On the Existence of Linear Differential Systems with All Positive Characteristic Exponents of the First Approximation and with Exponentially Decaying Perturbations and Solutions
N. A. Izobov, А. В. Ильин · Differential Equations · 2021
We prove the existence of $$n$$ -dimensional linear differential systems with the first approximation having all positive characteristic exponents, with exponentially decaying perturbations, and with exactly $$n-1 $$ linearly independent solutions with negative Lyapunov exponents. Thus, in the linear case we obtain an anti-Perron version—a version opposite to the well-known Perron effect of changing the values of negative exponents of the linear approximation to positive ones for solutions of a differential system with a perturbation of higher-order smallness in a neighborhood of the origin and admissible growth outside it.