Discrete approximations

When a dynamical system is simulated on a computer, a certain kind of approximation is made. A computer has only a finite set of numbers which it can represent. Given a point whose coordinates can be represented, the computer performs some arithmetic and arrives at an approximate image point. The true image point may not be representable, but it is usually safe to assume that the computer's approximation is close to the true one. If everything is working correctly, the computer will always compute the same approximate image point to a given initial point. The ideas discussed in this section are slight modifications of the original ideas found in a paper by Lax [10]. The reader is also referred to Rannou [12] and Hall [6] for further development of the area-preserving case.

To formulate the questions and give the corresponding answers we need the notion of a net and the notion of approximation for .

Suppose that the set of points representable by the computer is the -net . Since the computer can represent only those points in , an attempt to compute the map results in the -approximation .

Note that, if , then an -approximation to always exists. Henceforth, this inequality will be assumed. Indeed, since a -net is automatically an -net for any , it will be assumed that an -approximation occurs on an -net. Thus the phrase " is an -approximation for " means that is an -net for and is an -approximation for .

In the context of computer simulations, the -approximation is determined from the map by the computer arithmetic, the compiler, and the algorithm. The computer will always make the same error if it does the same computation. Thus the simulation of iteration of the original map on the computer is exactly the iteration of the map .

If the map has an attractor, one can ask whether one can expect to see the attractor in a computer simulation. This question can be interpreted as asking whether the map has an attractor corresponding to the attractor for .

The following theorem states that if the intensity of the attractor exceeds the computer's approximation error, then there exists a discrete representation for which the computer should be able to find by iterating .

Theorem 6.2 Let be an -approximation for , and let be an attractor for . If , then there exists an attractor for such that is a discrete representation of .

Proof.

This theorem gives a sufficient condition for the existence of a discrete representation, but is it a necessary one? In other words, if the intensity is less than the computer error, will the computer be unable to find the attractor? The answer is given by the following theorem, which states that one can find a discrete approximation and an orbit for the discrete approximation which starts in the attractor and leaves the domain of attraction. Of course, the discrete approximation might not be the one that the computer uses, but the theorem shows that, in general, one cannot expect to find attractors with small intensities.

Theorem 6.3 If is an attractor for , if , and if is any compact subset of , then there exists an -net , an -approximation , and an orbit of with and .

Proof.