Complexity 2008: Nonlinear Science 101

Welcome to the website that provides some background and additional material for the tutorial lecture entitled “Nonlinear Science 101,” which I prepared for the National Academies of Science Keck Foundation Initiative (NAKFI) conference on “Complex Systems.”

“Nonlinear science” is the study of those mathematical systems and natural phenomena that are not linear. One of the founders of the field, the mathematician Stan Ulam, famously remarked that this was “like defining the bulk of zoology by calling it the study of ‘non-elephant animals.” His point, clearly, was that the vast majority of natural phenomena and their mathematical models are nonlinear, with linearity being the exceptional, albeit important, case. In nonlinear systems, one cannot naively add solutions together, and this failure of superposition poses a daunting challenge to the construction of systematic methods to solve nonlinear problems. Indeed, historically, nonlinear problems were treated by ad hoc methods, often reinvented on a discipline by discipline basis.

Over the past several decades, researchers in “nonlinear science” have recognized and exploited the existence of certain “paradigms” of nonlinearity, which transcend discipline-specific applications and permit the transfer of insights gained in one field to many others. By “paradigm,” one means a central concept and an associated set of mathematical, computational, and experimental methods. In the tutorial “Nonlinear Science 101,” I provide detailed insight into three fundamental nonlinear paradigms — “deterministic chaos,” “solitons and coherent structures,” and “pattern formation and competition” — and will show how these paradigms apply in many different fields. I mention briefly two additional paradigms — “adaptation, evolution, and learning” and “networks” — that have become very important in recent years and will argue that they form a bridge between “nonlinear science” and “complex systems;” and that nonlinearity is an essential characteristic of complex systems.

The materials below provide considerably more detail than I was able to offer in my tutorial lecture that will enable them to explore a very wide range of inherent nonlinear systems.

Best,

David

Dr. David Campbell
University Provost, Boston University

Lecture

Slides of Nonlinear Science 101 Tutorial (PPT 7.8MB)

Copies of the actual powerpoint slides used in the tutorial lecture.

Background Material

1988 SFI Lectures on Nonlinear Science (PDF 24MB)

These lectures, entitled “Introduction to Nonlinear Dynamics,” were first offered at the Santa Fe Institute Summer School in 1988 and were published as pp 3–106 in Lectures in the Sciences of Complexity, SFI Studies in the Sciences of Complexity (Addison-Wesley Publishing Company, 1989).

Chaos-CHTO Delat (PDF 1MB)

This article is based on a plenary summary talk given at the International Conference on Nonlinear Science held in Monterey California in 1987. It provide both a summary of the presentations at the meeting and a number of predictions—a few of which were even proven correct!—of developments in chaos in the ensuring years. It appeared as pp. 541–562 in Nuclear Physics B (Proc. Suppl.) 2, (1987).

Nonlinear Science from Paradigms to Practicalities (PDF 3MB)

This article was one of the first attempting to create a coherent “field” of nonlinear science by identifying and describing three fundamental paradigms, providing examples of how they emerged in nonlinear systems in many different traditional disciplines, and illustrating some practical consequences. It appeared first as pp. 218–262 in a special issue of Los Alamos Science dedicated to the memory of Stan Ulam and was reprinted in From Cardinals to Chaos: Reflections on the Life and Legacy of Stanislaw Ulam (Cambridge University Press, 1989).

Localizing Energy Through Nonlinearity and Discreteness

This article discusses the important recent developments in the theory of “intrinsically localized modes,” nonlinear excitations that have been observed spatially extended, discrete nonlinear systems ranging from solid state materials through Josephson junctions, photonic crystals, and micromechanical oscillator to (possibly) biomacromolecules. It appeared as pp. 43–49 in Physics Today, January 2004.