Doubling of invariant curves and chaos in three-dimensional diffeomorphisms.

This paper gives a review of doubling bifurcations of closed invariant curves. We also discuss the role of the curve-doubling bifurcations in the formation of chaotic dynamics. In particular, we study scenarios of the emergence of discrete Lorenz and Shilnikov attractors in three-dimensional Hénon maps.

On scenarios of the onset of homoclinic attractors in three-dimensional non-orientable maps.

We study scenarios of the appearance of strange homoclinic attractors (which contain only one fixed point of saddle type) for one-parameter families of three-dimensional non-orientable maps. We describe several types of such scenarios that lead to the appearance of discrete homoclinic attractors including Lorenz-like and figure-8 attractors (which contain a saddle fixed point) as well as two types of attractors of spiral chaos (which contain saddle-focus fixed points with the one-dimensional and two-dimensional unstable manifolds, respectively). We also emphasize peculiarities of the scenarios and compare them with the known scenarios in the orientable case. Examples of the implementation of the non-orientable scenarios are given in the case of three-dimensional non-orientable generalized Hénon maps.