$J^{+}$-like Invariants under Bifurcations

We explore how the invariants $J^{+}$, $J^{-}$, ${\mathcal{J}}_1$ and ${\mathcal{J}}_2$ of immersions – generic (at most double points and only transverse intersections) planar smooth closed curves with non-vanishing derivative – change under $k$-bifurcations ($k\ge 2\in \mathbb{N}$), which are constructed by running $k$ times through an immersion and then perturbing it to be generic.