Functional Invariants in Semilocal Bifurcations

In [7] an open set of structurally unstable families of vector fields on a sphere was constructed. More precisely, a vector field with a degeneracy of codimension three was discovered whose bifurcation in a generic three-parameter family has a numeric invariant. This vector field has a polycycle and two saddles, one inside and one outside this polycycle; one separatrix of the outside saddle winds towards the polycycle and one separatrix of the inside saddle winds from it. Families with functional invariants were constructed also. In [2] a hyperbolic polycycle with five vertices and no saddles outside it was constructed whose bifurcations in its arbitrary narrow neighborhood (semilocal bifurcations in other words) have a numeric invariant and thus are structurally unstable. This paper deals with semilocal bifurcations. A hyperbolic polycycle with nine edges is constructed whose semilocal bifurcation in an open set of nine-parameter families has a functional invariant.