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

A linear version of the anti-Perron effect of changing all positive Lyapunov characteristic exponents to negative ones is realized. For arbitrary numbers and , the existence of the following -dimensional linear systems is proved: an original system , , with characteristic exponents , , and a perturbed system with perturbation matrix as and characteristic exponents , . Moreover, the coefficient matrices of the original and perturbed differential systems are bounded and infinitely differentiable on the half-line .