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 x˙ = A(t)x, t ≥ t0, 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 y˙ = A(t)y + f(t, y), y ∈ R2, t ≥ t0, has a solution y(t) with a negative
Lyapunov exponent.