Characterization of stable manifolds for nonuniform exponential dichotomies
With the aim to study perturbations of linear differential equation \(v'=A(t)\) (\(A(t)\) is a \(C^1\) function) that admit a nonuniform exponential dichotomy at sufficiently small \(C^1\) perturbation \(f(t,v)\) with bounded Lipschitz derivative the authors establish the following results for the perturbed equation: 1) the existence of \(C^1\) stable manifolds, 2) the exponential decay of the derivative of the flow defined by the autonomous equation \(t'=1, v'=A(t)v+f(t,v)\) along the stable manifold, in addition to the expected exponential decay of the flow along the stable manifold, 3) a characterization of the stable manifold as the set of points whose trajectories have a controlled exponential growth rate, namely those whose exponential growth rate is sufficiently far away from the positive Lyapounov exponents of the exponential dichotomy.
scientific article; zbMATH DE number 5315189