Nonuniform (h,k,μ,ν)-dichotomy with applications to nonautonomous dynamical systems

Abstract

The paper develops a more general notion of dichotomy, referred to as “nonuniform (h,k,μ,ν)-dichotomy”. The new notion unifies most versions of existing dichotomy in the literature, includes them in one comprehensive mathematics, and reveals more dichotomic behaviors of dynamical systems. Then we show that any linear nonautonomous system admits a nonuniform (h,k,μ,ν)-dichotomy if it has an (h,k) Lyapunov exponent with different signs or a Lyapunov function with different growth rates in the stable and unstable subspaces of solutions. This implies that the nonuniform (h,k,μ,ν)-dichotomy arises naturally and exists widely in the linear systems. Using the general notation, new versions of robustness, Hartman–Grobman theorem, and stable invariant manifold theorem for nonautonomous dynamical systems in Banach spaces will be presented. Especially, these new results not only generalize the corresponding ones in the literature, but also characterize the influences of different growth functions in the stable and unstable subspaces and in the uniform and nonuniform parts of a linear system for solving the above problems. This improves the application range of dichotomy in the nonautonomous systems.