Realization of the Perron effect whereby the characteristic exponents of solutions of differential systems change their values
S. K. Korovin, N. A. Izobov · Differential Equations · 2010
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 $$ \dot x $$ = 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 $$ \dot y = A(t)y + f(t,y), y \in R^2 $$ , 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.