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. We also describe the main geometric properties of the discrete Lorenz-like attractors including their homoclinic structures.