous to those discussed on page 319.

Other chaotic systems have a similar sensitivity to the details of computer arithmetic. But the simple behavior of the shift map turns out to be rather rare: in most cases—such as the multiplication by 3/2 shown in the pictures below—apparent randomness is produced, even on a typical computer.

It is important to realize however that this randomness has little to do with the details of the initial conditions. Instead, just as in other examples in this book, the randomness arises from an intrinsic process that occurs even with the simple repetitive initial condition shown in pictures (c) and (d) above.

Computer simulations of chaotic systems have been done since the 1950s. And it has often been observed that the sequences generated in these simulations look quite random. But as we now see, such randomness cannot in fact be a consequence of the chaos phenomenon and of sensitive dependence on initial conditions.

Nevertheless, confusingly enough, even though it does not come from sensitive dependence on initial conditions, such randomness is what makes the overall properties of simulations typically follow the idealized mathematical predictions of chaos theory. The point is that the presence of randomness makes the system behave on different steps as if it were evolving from slightly different initial conditions. But statistical averages over different initial conditions typically yield essentially the results one would get by evolution from a single initial condition containing an infinite number of randomly chosen digits.