• models of neural networks and to creation of the field of artificial life (mathematical models that simulate the dynamics of living systems).

Nonlinear science is inherently interdisciplinary, impacting traditional sciences, mathematics, and engineering, as well as the social sciences, notably economics and demographics. Any attempt to circumscribe artificially the scope of nonlinear science inevitably limits the insights it can provide.

The successful pursuit of nonlinear science requires the blending of four distinct methodological approaches:

1. Modeling, which seeks to improve the analytical foundation of the problem under consideration—improved models are particularly necessary for the emerging areas of nonlinear science that enable some of the new technologies discussed in this report;
2. Experimental mathematics, which involves the use of cleverly conceived computer-based numerical simulations to give qualitative insights into problems that are analytically intractable;
3. Novel analytical mathematical methods to treat functional recursion relations, to solve nonlinear partial differential equations, or to describe complex structures arising in chaotic systems; and
4. Experimental observations of similar nonlinear phenomena in natural and man-made systems arising in a variety of conventional disciplines.

One specific example that illustrates the interdisciplinary applicability of the paradigms of nonlinear science as well as its multicomponent methodology is the discovery, made in the late 1970s by Feigenbaum,¹ that the particular type of transition to chaos—a sequence of period doublings—observed in a very simple mathematical equation was universal in the sense that it was independent of the specific equation and should in fact arise in a wide class of physical, chemical, and biological systems. Feigenbaum's studies were initially computational, but he soon developed an analytic scaling theory. Nonetheless, skeptics considered his results to be a mere theoretical curiosity of no clear experimental significance until Libchaber² and others observed exactly the same period doubling dynamics in laboratory experiments on fluids and electric circuits. The ensuing efforts to prove various aspects of the theory rigorously stimulated large segments of the pure mathematics community. Conversely, in other cases involving new nonlinear phenomena, laboratory observations have stimulated and guided the development of theory and mathematical modeling. This close interaction among experimenters, theorists, and pure mathematicians—rare and refreshing in the recent age of increasingly specialized science—has also proved to be extremely powerful in aiding the rapid advance of nonlinear science.

Researchers in nonlinear science typically face particular difficulties with regard to obtaining support. Funding agencies are organized to evaluate proposals in the traditional scientific categories. Interdisciplinary funding of the sort likely to have an impact in nonlinear science is very difficult to obtain, especially with the post-Cold War cutbacks in scientific funding. Further, as a relatively new field, nonlinear science seems to pale in direct comparison to more mature fields (e.g., solid-state physics, space physics, or nuclear physics)