2007 A 3D symmetrical toroidal chaos

There are various types of chaotic dynamics. In three dimensions they have been distinguished by their global topologies, that is, the structure of the state space that contains their chaotic attractors. Among all known chaotic attractors produced by autonomous systems, there are very few toroïdal chaotic attractors [1], but none exhibits a symmetry. A set of ordinary differential equations proposed by Li produces a new chaotic attractor with a rotation symmetry and a nontrivial toroïdal structure.

 The system

The set of three ordinary differential equations recently proposed by Dequan Li [2] is :

This system of equations is invariant under the group of two-fold rotations about the symmetry axis in the state space \(\mathbb{R}^3 ({x,y,z})\) : \({\cal R}_z (pi) : ({x,y,z}) \mapsto ({-x,-y,+z})\). It was modeled after the Lorenz system, but contains two additional symmetry-preserving terms : \(dxz\) in the first equation and \(-ex^2\) in the third equation.

This system has three singular points, one located on the symmetry axis at the origin (0,0,0), and two symmetry-related singular points. If we define \(x_f\) and \(z_f\) by

the symmetry-related singular points are