Computational periodicity as observed in a simple system

Abstract

When the exact time-dependent solutions of a system of ordinary differential equations are chaotic, numerical solutions obtained by using particular schemes for approximating time derivatives by finite differences, with particular values of the time increment τ , are sometimes stably periodic. It is suggested that this phenomenon be called computational periodicity.

A particular system of three equations with a chaotic exact solution is solved numerically with an Nth-order Taylorseries scheme, with various values of N, and with values of τ ranging from near zero to just below the critical value for computational instability. When N = 1, the value of τ below which computational periodicity never appears is extremely small, and frequent alternations between chaos and periodicity occur within the range of τ . Computational periodicity occupies most of the range when N = 2 or 3, and about half when N = 4.

These solutions are compared with those produced by fourth-order Runge—Kutta and Adams—Bashforth schemes, and with numerical solutions of two other simple systems. There is some evidence that computational periodicity will more likely occur when the chaos in the exact solutions is not very robust, that is, if relatively small changes in the values of the constants can replace the chaos by periodicity.