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NAKFI 2008 - Complex Systems

Non-Linear Science
David K. Campbell

“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,” Campbell will 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 will mention briefly two additional paradigms—“adaptation, evolution, and learning” and “networks”—that have become very important in recent years and will  argue that 1) they form a bridge between “nonlinear science” and “complex systems;” and 2) that nonlinearity is an essential characteristic of complex systems.