Tubular neighborhoods and continuation of Morse decompositions

We prove a continuation result for Morse decompositions under tubular singular semiflow perturbations, which generalizes a corresponding result from Carbinatto and Rybakowski [Morse decompositions in the absence of uniqueness, II. Topol. Methods Nonlinear Anal.22 (2003), 15–51] and is applicable to cases in which the phase space of the perturbed semiflow is not necessarily homeomorphic to a product of metric spaces having as a factor the phase space of the limiting semiflow. We apply this result to singularly perturbed second-order differential equations on differential manifolds