Existence of an Anti-Perron Effect of Change of Positive Exponents of the Linear Approximation System to Negative Ones under Perturbations of a Higher Order of Smallness

Abstract

We prove the existence of a two-dimensional linear system \(\dot {x}=A(t)x \), \(t\geq t_0\), with bounded infinitely differentiable coefficients and all positive characteristic exponents, as well as an infinitely differentiable \(m\)-perturbation \(f(t,y) \) having an order \(m>1 \) of smallness in a neighborhood of the origin \(y=0 \) and an order of growth not exceeding \(m \) outside it, such that the perturbed system \(\dot {y}=A( t)y+\thinspace f(t,y)\), \(y\in \mathbb {R}^2 \), \(t\geq t_0\), has a solution \(y(t) \) with a negative Lyapunov exponent.