The talk is based on joint work with Yu.Ilyashenko.

Consider a smooth $k$-parametric family $v_\epsilon$ of vector fields on the sphere $S^2$. We say that for $\epsilon=0$ a bifurcation occurs if phase portraits of $v_\epsilon, \epsilon\approx 0$, are not topologically conjugate to the phase portrait of $v_0$. The bifurcation theory classifies and studies possible types of such bifurcations. Studying a bifurcation, it is reasonable to restrict ourselves to a subset of $S^2$ "where the bifurcation occurs" (to find a "bifurcation support"). This can drastically reduce the number of possible bifurcations. I am going to give a construction of such subset (a large bifurcation support) as well as the formalization of the phrase "where the bifurcation occurs". Surprizingly, the proof is almost purely topological, with only a few references to the smooth theory. We call the resulting set “a large bifurcation support”, because an earlier construction of a “bifurcation support” due to V.Arnold appeared to yield too small set, as recent studies show.