A. N. Sakharov

Nizhny Novgorod State Agrarian and Technological University named after L.Ya. Florentyev (Nizhny Novgorod, Russian Federation)

Abstract. In this paper, we construct a class of special flows on a multidimensional torus and a topological invariant of such flows, i.e. a rotation set. Such flows arise while reducing linear systems of differential equations with quasiperiodic coefficients to a triangular form. In the process of such a reduction, we obtain a system of nonlinear differential equations on a multidimensional torus, which generates a projective flow induced by the original linear system. In this paper, we use known results from the theory of matrix groups and Lie algebras and construct an algorithm for SO(n)-extension of a quasiperiodic linear system. The resulting system of equations admits a reduction in order, which allows us to write the right-hand sides as trigonometric polynomials in Euler angles on a sphere. The case n = 3 is considered separately. The equations defining the projective flow are written explicitly. The projective flow is defined on a torus of dimension m + 2, where m is the dimension of the original torus. The structure of this flow is determined by topological invariants of the flow. For example, a non-singular flow on a two-dimensional torus has a topological invariant – the rotation number (A. Poincare). Using M. Herman’s method, it is possible to prove the existence and uniqueness of the rotation vector (\rho 1, \rho 2) for the projective flow on \BbbT m+2. Using S. Schwartzman’s theory defining the rotation set for flows on compact metric spaces, it is shown that the component \rho 2 = 0. Here, the fact is used that the dimension of the maximal toric subalgebra of the algebra so(3) is equal to one.

Key Words: linear extensions, group extension, projective extension, toric subalgebra, rotation vector, asymptotic cycles

For citation: A. N. Sakharov. Rotation sets of SO(3)-extensions of quasiperiodic flows. Zhurnal Srednevolzhskogo matematicheskogo obshchestva. 27:2(2025), 171–184. DOI: https://doi.org/10.15507/2079-6900.27.202502.171-184

Information about the author:

Alexander N. Sakharov, Ph.D. (Phys. and Math.), Associate Professor of the Department of Applied Mechanics, Physics and Higher Mathematics, Nizhny Novgorod State Agrarian and Technological University named after L.Ya. Florentyev (10, Sibirtseva Str., Nizhny Novgorod, 603146, Russia), ORCID: https://orcid.org/0000-0002-4520-8062, ansakharov2008@yandex.ru