Nonlinear perturbations of nonuniform exponential dichotomy on measure chains
This paper deals with the following nonuniformly hyperbolic system on measure chains \[ x^\Delta=A(t)x, \] and its nonlinear perturbation \[ x^\Delta=A(t)x+f(t,x), \] where \(A\in C_{\text{rd}}(\mathbb{T},\mathcal{B}(\mathbb{X}))\) and \(f:\mathbb{T}\times\mathbb{X}\rightarrow\mathbb{X}\), while \(C_{\text{rd}}(\mathbb{T},\mathbb{X})\) denotes the set of rd-continuous functions \(g:\mathbb{T}\rightarrow\mathbb{X}\). By constructing the topological equivalence between the above systems, a generalized Grobman-Hartman theorem is established. In addition, stable invariant manifolds for sufficiently small nonlinear perturbations of a nonuniform exponential dichotomy are constructed. It is also shown that the stable invariant manifolds are Lipschitz in the initial values if the nonlinear perturbation is a sufficiently small Lipschitz perturbation.
scientific article; zbMATH DE number 5987799
scientific article