Conley index and tubular neighborhoods

We establish a new Conley index continuation principle, which generalizes the singular Conley index continuation principle from Carbinatto and Rybakowski (2002) [2] 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 smooth manifolds.