Topological robustness of non-saddle sets

We study in this paper preservation of dynamical and shape theoretical properties under continuation for parametrized families of flows. We show that, although attractors continue, the same does not hold for non-saddle sets. However, when they continue, their shape is preserved in quite general settings, which include differentiable families of flows and regular non-saddle sets for general flows, not necessarily differentiable.We also study how the continuation of a non-saddle set influences that of its dual non-saddle set.