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Mathematical Physics and Computer Simulation, 2017, Volume 20, Issue 5, Pages 27–31
DOI: https://doi.org/10.15688/mpcm.jvolsu.2017.5.3
(Mi vvgum203)
 

Mathematics and mechanics

On the structural stability relative to the space of linear differential equations with periodic coefficients

V. Sh. Roitenberg

Yaroslavl State Technical University
References:
Abstract: Let $\textrm{LE}^n_\omega$ be the Banach space of linear non-homogeneous differential equations of order $n$ with $\omega$-periodic coefficients. We prove the following statements. The equation $l\in \textrm{LE}^n_\omega$ is structurally stable in the phase space $\Phi^2:=\mathbf{R}^n\times\mathbf{R}/\omega \mathbf{Z}(n\geq2)$ if and only if its multiplicators do not belong to the unit circle. The set of all structurally stable equations is everywhere dense in $\textrm{LE}^n_\omega$. The equation $l\in \textrm{LE}^n_\omega$ is structurally stable in the phase space $\bar{\Phi}^2:=\mathbf{RP}^2\times\mathbf{R}/\omega \mathbf{Z}$ if and only if its multiplicators are real, different and distinct from $\pm 1$. We describe also the topological equivalence classis of structurally stable in $\bar{\Phi}^2$ equations.
Keywords: linear differential equations, periodic coefficients, projective plane, structurally stable equations, multiplicators.
Document Type: Article
UDC: 517.925.52 + 517.926
BBC: 22.161.6
Language: Russian
Citation: V. Sh. Roitenberg, “On the structural stability relative to the space of linear differential equations with periodic coefficients”, Mathematical Physics and Computer Simulation, 20:5 (2017), 27–31
Citation in format AMSBIB
\Bibitem{Roi17}
\by V.~Sh.~Roitenberg
\paper On the structural stability relative to the space of linear differential equations with periodic coefficients
\jour Mathematical Physics and Computer Simulation
\yr 2017
\vol 20
\issue 5
\pages 27--31
\mathnet{http://mi.mathnet.ru/vvgum203}
\crossref{https://doi.org/10.15688/mpcm.jvolsu.2017.5.3}
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