On a new type of bifurcations on manifolds

Palis and Pugh asked if there exists a one-parameter family of smooth vector fields on a compact manifold, having a closed orbit which depends continuously on the parameter but whose period is not bounded above (as a function of the parameter) and which disappears at a finite (positive) distance from the set of singular points of the vector field.
In this paper we answer this question affirmatively. Moreover, we formulate a condition for the existence of the corresponding bifurcation of a smooth vector field without singularities on a closed two-dimensional manifold, and we give concrete examples.
Bibliography: 4 titles.