Poincaré sections for a new three-dimensional toroidal attractor

A new 3D autonomous dynamical system proposed by Dequan Li [Physics Letters A, 372, 387, 2008.] produces a chaotic attractor whose global topological properties are unusual. The attractor has a rotation symmetry and only a single real fixed point for the parameters used in his study. The symmetric, complex pair of fixed points cannot be ignored: they play a major role in organizing the motion on a surface of peculiar toroidal type. We describe this attractor, propose a simple, intuitive model to understand it, show that it is of toroidal type and of genus three, construct a global Poincaré surface of section with two disjoint components and use this section to locate unstable periodic orbits and determine their topological period. We also show that its image attractor is of genus one and supports flow on a simple wrinkled torus. Finally, we use the interplay between the original covering attractor and its image as an aid to understand why the Li attractor is of genus-three type. PACS numbers: 05.45.-a Submitted to: J. Phys. A: Math. Gen. Poincaré sections for a new three-dimensional toroidal attractor 2