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The geometry of stability regions of linear canonical systems of periodic differential equations

TitleThe geometry of stability regions of linear canonical systems of periodic differential equations
AuthorsN. A. Mamaeva1, A. N. Sakharov1
1Nizhny Novgorod state agricultural academy
AnnotationA.M. Lyapunov and A. Poincare in the century before last built the theory of linear periodic systems of differential equations. The result of their research was the theorem of Floquet-Lyapunov and Poincare-Lyapunov. However, by the middle of the last century, there was a need to find opportunities for using this theory in applications. A number of wonderful works of M.G. Krein, I.M. Gelfand, V.B. Lidsky et al was appeared, in whose task of describing canonical periodic systems was reduced to a description of geometry the behavior of the eigenvalues of the monodromy operator. Since the monodromy operator in this case is a linear operator, his made it possible to use computational methods of linear algebra. For systems that depend on parameters, the behaviory of the multipliers is determined by the geometry of the regions of stability of solutions that appear and disappear when the system parameters change. At the end of the last century, the method of compactification of the phase space of the system was systematically used to describe the geometry of stability regions, which allowed us to use the results here theory of dynamical systems on compact spaces. Using two-dimensional systems as an example, both approaches are described and their partial topological classification is given.
Keywordscanonical linear systems, monodromy operator, multipliers, stability zones, rotation number, Arnold languages, topological equivalence
CitationMamaeva N. A., Sakharov A. N. ''The geometry of stability regions of linear canonical systems of periodic differential equations'' [Electronic resource]. Proceedings of the International Scientific Youth School-Seminar "Mathematical Modeling, Numerical Methods and Software complexes" named after E.V. Voskresensky (Saransk, October 8-11, 2020). Saransk: SVMO Publ, 2020. - pp. 225-236. Available at: http://conf.svmo.ru/files/2020/papers/article11.pdf. - Date of access: 06.03.2021.