Evolution of the main provisions of the theory of stability

R. R. Mukhin

Čebyševskij sbornik, Tome 23 (2022) no. 4, pp. 327-349.

Résumé

The aim of the work is to study the evolution of the concept of stability, which is a structure-forming concept in all areas of science and technology, and even beyond them. The stages of this long evolution corresponded to the dominant trends in the mathematics of their time. By the end of the XIX century. the complexity of the concept of stability was realized, the question arose of a mathematically rigorous approach to the problem. A general theory of motion stability was built on a solid mathematical foundation. This became a milestone not only in the development of the subject itself, but was one of the foundations for constructing a qualitative theory. Subsequently, the theory of stability was divided into two branches: one - the expansion of the theory in breadth based on old ideas, strengthening the links with applications; the other is stability in the context of the theory of dynamical systems. In the latter case, stable movements are considered in the series of all movements; in the stability-instability dichotomy both poles are equal and meaningful. Instability also turns out to be a complex concept that has a variety of forms. Instability has acquired a constructive meaning; it ensures innovation and development. Typical is the coexistence of stability and instability with a complex topology of such a structure. Diverse types of instability demonstrate the phenomenon of turbulence. The study of this phenomenon at the modern level requires the use of mathematics according to the canons of rigor adopted in mathematics itself. One can raise the question of the limits of applicability of the possibilities of the most qualitative description and the concept of stability. In this regard, there are first results, new ideas are required.

Détail

Keywords: stability, perturbation, stability criterion, dynamical system, instability, turbulence.

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