Yulij Ilyashenko

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.

In this paper, we give a full description of all possible singular points that occur in generic 2-parameter families of vector fields on compact 2-manifolds. This is a part of a large project aimed at a complete study of global bifurcations in two-parameter families of vector fields on the two-sphere.

A diffeomorphism is said to have a thick attractor provided that its Milnor attractor has positive but not full Lebesgue measure. We prove that there exists an open set in the space of boundary preserving step skew products with a fiber [0,1], such that any map in this set has a thick attractor.