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Who knows, does a theorem of such form exist:

Any one-parameter family of vector fields on the orientable surface may be slightly perturbed such that 1) all fields in the family except finite number are Morse-Smale; 2) at each non-Morse-Smale field a bifurcation from some list occurs (bifurcations are considered modulo topological equivalence).

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  • $\begingroup$ Where does this question arise from? $\endgroup$ – Stefan Kohl Jul 20 '14 at 15:38
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    $\begingroup$ A one-parameter family of vector field is an horizontal vector field on the product $\Sigma\times I$ where $\Sigma$ is the surface $I$ is an interval, and ``horizontal'' means tangent to the slices $\Sigma\times\{t\}$. Now, did you try to apply classical genericity results on this horizontal vector field (and then projects the obtained vector field to an horizontal one)? $\endgroup$ – Benoît Kloeckner Jul 20 '14 at 15:49
  • $\begingroup$ Stefan, I have a family of embedded surfaces of certain type into a manifold with a contact structure and I want to prove that contact folitations on the initial surface and the terminal one are related by some moves. So I need much control over foliation on each surface in the family. $\endgroup$ – Maxim Prasolov Jul 20 '14 at 19:13
  • $\begingroup$ Benoit, I do not know which genericity results are to apply in 3d-case. For example, Morse-Smale vector fields are no longer dense on 3-manifolds. $\endgroup$ – Maxim Prasolov Jul 20 '14 at 19:29
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    $\begingroup$ @MaximPrasolov: your last comments answers mine perfectly. Considering the previous one, it rings a bell; did you try to apply the methods of Emmanuel Giroux? I guess you know about them, but just in case I think that the most relevant paper is "Structures de contact en dimension trois et bifurcations des feuilletages de surfaces." [Contact structures in dimension three and bifurcations of surface foliations] Invent. Math. 141 (2000), no. 3, 615–689. It is in French but there should be some accounts of these works in english, at least I know who to ask. $\endgroup$ – Benoît Kloeckner Jul 21 '14 at 11:48
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The paper you are looking for is Generic one-parameter families of vector fields on two-dimensional manifolds. by J. Sotomayor, Inst. Hautes Études Sci. Publ. Math. No. 43 (1974), 5–46.

You cannot hope for a finite number of bifurcations. Some 1-parameter families of vector fields with infinitely many bifurcations are stable (as 1-parameter families of course). Think of infinitely many longer and longer saddle connections accumulating on a degenerate closed leave. Also you cannot hope to get rid of non-trivial recurrence by perturbation. If you can ensure absence of recurrence then the only bifurcations are what you think: parabolic singularities, weakly degenerate closed leaves and saddle connections.

As Benoît pointed out, the most relevant reference in contact topology is Giroux's paper Structures de contact en dimension trois et bifurcations des feuilletages de surfaces, Invent. Math. 141 (2000), no. 3, 615–689 but also the discretization lemma proved in Sur les transformations de contact au-dessus des surfaces, Essays on geometry and related topics, Vol. 1, 2, 329–350, Monogr. Enseign. Math., 38, Enseignement Math. (2001) which explains how to get rid of all but finitely many bifurcations through a large modification of the family.

If you have trouble reading French, I wrote an introductory discussion of part of this in a chapter of Contact and Symplectic Topology, edited by Bourgeois, Colin and Stipsicz, Bolyai Society Mathematical Studies (2014). An early version can be found on arxiv. You will also find some discussion of generic singularities in the contact context in Section 2 of Thomas Vogel's Uniqueness of the contact structure approximating a foliation.

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