Dynamical system of scalar field from 2-dimension to 3-D and its cosmological implications

We give the three-dimensional dynamical autonomous systems for most of the popular scalar field dark energy models including (phantom) quintessence, (phantom) tachyon, K-essence, and general non-canonical scalar field models, change the dynamical variables from variables \((x, y, \lambda )\) to observable related variables \((w_{\phi }, \Omega _{\phi }, \lambda )\), and show the intimate relationships between those scalar fields that the three-dimensional system of K-essence can reduce to (phantom) tachyon, general non-canonical scalar field can reduce to (phantom) quintessence and K-essence can also reduce to (phantom) quintessence for some special cases. For the applications of the three-dimensional dynamical systems, we investigate several special cases and give the exactly dynamical solutions in detail. In the end of this paper, we argue that it is more convenient and also has more physical meaning to express the differential equations of dynamical systems in \((w_{\phi }, \Omega _{\phi }, \lambda )\) instead of variables \((x, y, \lambda )\) and to investigate the dynamical system in three dimensions instead of two dimensions. We also raise a question about the possibility of the chaotic behavior in the spatially flat single scalar field FRW cosmological models in the presence of ordinary matter.