David Campbell, Gottfried Mayer-Kress

Paper #: 91-09-032

We discuss the extent to which recent improvements in our understanding of inherently nonlinear phenomena present challenges to the use of mathematical models in the analysis of environmental and socio-political issues. In particular, we demonstrate that the “deterministic chaos” present in many nonlinear systems can impose fundamental limitations on our ability to predict behavior even when precisely defined mathematical models exist. On the other hand, results from chaos theory can provide means for better accuracy for short-term predictions even for systems which appear to behave completely randomly. Chaos also provides a new paradigm of a complex temporal evolution with bounded growth and limited resources without the equivalent of stagnation and decay. This is in contrast to a traditional view of historical evolution which is perhaps best expressed by the phrase: “if something stops growing, it starts rotting.” The exploration of a large number of states by a single deterministic solution creates the potential for adaptation and evolution. In the context of artificial life models this has led to the notion of “Life at the edge of chaos” expressing the principle that a delicate balance of chaos and order is optimal for successful evolution. Since our primary aim is didactic, we make no attempt to treat realistic models for complex issues but rather introduce a sequence of simple models which illustrate the increasingly complicated behavior that can arise when the nonlinearity is properly taken into account. We begin with the familiar elementary model of population growth originally due to Malthus and indicate how the incorporation of nonlinear effects alters dramatically the expected dynamics of the populations. We then discuss models which are caricatures of two issues--weather prediction and international arms races. Among the arms race models we consider a special class which is related to population dynamcis and which is first introduced by L. F. Richardson after WW I. The examples we discuss, however, have discrete time dependence. For certain ranges of their control parameters, these models exhibit “deterministic chaos,” and we discuss how this behavior limits our ability to anticipate and predict the outcomes of various situations. We then briefly describe methods to exploit the high sensitivity of chaotic systems to dramatically increase the capability of both forecasting and control of chaotic systems. We show that many different solutions can coexist even in simple models and how machine learning methods such as neural nets and genetic algorithms can be used to find classes of optimal solutions. We speculate on some generalizations of arms control models into object oriented frameworks which allow simultaneous modeling on different levels of quantitative formalizations: In a computational network we can have nodes which represent purely conceptual models of areas where quantitative analysis would be inappropriate and other nodes for which a hierarchical structure of mls of arbitrary quantitative detail and sophistication can be generated. Finally, we close with a few remarks on our general theme, stressing that, despite its limitations and because of its challenges, mathematical modeling of complex environmental and socio-political issues is crucial to any efforts to use technology to enhance international stability and cooperation.