Abstract
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 3-Dimension instead of 2-Dimension. 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.
Highlights
Phase-plane analysis is a very useful and common method to study the dynamical evolution of those scalar fields models and their cosmological implications. Most of those works only focus on the quintessence models with unique exponential potential and tachyon models with inverse square potential, and correspondingly, the dynamical systems are two-dimensional autonomous systems
When the potentials are beyond the special type such as exponential or inverse square potentials, the dynamical systems become a three-dimensional autonomous systems
There is very few work focusing on the dynamical behavior of the scalar field with a general modified kinetic term, such as K-essence (L = V (φ)F(X )) and general non-canonical scalar field (L = F(X ) − V (φ))
Summary
Phase-plane analysis is a very useful and common method (see Ref. [4] and recent papers, e.g. [5,6,7,8]) to study the dynamical evolution of those scalar fields models and their cosmological implications. Josue De-Santiago et al analyzed the dynamical system of general non-canonical scalar field with the lagrangian L = F(X ) − V (φ) and studied the phase plane after a suitable choice of variables [40]. They obtained the three-dimensional autonomous system of this non-canonical scalar field after specifying the kinetic term as F(X ) = AX η and choosing the potential as V (φ) = V0(φ − φ0)1/(1− ) We raise a question about the possibility of chaotic behavior in the spatially flat single scalar field FRW cosmological models in the presence of the ordinary matter
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