Linear Version of the Anti-Perron Effect of Change of Positive Characteristic Exponents to Negative Ones

Abstract

A linear version of the anti-Perron effect of changing all positive Lyapunov characteristic exponents to negative ones is realized. For arbitrary numbers $\lambda _n\geq \ldots \geq \lambda _1>0$ and $\mu _1\leq \ldots \leq \mu _n<0$, the existence of the following $n$-dimensional linear systems is proved: an original system $\dot {x}= A(t)x$, $t\geq t_0$, with characteristic exponents $\lambda _i(A)=\lambda _i$, $i={1,\dots ,n}$, and a perturbed system $\dot {y}= A(t)y+Q(t)y$ with perturbation matrix $Q(t)\to 0$ as $t\to \infty$ and characteristic exponents $\lambda _i(A+Q)=\mu _i$, $i ={1,\dots ,n}$. Moreover, the coefficient matrices of the original and perturbed differential systems are bounded and infinitely differentiable on the half-line $[t_0,+\infty )$.