Key words: faint attractor, twin spatial limit closed curves, rotation number of a spatial closed curve, bifurcation ofrotation numbers, spatial limit circle

Abstract:  Based on both qualitative method and numerical tests for a series of particular cases in the parameter region, $a=1$ ,  $0 \leq b < 1$ , it is shown that the 3-dimensional dynamical system (2) may have a series of interesting phenomenaon 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 curves, the bifurcation of rotation numbers, and the spatial limit circle, etc.. The system (2) is a very rich source in the study of dynamical system theory.

Since there are 47 figures , a lot of mathematical symbols and complicated expresions, it is difficult for me to input them through the editor here. So, the full text of this paper is compiled as a pdf file, attached below.

Non-trivial Attractor.pdf