Nonautonomous equations with arbitrary growth rates: A Perron-type theorem

For a linear equation x′=A(t)xx′=A(t)x, we show that the asymptotic behavior of its solutions is reproduced by the solutions of the nonlinear equation x′=A(t)x+f(t,x)x′=A(t)x+f(t,x) for any sufficiently small perturbation ff. More precisely, we show that if the Lyapunov exponents of the linear equation are limits, even for general exponential rates ecρ(t)ecρ(t) for an arbitrary function ρρ, then the same happens with the Lyapunov exponents of the solutions of the nonlinear equations, without introducing new values. Our approach is based on Lyapunov’s theory of regularity.