Topological decoupling, linearization and perturbation on inhomogeneous time scales

We derive a linearization theorem in the framework of dynamic equations on time scales. This extends a recent result from [Y. Xia, J. Cao, M. Han, A new analytical method for the linearization of dynamic equation on measure chains, J. Differential Equations 235 (2007) 527–543] in various directions: Firstly, in our setting the linear part need not to be hyperbolic and due to the existence of a center manifold this leads to a generalized global Hartman–Grobman theorem for nonautonomous problems. Secondly, we investigate the behavior of the topological conjugacy under parameter variation. These perturbation results are tailor-made for future applications in analytical discretization theory, i.e., to study the relationship between ODEs and numerical schemes applied to them.