Research Briefings 1987 (1988) / Chapter Skim
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Report of the Research Briefing Panel on Order, Chaos, and Patterns: Aspects of Nonlinearity
Report of the Research Briefing Panel on Order, Chaos, and Patterns: Aspects of Nonlinearity
Pages 39-54
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From page 39... ...
Report of the Research Bnefing Panel on Order, Chaos, and Patterns: Aspects of Nonlinearity
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Newell, Professor of Mathematics University of Arizona, Tucson Steven Orszag, Professor of Applied and Computational Mathematics, Princeton University H Eugene Stanley, Professor of Physics, Boston University lames Yorke, Acting Director, Institute for Physical Science and Technology, University of Maryland, College Park Staff Donalcl C
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From page 41... ...
Many other nonlinear phenomena exhibit sharp and unstable boundaries, erratic or chaotic motion, and dramatic responses to very small influences. Such properties typically defy full analytical treatment and make even quantitative numerical description a daunting task.
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From page 42... ...
Rather it relies on the identification and exploitation of paradigms, namely, unifying concepts anct associated methodologies that are broadly applicable in many different fielcls. This report focuses on three of the central paradigms of nonlinearity: coherent structures, chaos, and complex configurations and pattern selection.
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From page 43... ...
A wide variety of soliton equations has been discovered, allowing a broad range of applications to natural phenomena. In fiber optics, Tosephson transmission lines, conducting polymers and other chainlike solids, and plasma ''cavitons," the prevailing mathematical moclels are slight modifications of soliton equations.
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From page 44... ...
Fluid vortices a classic example of which is provided by the Red Spot of Jupiter (Figure 1) cars be used to illustrate the essential role of general coherent structures in nonlinear systems.
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it has recently been shown that nonlinear resonance of this type must occur in three-phase flow in oil reservoirs, and a systematic program is well under way to identify and classify all possible types of nonlinear wave interaction and to assess their importance for oil recovery methods. Given the ubiquity and importance of coherent structures in nonlinear phenomena, it is gratifying that recent years have witnessed remarkable progress in studying them and that there is great promise for still deeper insights.
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From page 46... ...
On the other hand economists, motivated by the ideas of chaotic dynamics, have developed new and powerful statistical tests for analyzing time series, which may be useful in other areas of nonlinear science. As this brief listing suggests, deterministic chaos is essential to the understanding of 46 many reai-worId nonlinear phenomena.
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In this regard, an important and generally open problem is to establish the relation, if any, between coherent structures observed in a given motion and the recluced modes that in principle characterize the motion. In certain specific problems, notably perturbed soliton equations and models for chemical-diffusion fronts, progress has been made, but much further research is required.
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From page 48... ...
COMPLEX CONFIGURATIONS AND PATTERN SELECTION When an extended nonlinear system is driven far from equilibrium, the many localized coherent structures that typically appear in it can organize into an enormous range of spatial patterns, regular or random. This process is familiar in turbulent fluid flows (note the complex pattern surrouncling the Red Spot in Figure I)
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From page 49... ...
Another important instance of interracial instability, with potential technological implications for metallurgical processes and crystal growth problems, occurs in phase transitions in supersaturated or metastable media. Here nonuniform growth of the stable phase produces fingers, known as clendrites, which compete, grow irregu Figure 3 Results of a numerical simulation of vortex sheet model for the shear layer that forms in an aircraft wake.
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To estimate the fractal dimension, imagine covering the image of the viscous fingering with square cells of side ~ and calculating, for a given I, the number of cells required to cover the object entirely. As the length of the side Z goes to zero, the number of cells required grows as Ilk, where ~ is the fractal dimension.
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From page 51... ...
Finally, a fascinating class of discrete nonlinear dynamic systems, known as cellular automata, exhibit remarkable pattern formation properties and are currently being subjected to rigorous mathematical scrutiny. At a more speculative level, these highly discrete systems have suggested novel computational algorithms often called lattice-gas models for solving certain continuum nonlinear partial differential equations.
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From page 52... ...
scientific community has not been comparable to that seen elsewhere, especially in the Soviet Union and France. In both those countries longstanding traditions in mathematical physics and applied mathematics have helped to stimulate interest in nonlinear phenomena, and the high level of importance that many leading scientists attach to this enterprise is readily noticed in their public comments and in their contributions to the field, particularly in the analytic and experimental areas.
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From page 53... ...
Coursework in differential equations should inclucle more modern dynamic-systems ideas; calculus should more regularly be followed by deeper courses in analysis; mechanics courses should stress the limitations of perturbation theory and the omnipresence of nonintegrability. A course in numerical methods that leads to intuitive algorithm development based on deep understanding could prepare a researcher to perform meaningful experimental mathematics.
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From page 54... ...
In particular, they can marshal the resources necessary to respond to the challenging research opportunities. in any effort to guide this research, however, it is imperative that nonlinear science be recognized for what it 54 is: an inherently interctisciplinary effort not suited to confinement within any single conventional ctiscipline or department.
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