Abstract
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.