Nonuniform Dichotomies via Lyapunov Functions

For a nonautonomous linear equation x′ =  A(t)x we show how to characterize a nonuniform exponential dichotomy using strict Lyapunov functions. In particular, the stable and unstable subspaces are obtained from invariant families of cones determined by each Lyapunov function. We also obtain converse theorems, constructing explicitly a family of strict Lyapunov functions for each nonuniform exponential dichotomy. We emphasize that nonuniform exponential dichotomies include as a very particular case (uniform) exponential dichotomies.