This fall Yulij Ilyashenko gave a talk about new effects in the theory of bifurcations of planar vector fields. Recall that a conjecture due to V. Arnold states that a generic finite-parametric family of vector fields on the two-sphere is structurally stable. Recently Yulij Ilyashenko, Ilya Schurov and I disproved this conjecture [arXiv:1506.06797]. It turns out that 3-parametric families of vector fields may have numerical invariants of moderate topological classification, and 6-parametric families may have functional invariants.

I will discuss the main construction, then discuss a work in progress of Nataliya Goncharuk and myself motivated by John Hubbard's talk about enriched dynamics. It seems that slightly changing the construction, for any $n$ we can obtain a 3-parametric family with $n$-dimensional numerical invariant. It is still unclear if this modification leads to functional invariants in 4-parametric families.