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Section
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Mathematics
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Title
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An example of a linear discrete system with unstable Lyapunov exponents
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Author(-s)
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Banshchikova I.N.ab
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Affiliations
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Izhevsk State Agricultural Academya,
Udmurt State Universityb
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Abstract
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We consider a discrete time-varying linear system
$$x(m+1)=A(m)x(m),\quad m\in\mathbb Z,\quad x\in\mathbb R^n,\qquad\qquad (1)$$
where $A(\cdot)$ is completely bounded on $\mathbb N$, i.e., $\sup_{m\in\mathbb N}\bigl(\|A(m)\|+\|A^{-1}(m)\|\bigr)<\infty$. Let $\lambda_1(A)\leqslant\ldots\leqslant\lambda_n(A)$ be the Lyapunov spectrum of the system (1). It is called stable if for any $\varepsilon>0$ there exists a $\delta>0$ such that for every completely bounded $n\times n$-matrix $R(\cdot)$, $\sup_{m\in\mathbb N}\|R(m)-E\|<\delta$, the inequality $$\max_{j=1,\ldots,n}|\lambda_j(A)-\lambda_j(AR)|<\varepsilon $$ holds. We construct an example of the system (1) with unstable Lyapunov spectrum.
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Keywords
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discrete time-varying linear system, Lyapunov exponents, perturbations of coefficients
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UDC
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517.929.2
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MSC
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39A06, 39A30
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DOI
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10.20537/vm160203
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Received
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1 May 2016
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Language
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Russian
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Citation
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Banshchikova I.N. An example of a linear discrete system with unstable Lyapunov exponents, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2016, vol. 26, issue 2, pp. 169-176.
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References
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