Multidimensional analog of the two-dimensional Perron effect of sign change of characteristic exponents for infinitely differentiable differential systems (Q1939468)

\textit{O. Perron} noted [Math. Z. 32, 703--728 (1930; JFM 56.1040.01)] that there exist two-dimensional linear systems \[ \dot{x}=A(t)x \] for which all the characteristic exponents are negative, while there are nonlinear perturbations \[ \dot{x}=A(t)x+f(t,x) \] of them with nonlinearities \(f\) of second order of smallness such that almost all solutions of the perturbed system have positive characteristic exponents. The authors prove a multidimensional generalization of this result. Let \(\lambda_1\leq\dots\leq\lambda_n<0\), \([b_i,d_i)\subset [\lambda_i,\infty)\), and \(m>1\) be fixed. There exists an \(n\)-dimensional linear system with characteristic exponents \(\lambda_1\leq\dots\leq\lambda_n\) and its nonlinear perturbation of class \(C^\infty\) and of smallness order \(m\) having solutions \(x_i(t)\) whose characteristic exponents belong to \([b_i,d_i)\).