Perron Effect of Infinite Change of Values of Characteristic Exponents in Any Neighborhood of the Origin

We develop our earlier generalizations of the Perron effect of change of values of characteristic exponents for arbitrary parameters m > 1 and lambda(1) <= lambda(2) < 0 and an arbitrary bounded countable set beta subset of, [lambda(1),+infinity), beta boolean AND [lambda(2),+infinity) not equal empty set, and show that there exists a two-dimensional differential system of linear approximation with bounded coefficients infinitely differentiable on the positive half-line and with characteristic exponents lambda(1) and lambda(2) and an infinitely differentiable perturbation infinitesimal of order m > 1 in a neighborhood of the origin and possibly growing outside the neighborhood such that the nontrivial solutions of the perturbed system are infinitely extendible and the characteristic exponents of solutions issuing from any neighborhood of the origin form exactly the set beta. In addition, we generalize this infinite version of the Perron effect in a neighborhood of the origin to other points of the plane of initial values of solutions.