Parameter Dependence of Stable Invariant Manifolds for Delay Differential Equations under (μ, ν)-Dichotomies. (27th October 2014) - Pan, Lijun
Abstract : We obtain the existence of stable invariant manifolds for the nonlinear equation x ′ = L ( t ) x t + f ( t, x t, λ ) provided that the linear delay equation x ′ = L ( t ) x t admits a nonuniform ( μ, ν ) -dichotomy and f is a sufficiently small Lipschitz perturbation. We show that the stable invariant manifolds are dependent on parameter λ . Namely, the stable invariant manifolds are Lipschitz in the parameter λ . In addition, we also show that nonuniform ( μ, ν ) -contraction persists under sufficiently small nonlinear perturbations.