On discrete Lorenz-like attractors in three-dimensional maps with axial symmetry.

We describe a class of three-dimensional maps with axial symmetry {x→-x,y→-y,z→z} and the constant Jacobian. We study bifurcations and chaotic dynamics in quadratic maps from this class and show that these maps can possess discrete Lorenz-like attractors of various types. We give a description of bifurcation scenarios leading to such attractors and show examples of their implementation in our maps.

Conjoined Lorenz twins-a new pseudohyperbolic attractor in three-dimensional maps and flows.

We describe new types of Lorenz-like attractors for three-dimensional flows and maps with symmetries. We give an example of a three-dimensional system of differential equations, which is centrally symmetric and mirror symmetric. We show that the system has a Lorenz-like attractor, which contains three saddle equilibrium states and consists of two mirror-symmetric components that are adjacent at the symmetry plane.

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.