Ivan I. Ovsyannikov (Institute for Applied Mathematics and Cybernetics, Nizhny Novgorod, Russia): Global bifurcations of 3D diffeomorphisms with non-rough homoclinic and heteroclinic trajectories

Global bifurcations leading to appearance of stable periodic orbits,
invariant tori and strange attractors in multi-dimensional dynamical
systems are studied. These problems attract high interest provided by
recent discovery of strange attractors of a new type: wild hyperbolic
attractors. This kind of strange attractor, unlike well-known Lorenz
attractor and hyperbolic attractor allows homoclinic tangencies inside
it but, of course, as a true strange attractor does not contain stable
periodic orbits. Another important property of such attractors is that
they can be met in applications and models. The examples are: spiral
attractor in the Turaev-Shil'nikov model, attractors obtained as
periodic perturbations of Lorenz-like systems etc. But when observing an
attractor in application, it is hard (or even impossible when using
numerical methods only) to distinguish true strange attractor from a
quasiattractor (which can contain stable orbits).

Some cases related to this problem are considered. Namely, bifurcations
of homoclinic tangencies to saddle and saddle-focus fixed points of a
neutral type are investigated. It is shown that these cases are
principally different. Explanation of the "invisibility effect" of
stable periodic orbits in one-parametric families is provided.

Regarding the heteroclinic bifurcations, a contracting-expanding case of
a contour consisting of two saddle-foci is investigated. It is shown
that there may co-exist an infinite number of wild hyperbolic
Lorenz-like attractors.