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This article is cited in 4 scientific papers (total in 4 papers)
MATHEMATICS
An example of a linear discrete system with unstable Lyapunov exponents
I. N. Banshchikova Udmurt State University, ul. Universitetskaya, 1, Izhevsk, 426034, Russia
Abstract:
We consider a discrete time-varying linear system
\begin{equation}
x(m+1)=A(m)x(m),\quad m\in\mathbb Z,\quad x\in\mathbb R^n,
\tag{1}
\end{equation}
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 $\lambda_1(A)\le\ldots\le\lambda_n(A)$ 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.
Received: 01.05.2016
Citation:
I. N. Banshchikova, “An example of a linear discrete system with unstable Lyapunov exponents”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 26:2 (2016), 169–176
Linking options:
https://www.mathnet.ru/eng/vuu528 https://www.mathnet.ru/eng/vuu/v26/i2/p169
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