These highly osculating curves were studied, in particular by V.I. Arnol'd. One of the important references 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 studied are the points of the curve, where the level of its tangency with (say) conics is higher than expected. I guess these are exactly the points that (using your terminology) separate the elliptic part of the curve from the hyperbolic part.
The key words for this research are Extactic points (terminology proposed by D. Eisenbud). 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.
This article contains some generalisations of the four vertex theorem. http://en.wikipedia.org/wiki/Four-vertex_theorem
One more nice reference is a paper of Tabachnikov and Timorin. https://arxiv.org/abs/math/0602317
