The Theoretician Grapples with Complex Systems

The title echoes ``The Engineer Grapples with Nonlinear Problems,'' a classical paper of von Karman (1940) (who was hardly a ``mere'' engineer!). Much has been accomplished in the half century since Karman wrote. A great deal remains to be done, however, and nonlinear problems will be of major interest for the forseeable future. Yet in the last years there has been a subtle shift of emphasis; in many instances nonlinear problems have become embedded in the study of complex systems. The present, semi-popular, essay describes something of how theoreticians view complex systems, with emphasis on those systems with which I am familiar. It begins with a description of what a complex system is and what is meant by ``understanding'' how such a system works. Next the two principal approaches to the theoretical analysis of complex systems are discussed (models and simulations). Given the variety of possible types of models, how does one go about constructing one? From this large subject we limit ourselves to an illustration of how the hierarchical nature of complex systems leads to the notion of a phenomenological model. Comprehensive advice on solving the equations that result from a typical mathematical model is of course out of the question. Attention is focussed on the importance of concept development in the process of mathematical analysis. Comparison of theory and experiment is the next step. Here the advantages of a hand-made approach are stressed. The discourse concludes with what I believe is a novel and conceptually useful way of evaluating just how complex a given system really is.