Chapter 19 Nonlinear Systems

Nonlinearity is ubiquitous in physical phenomena. Fluid and plasma mechanics, gas dynamics, elasticity, relativity, chemical reactions, combustion, ecology, biomechanics, and many, many other phenomena are all governed by inherently nonlinear equations. (The one notable exception is quantum mechanics, which is a fundamentally linear theory. Recent attempts at grand unification of all fundamental physical theories, such as string theory and conformal field theory, [85], do venture into the nonlinear wilderness.) For this reason, an ever increasing proportion of modern mathematical research is devoted to the analysis of nonlinear systems. Why, then, have we devoted the overwhelming majority of this text to linear mathematics? The facile answer is that nonlinear systems are vastly more difficult to analyze. In the nonlinear regime, many of the most basic questions remain unanswered: existence and uniqueness of solutions are not guaranteed; explicit formulae are difficult to come by; linear superposition is no longer available; numerical approximations are not always sufficiently accurate; etc., etc. A more intelligent answer is that a thorough understanding of linear phenomena and linear mathematics is an essential prerequisite for progress in the nonlinear arena. Therefore, in this introductory text on applied mathematics, we have no choice but to first develop the proper linear foundations in sufficient depth before we can realistically confront the untamed nonlinear wilderness. Moreover, many important physical systems are “weakly nonlinear”, in the sense that, while nonlinear effects do play an essential role, the linear terms tend to dominate the physics, and so, to a first approximation, the system is essentially linear. As a result, such nonlinear phenomena are best understood as some form of perturbation of their linear approximations. The truly nonlinear regime is, even today, only sporadically modeled and even less well understood. The advent of powerful computers has fomented a veritable revolution in our understanding of nonlinear mathematics. Indeed, many of the most important modern analytical techniques drew their inspiration from early computer forays into the uncharted nonlinear wilderness. However, despite dramatic advances in both hardware capabilities and sophisticated mathematical algorithms, many nonlinear systems — for instance, Einsteinian gravitation — still remain beyond the capabilities of today’s computers. Space limitations restrict us to providing just a brief overview of some of the most important ideas, mathematical techniques, and new physical phenomena that arise when venturing into the nonlinear realm. In this chapter, we start with iteration of nonlinear functions. Building on our experience with iterative linear systems, as developed in Chapter 10, we will discover that functional iteration, when it converges, provides a powerful mechanism for solving equations and for optimization. On the other hand, even very