Vestnik Volgogradskogo gosudarstvennogo universiteta. Seriya 1. Mathematica. Physica
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 Vestnik Volgogradskogo gosudarstvennogo universiteta. Seriya 1. Mathematica. Physica, 2017, Issue 1(38), Pages 13–21 (Mi vvgum159)

Mathematics

On the structure of the space of linear sistems of differential equations with periodic coeffiñients

V. Sh. Roitenberg

Yaroslavl State Technical University

Abstract: We examine linear systems of differential equations
$$l: \dot {x}_i = \sum_{j=1}^{n} a_{ij}(t) x_j + b_j(t), i=1,\ldots,n$$
with continuous $\omega$-periodic coefficients. The system $l$ induce the autonomous system $l_p: \dot {x}_i = \sum_{i=1}^{n} a_{ij}(s) x_j + b_j(s), \dot s = 1$ on ${\mathbf{R}}^n \times \mathbf{S}^1$, where $\mathbf{S}^1 = \mathbf{R}/\omega \mathbf{Z}$. The system $l_p$ has the unique extension $\overline{l}_p$ on ${\mathbf{RP}}^n \times \mathbf{S}^1$. By trajectories of system $l$ in ${\mathbf{R}}^n \times \mathbf{S}^1$ (${\mathbf{RP}}^n \times \mathbf{S}^1$) we will mean trajectories of system $l_p$ ($\overline{l}_p$). Let us consider linear systems $l$ as elements of Banach space $L S^n_{\omega}$ of continuous $\omega$-periodic functions $(a_{11}, \ldots, a_{nn}, b_1,\ldots,b_n)\colon \mathbf{R} \to \mathbf{R}^{n^2+n}$ with norm $\|l\|:=\max_{i,j}\max_{t}\max {|a_{ij}(t)|,|b_i(t)|}$. The system $l \in L S^n_{\omega}$ is said to be structurally stable in ${\mathbf{R}}^n \times \mathbf{S}^1$ (in ${\mathbf{RP}}^n \times \mathbf{S}^1$) if $l$ has a neighborhood $V$ in $l \in L S^n_{\omega}$ such that for any system $\widetilde{l} \in V$ we may find a homeomorphism $h \colon {\mathbf{R}}^n \times \mathbf{S}^1 \to {\mathbf{R}}^n \times \mathbf{S}^1$ ( $h \colon {\mathbf{RP}}^n \times \mathbf{S}^1 \to {\mathbf{RP}}^n \times \mathbf{S}^1$, $h (\mathbf{R}^n \times \mathbf{S}^1) = \mathbf{R}^n \times \mathbf{S}^1$) which maps oriented trajectories of system $\widetilde{l}$ onto oriented trajectories of system $l$.
Let $\Sigma_0 L S^n_{\omega}$ be the set of systems $l \in L S^n_{\omega}$ whose multiplicators do not belong to the unit circle.
Theorem 1. The set $\Sigma_0 L S^n_{\omega}$ is open and everywhere dense in $L S^n_{\omega}$ . A system $l \in L S^n_{\omega}$ is structurally stable in $\mathbf{R}^n \times \mathbf{S}^1$ if and only if it belong to the set $\Sigma_0 L S^n_{\omega}$.
Let $\Sigma L S^2_{\omega}$ be the set of systems $l \in L S^2_{\omega}$ whose multiplicators are real, distinct and different from $-1$ and $1$. Let $\Sigma^{+}_s$, $\Sigma^{-}_s$, $\Sigma^{+}_{ns}$, $\Sigma^{-}_{ns}$, $\Sigma^{+}_{nu}$ and $\Sigma^{-}_{nu}$ be subsets of $\Sigma L S^2_{\omega}$ consisting of systems $l$ with multiplicators $\mu_1$, $\mu_2$ for which $\mu_1 < 1 < \mu_2$ ( $\mu_2 < -1 < \mu_1$) if $l \in \Sigma^{+}_s$ ($l \in \Sigma^{-}_s$) , $0 < \mu_1 < \mu_2 < 1$ ( $-1 < \mu_1 < \mu_2 < 0$) if $l \in \Sigma^{+}_{ns}$ ($l \in \Sigma^{-}_{ns}$), $1 < \mu_1 < \mu_2$ ( $\mu_1 < \mu_2 < 1$) if $l \in \Sigma^{+}_{nu}$ ($l \in \Sigma^{-}_{nu}$).
Theorem 2. 1) A system $l \in L S^2_{\omega}$ is structurally stable in $\mathbf{RP}^2 \times \mathbf{S}^1$ if and only if it belong to the set $\Sigma L S^2_{\omega}$. 2) For any system $l \in \Sigma L S^2_{\omega}$ the corresponding system $\overline{l}_p$ in $\mathbf{RP}^2 \times \mathbf{S}^1$ is a Morse–Smale system. 3) The sets $\Sigma^{+}_s$, $\Sigma^{-}_s$, $\Sigma^{+}_{ns}$, $\Sigma^{-}_{ns}$, $\Sigma^{+}_{nu}$ and $\Sigma^{-}_{nu}$ are classes of topological equivalence in $\Sigma L S^2_{\omega}$.
The paper also describes bifurcation manifolds of codimension one in the space $L S^2_{\omega}$.

Keywords: linear periodic systems of differential equations, projective plane, structural stability, bifurcation manifolds, multiplicators.

DOI: https://doi.org/10.15688/jvolsu1.2017.1.2

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Citation: V. Sh. Roitenberg, “On the structure of the space of linear sistems of differential equations with periodic coeffiñients”, Vestnik Volgogradskogo gosudarstvennogo universiteta. Seriya 1. Mathematica. Physica, 2017, no. 1(38), 13–21

Citation in format AMSBIB
\Bibitem{Roi17} \by V.~Sh.~Roitenberg \paper On the structure of the space of linear sistems of differential equations with periodic coeffiñients \jour Vestnik Volgogradskogo gosudarstvennogo universiteta. Seriya 1. Mathematica. Physica \yr 2017 \issue 1(38) \pages 13--21 \mathnet{http://mi.mathnet.ru/vvgum159} \crossref{https://doi.org/10.15688/jvolsu1.2017.1.2} 

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This publication is cited in the following articles:
1. V. Sh. Roitenberg, “O grubosti otnositelno prostranstva lineinykh differentsialnykh uravnenii s periodicheskimi koeffitsientami”, Matematicheskaya fizika i kompyuternoe modelirovanie, 20:5 (2017), 27–31
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