Structurally stable linear extensions quasi-periodic flows on the torus

TitleStructurally stable linear extensions quasi-periodic flows on the torus
AuthorsA. N. Sakharov1
1Nizhny Novgorod State Agrarian and Technological University
AnnotationThe paper considers the problem of the geometry of the regions of stability (instability) of linear canonical systems of the 2nd order with quasi-periodic coefficients depending on the parameters. Any such system generates a flow, which is usually called a linear extension of a quasi-periodic flow on a torus. The basis for solving this problem is the transition to a system on an induced projective bundle. If the base torus is ${mathbb T}^2$, then the projective bundle three-dimensional torus ${mathbb T}^2 imes S^1$. In the periodic case, such a transition leads to a system on the torus that does not have singular points and Reeb cells, which makes it possible to use the classical theory Poincaré-Denjoy. When changing the parameters, the stability regions alternate with the instability regions, which correspond to integer rotation number values. The boundaries of these regions are curves in the parameter space, the smoothness of which depends on the smoothness the system under study. In the quasi-periodic case, there is also a characteristic similar to A. Poincaré's rotation number: {em fiber rotation number}. However, it is impossible to obtain a complete analogy with the periodic case here, as evidenced, for example, by the existence of Lyapunov’s wrong canonical systems. It is shown that structurally stable linear extensions correspond to projective flows having two invariant normally hyperbolic torus (stable and unstable). Moreover, in systems depending on the parameter, the intervals of constancy of the rotation number of the layer correspond to structurally stable linear extensions.
Keywordslinear extension, Lyapunov exponents, projective flow, fiber rotation number, normally hyperbolic invariant manifold
CitationSakharov A. N. ''Structurally stable linear extensions quasi-periodic flows on the torus'' [Electronic resource]. Proceedings of the XVI International scientific conference "Differential equations and their applications in mathematical modeling". (Saransk, July 17-20, 2023). Saransk: SVMO Publ, 2023. - pp. 213-219. Available at: https://conf.svmo.ru/files/2023/papers/paper34.pdf. - Date of access: 02.12.2023.