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Russia Izhevsk
Section Mathematics
Title An example of a linear discrete system with unstable Lyapunov exponents
Author(-s) Banshchikova I.N.ab
Affiliations Izhevsk State Agricultural Academya, Udmurt State Universityb
Abstract 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.
Keywords discrete time-varying linear system, Lyapunov exponents, perturbations of coefficients
UDC 517.929.2
MSC 39A06, 39A30
DOI 10.20537/vm160203
Received 1 May 2016
Language Russian
Citation 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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