Global and local bifurcations, three-dimensional Henon maps and discrete Lorenz attractors

TL;DR

This paper investigates bifurcations in multidimensional diffeomorphisms leading to discrete Lorenz attractors, providing a comprehensive list of such bifurcations and demonstrating their persistence under time reversal, advancing the understanding of chaos.

Contribution

It offers a complete classification of bifurcations resulting in discrete Lorenz attractors and shows their robustness under time reversal in complex dynamical systems.

Findings

Identified all bifurcations leading to Lorenz attractors in the studied systems.

Proved the existence of Lorenz attractors in time-reversed systems.

Enhanced understanding of chaos and hyperchaos in multidimensional maps.

Abstract

Lorenz attractors play an important role in the modern theory of dynamical systems. The reason is that they are robust, i.e. preserve their chaotic properties under various kinds of perturbations. This means that such attractors can exist in applied models and be observed in experiments. It is known that discrete Lorenz attractors can appear in local and global bifurcations of multidimensional diffeomorphisms. However, to date, only partial cases were investigated. In this paper bifurcations of homoclinic and heteroclinic cycles with quadratic tangencies of invariant manifolds are studied. A full list of such bifurcations, leading to the appearance of discrete Lorenz attractors is provided. In addition, with help of numerical techniques, it was proved that if one reverses time in the diffeomorphisms described above, the resulting systems also have such attractors. This result is an important step in the systematic studies of chaos and hyperchaos.