Nonautonomous equations and almost reducibility sets

Abstract

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.