### 9. CONCLUSION

In spite of the practical problems which remain to be solved, the
ideas developed in this review have obvious applications in
many areas.

The most important applications, however, may be pedagogical.

The elegant body of mathematical theory pertaining to linear
systems (Fourier analysis, orthogonal functions, and so on),
and its successful application to many fundamentally linear
problems in the physical sciences, tends to dominate even
moderately advanced University courses in mathematics and
theoretical physics. The mathematical intuition so developed
ill equips the student to confront the bizarre behaviour exhibited
by the simplest of discrete nonlinear systems, such as
equation (3). Yet such nonlinear systems are surely the rule,
not the exception, outside the physical sciences.

I would therefore urge that people be introduced to, say,
equation (3) early in their mathematical education. This
equation can be studied phenomenologically by iterating it on
a calculator, or even by hand. Its study does not involve as
much conceptual sophistication as does elementary calculus.
Such study would greatly enrich the student's intuition about
nonlinear systems.

Not only in research, but also in the everyday world of
politics and economics, we would all be better off if more
people realised that simple nonlinear systems do not necessarily
possess simple dynamical properties.

I have received much help from F. C. Hoppensteadt, H. E.
Huppert, A. I. Mees, C. J. Preston, S. Smale, J. A. Yorke, and
particularly from G. F. Oster. This work was supported in part
by the NSF.