Discrete Lorenz attractors of new types in three-dimensional maps with axial symmetry

Abstract

We propose a new class of quadratic three-dimensional maps with the axial symmetry and constant Jacobian that is additional to the well-known class of quadratic three-dimensional Hénon maps. We study dynamical properties and bifurcations in three-parameter families of these axial symmetric maps and show that such maps can have discrete Lorenz attractors, including those of new types in which one-dimensional unstable separatrices of a saddle fixed point are twisted. We also demonstrate typical scenarios that start with a stable fixed point and lead to the appearance of discrete Lorenz attractors of various types. Additionally, we show numerically that such attractors can be robustly chaotic, i.e. they preserve the maximum Lyapunov exponent to be positive for all smooth perturbations.

Keywords

• The Lorenz attractor
• Homoclinic bifurcations
• Three-dimensional map
• Axial symmetry
• Bifurcation scenario
• Twisted attractor