Polynomial contractions and Lyapunov regularity
The article deals with a nonautonomous linear dynamics with discrete time for systems defined by the product of a sequence of matrices \((A_n)_{n \in {\mathbb N}}\) (\(x_{n+1} = A_nx_n\), \(n = 0,1,\dots\)) and satisfying the estimate \[ \|A_{m-1} \cdots A_n\| \leq D(m / n)^\lambda n^a, \quad m \geq n \] for some \(\lambda < 0\), \(D \geq 0\), \(a \geq 0\). If \(a = 0\) then such a system admit a \textit{uniform polynomial contraction}; if \(a > 0\) then such a system admits a \textit{nonuniform polynomial contraction}. In the article, it is presented necessary and sufficient conditions under that the system \((A_n)_{n \in {\mathbb N}}\) admits nonuniform polynomial contraction in terms of ``polynomial'' Lyapunov exponents, sharp lower and upper bounds for constants \(\lambda\) and \(a\) in terms of matrices \(A_n\), and some results about the stability of systems with nonuniform polynomial behavior under small linear and nonlinear perturbations. Unfortunately, in the article there is no examples of systems admitting uniform and nonuniform polynomial contraction.
scientific article; zbMATH DE number 5718402
scientific article