Shadowing for Nonautonomous Dynamics

Abstract

Abstract We prove that whenever a sequence of bounded operators ( A m ) m ∈ ℤ {(A_{m})_{m\in\mathbb{Z}}} acting on a Banach space X admits an exponential dichotomy and a sequence of differentiable maps f m : X → X {f_{m}\colon X\to X} , m ∈ ℤ {m\in\mathbb{Z}} , has bounded and Hölder derivatives, the nonautonomous dynamics given by x m + 1 = A m ⁢ x m + f m ⁢ ( x m ) {x_{m+1}=A_{m}x_{m}+f_{m}(x_{m})} , m ∈ ℤ {m\in\mathbb{Z}} , has various shadowing properties. Hence, we extend recent results of Bernardes Jr. et al. in several directions. As a nontrivial application of our results, we give a new proof of the nonautonomous Grobman–Hartman theorem.