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## Archive of Issues

Russia Izhevsk
Year
2016
Volume
26
Issue
2
Pages
169-176
 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. References Demidovich V.B. On a criterion of stability for difference equations, Differ. Uravn., 1969, vol. 5, no. 7, pp. 1247-1255 (in Russian). Gaishun I.V. Sistemy s diskretnym vremenem (Discrete-time systems), Minsk: Institute of Mathematics of the National Academy of Sciences of Belarus, 2001, 400 p. Banshchikova I.N., Popova S.N. On the spectral set of a linear discrete system with stable Lyapunov exponents, Vestn. Udmurt. Univ. Mat. Mekh. Komp'ut. Nauki, 2016, vol. 26, no. 1, pp. 15-26 (in Russian). Czornik A. Perturbation theory for Lyapunov exponents of discrete linear systems, Kraków: AGH University of Science and Technology Press, 2012, 110 p. Izobov N.A. Linear systems of ordinary differential equations, Journal of Soviet Mathematics, 1976, vol. 5, issue 1, pp. 46-96. Izobov N.A. Vvedenie v teoriyu pokazatelei Lyapunova (Introduction to the theory of Lyapunov exponents), Minsk: Belarusian State University, 2006, 319 p. Perron O. Die Ordnungszahlen linearer Differentialgleichungssysteme, Math. Z., 1930, bd. 31, s. 748-766 (in German). Adrianova L.Ya. Vvedenie v teoriyu lineinykh sistem differentsial'nykh uravnenii (Introduction to the theory of linear systems of differential equations), St. Petersburg: Saint Petersburg State University, 1992, 240 p. Horn R.A., Johnson C.R. Matrix analysis, Cambridge: Cambridge University Press, 1986. Translated under the title Matrichnyi analiz, Moscow: Mir, 1989, 655 p. Full text