Begell House Inc.
International Journal of Fluid Mechanics Research
FMR
2152-5102
22
5-6
1995
Introductory Remarks. Expectations that are Coming True
1-6
S. P.
Kurdyumov
G. G.
Malinetskii
The Multifaceted Turbulence
7-74
Yu. S.
Sigov
A. S.
Bakai
The title of our story contains the word "turbulence" (from the Latin turbulentus - stormy, disorderly), which is customarily applied to disorderly chaotic flow of fluids. Each one of us has at some time observed various vortical, stormy flows in nature. In spite of their large variety and multifaceted forms of manifestations, their nature is such that it is possible to combine them and to learn to "recognize" familiar features also in the relationships governing a simple mechanical system such as a rolling in a trough with depressions, and in the specifics of water flowing rapidly in the bed of a river, and in plasma pulsating in the prototype of a nuclear reactor. This paper is concerned with precisely these problems.
Nonstationary Structures, Dynamic Chaos, Cellular Automata
75-133
S. P.
Kurdyumov
G. G.
Malinetskii
A. B.
Potapov
Nonlinearity. New Problems, New Opportunities
134-154
G. G.
Malinetskii
A. B.
Potapov
Work done on mathematical modeling and applied mathematics makes it currently possible to solve a large number of applied problems. However, it is at times useful to look around and ask several general questions. What development in the discipline under study changes the investigators' view of nature and affects their world outlook? Which new ideas have recently appeared? What would be interesting to tell scientists working in related fields, from a bird's eye view of one's own field?
This is particularly important for approaches that arose relatively recently and are currently developing at a high rate. Such a discussion will assist in understanding which hopes that were initially pinned on the new approach have justified themselves, and what can be expected in the future.
Nonlinear science is precisely such an approach. One of its purposes is to find universal relationships governing the behavior of nonlinear systems. These relationships manifest themselves in the general nature of mathematical description of a very large number of objects in physics, biology, technology, chemistry and most likely also in social sciences. Nonlinear science attempts to see a new, higher level of the unity of nature behind the enormous number of equations, models and problems.
Such a program of study is highly attractive. If general relationships indeed exist in the nonlinear world, then they can be found and investigated employing elementary models.
The hopes that were pinned on nonlinear science were discussed almost ten years ago by one of the present authors (G. M.) with his coworkers [1]. Many of the problems listed in that article have found their way from scientific journals into books and are on the way to textbooks. A survey of many relatively recent results, can be found, among others, in [2, 3]. It is hence sensible to single out ever new ideas and discuss the future of nonlinear science.
Nonlinear Science in the Problem of Safety
155-175
N. A.
Mitin
G. G.
Malinetskii
Nonlinear science is the term employed to describe an interdisciplinary approach currently applied to investigation of relations common to a large number of natural and man-made systems.
Nonlinear science is based on a profound analogy that was found to exist between nonlinear mathematical models arising in natural sciences and in many fields of engineering. The existence of common features in the description of various phenomena makes it possible to construct elementary models which clearly and graphically reflect the substance of such processes. These elementary models, known as base models, can be used, just as cubes in children's games, for constructing models of actual systems and processes, by introducing the necessary refinements into the mathematical description.
One of the major places in nonlinear science is occupied by the study of jumps, phase transitions, high-rate processes and qualitative changes in the state of the objects under study. And it is precisely this which is usually the main component in the analysis of catastrophes in natural and man-made environments. We are faced with a paradoxical situations when new, effective, and most probably useful tools are not used in the field for which they have been predominantly designed. In our opinion the main reason for this is that specialists in different fields speak in different languages, which prevents them from understanding one another.
The main purpose of this article is to serve as a kind of dictionary and phrase book that would ensure contact between safety experts and their colleagues in nonlinear science. This dictates the style of the article. In each section we shall formulate an elementary conceptual model of a catastrophe, then its mathematical description and finally, we shall turn to specific studies, where these tools are used for analyzing safety problems.
Problems in a Course of Nonlinear Science
176-218
G. G.
Malinetskii
Mathematical problems, are assigned a place of paramount importance in high school as a tool for the development of thinking. The problems that were carefully selected in the course of the past two millennia were highly instrumental in developing the ability of logical deliberation, intuition and common sense.
The importance of problems is not diminished on the university level. However, their role here is different. A professional differs from an amateur, among others, by the former's understanding of the limits of his expertise. It is imperative that a specialist be able to distinguish between elementary problems, or problems that have already been solved and those in his field that still require solution. The simplest method to become versant in what "is inconvenient not to know" is to suggest a specially selected set of problems. These problems should be surprising and interesting and their solution should require a certain intellectual effort rather than elementary application of familiar theorems and methods, but at the same time be within the ability of the student.
In addition, problems allow avoiding a rather frequent situation which was aphoristically formulated by one of my students: "When you are explaining, everything is clear, when you start asking everything is incomprehensible".