Complex dynamics of a simple 3d autonomous chaotic system with four-wing

Complex dynamics of a simple 3D autonomous chaotic system with four-wing

The present paper revisits a three dimensional (3D) autonomous chaotic system with four-wing occurring in the known literature Nonlinear Dyn (2010) 60(3): 443--457] with the entitle ``A new type of four-wing chaotic attractors in 3-D quadratic autonomous systems'' and is devoted to discussing its complex dynamical behaviors, mainly for its non-isolated equilibria, Hopf bifurcation, heteroclinic orbit and singularly degenerate heteroclinic cycles, etc. Firstly, the detailed distribution of its equilibrium points is formulated. Secondly, the local behaviors of its equilibria, especially the Hopf bifurcation, are studied. Thirdly, its such singular orbits as the heteroclinic orbits and singularly degenerate heteroclinic cycles are exploited. In particular, numerical simulations demonstrate that this system not only has four heteroclinic orbits to the origin and other four symmetry equilibria, but also two different kinds of infinitely many singularly degenerate heteroclinic cycles with the corresponding two-wing and four-wing chaotic attractors nearby.