An Introduction to the Properties of One-Dimensional Difference Equations
Until quite recently most physical systems have been described by continuous equations, either differential or integral ones. Difference equations have been studied, but usually as a finite-difference approximation to be used as a computational algorithm for the continuous system. In this case the approximation is designed such that the difference equation mirrors the corresponding continuous system. However, it is now realized that simple, but nonlinear, difference equations can have very complicated solutions, a complexity not found in what one would consider the analogous differential equation. This was brought to the attention of scientists by May.(1) Although the original application involved problems in ecology, where the difference equations have an immediate interpretation, it was soon realized that they had a much wider range of applicability. In particular, the pioneering work of Feigenbaum showed their relevance to such problems as turbulence. However, it is not the purpose of this chapter to relate the mathematical results to fields of application. Here, some of the mathematical properties of one-dimensional difference equations will be considered. Surprisingly, even for such a restricted class of equations, there is a bewildering range of properties.
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