On discrete Lorenz attractors of various types

Abstract

The purpose of this work is to develop the theory of discrete attractors of Lorenz type in the case of three-dimensional maps. In this case, special attention will be paid to standard discrete Lorenz attractors, as well as discrete Lorenz attractors with axial symmetry (i.e. with symmetry x → -x, y → -y, z → -z characteristic of flows with the Lorenz attractors). The main results of the work are related to the construction of elements of classification of such attractors. For various types of discrete Lorenz attractors, we will describe their basic geometric and dynamical properties, and also present the main phenomenological bifurcation scenarios in which they arise. In the work we also consider specific examples of discrete Lorenz attractors of various types in three-dimensional quadratic maps such as three-dimensional Henon maps and quadratic maps with axial symmetry and constant Jacobian. For the latter, their normal forms will be constructed — universal maps, to which any map from a given class can be reduced by means of linear coordinate transformations.

Keywords

• lorenz attractor
• bifurcation
• three-dimensional henon map
• global symmetry
• bifurcation scenario