Chapter 2
Background

What Is Nonlinear Science?

Basic Concepts and Definitions

At the end of the last century, scientists discovered that the conventional understanding of classical mechanics did not capture the surprisingly irregular motions of systems of only a few interacting particles or the surprisingly organized collective behavior of systems of many interacting particles. The advent of new mathematical and computational techniques during the last quarter of this century has rendered these phenomena more amenable to analysis and has thus engendered renewed interest in this area. The ideas and techniques of nonlinear science have been developed to describe, categorize, and understand these surprising behaviors.

In the past two decades, the new science known popularly as chaos has given us deep insights into previously intractable, inherently nonlinear, phenomena. Chaos has caused a fundamental reassessment of the way in which we view aspects of the physical world. For instance, certain seemingly simple natural nonlinear processes, for which the laws of motion are known and completely deterministic, can exhibit enormously complex behavior, often appearing as if they were evolving under random forces rather than deterministic laws. One consequence is the remarkable result that these processes, although completely deterministic, are essentially unpredictable for long times.

Practitioners of nonlinear science, as chaos has become known among experts, also recognize that nonlinear phenomena can exhibit equally surprising orderliness. For example, certain a priori very complex nonlinear systems, involving many interacting components, can exhibit great regularity in their motion, and coherent structures, such as the Red Spot of Jupiter, can emerge from a highly disordered background.

Paradigms

Researchers in this new nonlinear science have learned to accept the seemingly contradictory manifestations of chaos and order as two fundamental features of inherently nonlinear phenomena. Indeed, deterministic chaos and coherent structures are often referred to as two paradigms of nonlinear science, in the sense that they represent archetypal aspects of nonlinear phenomena, independent of the conventional discipline in which they are observed. Two other paradigms that have emerged from recent studies of nonlinear phenomena can be termed (1) pattern formation, competition, and selection and (2) adaptation, evolution, and learning. It is perhaps most convincing to clarify the impact of these paradigms by presenting examples of their interdisciplinary relevance.

  • Deterministic chaos can be observed in electrical activity from biological systems, in the transition of a fluid to turbulent motion, and in the motion of the moons of the giant planets.
  • Coherent structures arise in the turbulent atmosphere of Jupiter, in giant Earth ocean waves (e.g., tsunamis), in the spatial spread of certain epidemics, and, on a microscopic scale, in the behavior of certain unusual solid-state materials.
  • Pattern formation, competition, and selection occur in very similar ways in such seemingly disparate phenomena as instabilities in secondary oil recovery techniques, laser-plasma interactions designed to control fusion energy, and biological morphogenesis.
  • Recent attempts to isolate the conceptual, as opposed to the biological, essence of life have identified and clarified the paradigm of adaptation, evolution, and learning and have led to extensive studies of mathematical


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Chapter 2 Background What Is Nonlinear Science? Basic Concepts and Definitions At the end of the last century, scientists discovered that the conventional understanding of classical mechanics did not capture the surprisingly irregular motions of systems of only a few interacting particles or the surprisingly organized collective behavior of systems of many interacting particles. The advent of new mathematical and computational techniques during the last quarter of this century has rendered these phenomena more amenable to analysis and has thus engendered renewed interest in this area. The ideas and techniques of nonlinear science have been developed to describe, categorize, and understand these surprising behaviors. In the past two decades, the new science known popularly as chaos has given us deep insights into previously intractable, inherently nonlinear, phenomena. Chaos has caused a fundamental reassessment of the way in which we view aspects of the physical world. For instance, certain seemingly simple natural nonlinear processes, for which the laws of motion are known and completely deterministic, can exhibit enormously complex behavior, often appearing as if they were evolving under random forces rather than deterministic laws. One consequence is the remarkable result that these processes, although completely deterministic, are essentially unpredictable for long times. Practitioners of nonlinear science, as chaos has become known among experts, also recognize that nonlinear phenomena can exhibit equally surprising orderliness. For example, certain a priori very complex nonlinear systems, involving many interacting components, can exhibit great regularity in their motion, and coherent structures, such as the Red Spot of Jupiter, can emerge from a highly disordered background. Paradigms Researchers in this new nonlinear science have learned to accept the seemingly contradictory manifestations of chaos and order as two fundamental features of inherently nonlinear phenomena. Indeed, deterministic chaos and coherent structures are often referred to as two paradigms of nonlinear science, in the sense that they represent archetypal aspects of nonlinear phenomena, independent of the conventional discipline in which they are observed. Two other paradigms that have emerged from recent studies of nonlinear phenomena can be termed (1) pattern formation, competition, and selection and (2) adaptation, evolution, and learning. It is perhaps most convincing to clarify the impact of these paradigms by presenting examples of their interdisciplinary relevance. Deterministic chaos can be observed in electrical activity from biological systems, in the transition of a fluid to turbulent motion, and in the motion of the moons of the giant planets. Coherent structures arise in the turbulent atmosphere of Jupiter, in giant Earth ocean waves (e.g., tsunamis), in the spatial spread of certain epidemics, and, on a microscopic scale, in the behavior of certain unusual solid-state materials. Pattern formation, competition, and selection occur in very similar ways in such seemingly disparate phenomena as instabilities in secondary oil recovery techniques, laser-plasma interactions designed to control fusion energy, and biological morphogenesis. Recent attempts to isolate the conceptual, as opposed to the biological, essence of life have identified and clarified the paradigm of adaptation, evolution, and learning and have led to extensive studies of mathematical

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models of neural networks and to creation of the field of artificial life (mathematical models that simulate the dynamics of living systems). Interdisciplinary Nature of Nonlinear Science Nonlinear science is inherently interdisciplinary, impacting traditional sciences, mathematics, and engineering, as well as the social sciences, notably economics and demographics. Any attempt to circumscribe artificially the scope of nonlinear science inevitably limits the insights it can provide. The successful pursuit of nonlinear science requires the blending of four distinct methodological approaches: Modeling, which seeks to improve the analytical foundation of the problem under consideration—improved models are particularly necessary for the emerging areas of nonlinear science that enable some of the new technologies discussed in this report; Experimental mathematics, which involves the use of cleverly conceived computer-based numerical simulations to give qualitative insights into problems that are analytically intractable; Novel analytical mathematical methods to treat functional recursion relations, to solve nonlinear partial differential equations, or to describe complex structures arising in chaotic systems; and Experimental observations of similar nonlinear phenomena in natural and man-made systems arising in a variety of conventional disciplines. An Example: Period Doubling One specific example that illustrates the interdisciplinary applicability of the paradigms of nonlinear science as well as its multicomponent methodology is the discovery, made in the late 1970s by Feigenbaum,1 that the particular type of transition to chaos—a sequence of period doublings—observed in a very simple mathematical equation was universal in the sense that it was independent of the specific equation and should in fact arise in a wide class of physical, chemical, and biological systems. Feigenbaum's studies were initially computational, but he soon developed an analytic scaling theory. Nonetheless, skeptics considered his results to be a mere theoretical curiosity of no clear experimental significance until Libchaber2 and others observed exactly the same period doubling dynamics in laboratory experiments on fluids and electric circuits. The ensuing efforts to prove various aspects of the theory rigorously stimulated large segments of the pure mathematics community. Conversely, in other cases involving new nonlinear phenomena, laboratory observations have stimulated and guided the development of theory and mathematical modeling. This close interaction among experimenters, theorists, and pure mathematicians—rare and refreshing in the recent age of increasingly specialized science—has also proved to be extremely powerful in aiding the rapid advance of nonlinear science. The Challenges of Pursuing Nonlinear Science Researchers in nonlinear science typically face particular difficulties with regard to obtaining support. Funding agencies are organized to evaluate proposals in the traditional scientific categories. Interdisciplinary funding of the sort likely to have an impact in nonlinear science is very difficult to obtain, especially with the post-Cold War cutbacks in scientific funding. Further, as a relatively new field, nonlinear science seems to pale in direct comparison to more mature fields (e.g., solid-state physics, space physics, or nuclear physics) 1   Mitchell J. Feigenbaum, ''Universal Behavior in Nonlinear Systems," in Los Alamos Science, Summer 1980, pp. 4-27, reprinted in Physics D, Vol. 7, 1983, pp. 16-39. 2   Albert Libchaber and J. Maurer, "Effect of the Prandtl Number on the Onset of Turbulence in Liquid He4," J. Phys. Lett. (France), Vol. 41, No. 21, 1 Nov. 1980, pp. 515-518.

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in areas such as number of publications or (claimed) proximity to applications. It is important to remember that such comparisons are appropriate only at similar stages of development. Finally, even well-established researchers often face considerable intellectual opposition in venturing into new interdisciplinary areas. In an application discussed below (solitons in telecommunications), a leading researcher in the field had considerable difficulty in convincing engineers that solitons were not just some mathematical construct. Thus, the interdisciplinary nature of nonlinear science requires special administrative and financial mechanisms to ensure that advances in basic research can be transferred to applied areas. Managers responsible for guiding research and development must pay close attention to improving mechanisms for facilitating this intellectual technology transfer from basic research in nonlinear science to the programmatic areas of their organizations.