Christophe LETELLIER
17/04/2009

Dequan Li

There are various types of chaotic dynamics.
In three dimensions they have been distinguished
by their global topologies, that is, the structure
of the phase space that contains their chaotic attractors.
Among all known chaotic attractors produced
by *autonomous systems*, there are very few
toroidal 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 toroidal 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 phase space R^{3} (*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 fixed points, one located on the symmetry axis at the origin (0,0,0), and two symmetry-related fixed points. If we define and by

the symmetric fixed points are .

**Fig. 1 : A non trivial quasi-periodic regime**

[1] **Bo Deng**, Constructing homoclinic orbits and chaotic attractors, *International Journal of Bifurcation & Chaos*, **4**, 823-841, 1994.

[2] **Dequan Li**, A three-scroll chaotic attractor, *Physics Letters A*, **372**, 387-393, 2007.