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, although recent attempts at grand unification of all fundamental physical theories, such as string theory and conformal field theory, [8], are nonlinear.) For this reason, an ever increasing proportion of modern mathematical research is devoted to the analysis of nonlinear systems and nonlinear phenomena. Why, then, does one devote so much time studying 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, one must 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-aided investigations of nonlinear systems. However, despite dramatic advances in both hardware capabilities and sophisticated mathematical algorithms, many nonlinear systems — for instance, fully general Einsteinian gravitation, or the Navier–Stokes equations of fluid mechanics at high Reynolds numbers — still remain beyond the capabilities of today’s computers. The goal of these lecture notes is to provide a brief overview of some of the most important ideas, mathematical techniques, and new physical phenomena in the nonlinear realm. We start with iteration of nonlinear functions, also known as discrete dynamical systems. Building on our experience with iterative linear systems, as developed in Chapter 10 of [14], we will discover that functional iteration, when it converges, provides a powerful mechanism for solving equations and for optimization. On the other hand, even