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 mathe
matics? 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 under
standing 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 so
phisticated 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 Chap
ter 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
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 Fall '09
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 Numerical Analysis, Linear Systems, The Land, Nonlinear system, Iterated function, Peter J. Olver

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