Non-trivial Local Attractors of a Three-dimensional Dynamical System

Based on both qualitative method and numerical tests for a series of particular cases in the parameter region, a=1, 0<b <1, it is shown that the three-dimensional system (2) may have a series of interesting phenomena on the non-trivial local attractors, such as the faint attractor (this term is suggested by the author), the local attractor with complex structure, twin spatial limit closed orbits, the bifurcation of rotation numbers, and the spatial limit cycle, etc… The system (2) is a very rich source in the study of dynamical system theory.