These highly osculating curves were studied, in particular by V.I. Arnol'd. One of the important refferences will be:
Topological invariants of plane curves and caustics. Dean Jacqueline B. Lewis Memorial Lectures presented at Rutgers University, New Brunswick, New Jersey. University Lecture Series, 5. American Mathematical Society, Providence, RI, 1994.
More precisely, what was studdied are the points of the curve, where the level of it tangency with (say) conics is higher than expected. I guess these are exactly the points that (using your terminology) separate elliptic part of the curve form hyperbolic.
The key words for these research are Extatic points (therminology proposed by D. Esenbud). Using google scholar you can find a complete text of Arnol'd, called
Remarks on the extatic points of plane curves V.I. Arnold - The Gelfand Mathematical Seminars, 1993-1995.
These article contains some genearlisations of four vertex theorem. http://en.wikipedia.org/wiki/Four-vertex_theorem