Leonid Shilnikov

We study questions of chaotic dynamics of three-dimensional smooth maps (diffeomorphisms). We show that there exist two main scenarios of chaos developing from a stable fixed point to strange attractors of various types: a spiral attractor, a Lorenz-like strange attractor or a «figure-8» attractor. We give a qualitative description of these attractors and define certain condition when these attractors can be «genuine» ones (pseudohyperbolic strange attractors). We include also the corresponding results of numerical analysis of attractors in three-dimensional Hénon maps.

Let a $C^r$-smooth $r \geqslant 5$ two-dimensional diffeomorphism $f$ have a non-transversal heteroclinic cycle containing several saddle periodic and heteroclinic orbits and, besides, some of the heteroclinic orbits are non-transversal, i.e. at the points of these orbits the invariant manifolds of the corresponding saddles intersect non-transversally. Suppose that a cycle contains at least two saddle periodic orbits such that the saddle value (the absolute value of product of multipliers) of one orbit is less than 1 and it is greater than 1 for the other orbit. We prove that in any neighbourhood (in $C^r$-topology) of $f$ in the space of $C^r$-diffeomorphisms, there are open regions (so-called Newhouse regions with heteroclinic tangencies) where diffeomorphisms with infinitely many stable and unstable invariant circles are dense. For three-dimensional flows, this result implies the existence of Newhouse regions where flows having infinitely many stable and unstable invariant two-dimensional tori are dense.