Chapter 4: Systems Based on Numbers

Section 9: Partial Differential Equations

Ordinary differential equations

It is also possible to set up systems which have a finite number of continuous variables (say a[t], b[t], etc.) that change continuously with time. The rules for such systems correspond to ordinary differential equations. Over the past century, the field of dynamical systems theory has produced many results about such systems. If all equations are of the form a'[t] == f[a[t], b[t],…], etc. then it is known for example that it is necessary to have at least three equations in order to get behavior that is not ultimately fixed or repetitive. (The Lorenz equations are an example.) If the function f depends explicitly on time, then two equations suffice. (The van der Pol equations are an example.)

Just as in iterated maps, a small change in the initial values a[0] etc. can often lead to an exponentially increasing difference in later values of a[t], etc. But as in iterated maps, the main part of this process that has been analyzed is simply the excavation of progressively less significant digits in the number a[0].

(Note that numerical simulations of ODEs on computers must approximate continuous time by discrete steps, making the system essentially an iterated map, and often yielding spurious complicated behavior.)

From Stephen Wolfram: A New Kind of Science [citation]