From the history of the concept of structural stability

R. R. Mukhin

Čebyševskij sbornik, Tome 22 (2021) no. 2, pp. 417-436.

Résumé

Aim. The aim of the work is to study the history of ideas about coarseness (structural stability), which is not only one of the most important concepts of the theory of nonlinear systems, but lies at the heart of our worldview. To nowаdays, structural stability has been considered in historical terms only fragmentarily (mainly in connection with the Andronov school) and has not been the subject of a consistent historical study. Method. The study is based on an analysis of original works, historical and scientific literature with the involvement of the memories of participants in the events described. Results. In Andronov’s school, in the context of applied problems, two-dimensional systems for which structural stability is a typical property have been exhaustively studied. Since the late 1950s there is a shift in research on structural stability in the context of applied problems towards the theory of dynamical systems. M. Peixoto studied structural stability on closed two-dimensional manifolds and proved the density of such systems. S. Smale hypothesized the existence of structurally stable systems in the multidimensional case ($n\geqslant3$) . Such systems exist (Morse-Smale systems), but he himself established their atypicality, they do not constitute a dense set. Multidimensional systems are characterized by complex behavior; an example of such a system (Smale's horseshoe) was built. The study of systems with complex behavior stimulated the development of hyperbolic theory. Discussion. Structural stability was an important factor in the discovery of the complex behavior of dynamical systems already in the three-dimensional case; it continues to play a significant role in the modern theory of dynamical systems. Structural stability is of general scientific importance, played a key role in the construction of catastrophe theory, it went beyond the framework of the theory of dynamical systems and mathematics itself, penetrates into other areas of science, including the humanitarian sphere.

Détail

Keywords: dynamical system, coarseness, structural stability, topological equivalence, typicality, transversality, dense set.

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