Realization of the Perron effect whereby the characteristic exponents of solutions of differential systems change their values

We realize the Perron effect of change of values of characteristic exponents: for arbitrary parameters λ 1 1, we prove the existence of a linear differential system = A ( t ) x , x ∈ R 2 , t ≥ t 0 , with bounded infinitely differentiable coefficients and with characteristic exponents λ 1 ( A ) = λ 1 1), satisfying the condition ‖ f ( t, y )‖ ≤ const × ‖ y ‖ m , y ∈ R 2 , t ≥ t 0 , and such that all nontrivial solutions y ( t, c ) of the perturbed system , have Lyapunov exponents λ [ y (·, c )] = β 1 for c 1 = 0 and λ [ y (·, c )] = β 2 for c 1 ≠ 0. In particular, this effect contains the Perron effect whereby the characteristic exponents of an exponentially stable linear differential system change their sign under perturbations of higher-order smallness.