A 3-D spherical chaotic attractor

A simple smooth chaotic system, which showed a 3-layer sphere chaotic attractor, is investigated. It is found that this chaotic attractor is a limit cycle instead of chaotic attractor. This situation was caused by the simulation time which is too short to reach its real status. It also shows that it is not reliable to construct chaotic system based only on the Šhilnikov criterion without finding the exact homoclinic orbits. Then a chaotic system with the real sphere shape is proposed. This proposed system is investigated through numerical simulations and analyses including time phase portraits, Lyapunov exponents, bifurcation diagrams and Poincaré section.