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
N. A. Izobov, А. В. Ильин · Differential Equations · 2023
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.