On the Stability of Bifurcation Diagrams of Vanishing Flattening Points

On a smooth surface in Euclidean $3$-space, we consider vanishing curves whose projections on a given plane are small circles centered at the origin. The bifurcations diagram of a parameter-dependent surface is the set of parameters and radii of the circles corresponding to curves with degenerate flattening points.
Solving a problem due to Arnold, we find a normal form of the first nontrivial example of a flattening bifurcation diagram, which contains one continuous invariant.