Nonautonomous equations and almost reducibility sets

For a nonautonomous differential equation, we consider the almost reducibility property that corresponds to the reduction of the original equation to an autonomous equation via a coordinate change preserving the Lyapunov exponents. In particular, we characterize the class of equations to which a given equation is almost reducible. The proof is based on a characterization of the almost reducibility to an autonomous equation with a diagonal coefficient matrix. We also characterize the notion of almost reducibility for an equation $x'=A(t,\theta) x$ depending continuously on a real parameter $\theta$. In particular, we show that the almost reducibility set is always an $F_{\sigma\delta}$-set and for any $F_{\sigma\delta}$-set containing zero we construct a differential equation with that set as its almost reducibility set.