Dynamical System of Scalar Field from 2-Dimension to 3-D and its Cosmological Implication

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 $(w_{\varphi },{\Omega }_{\varphi },\lambda )$, 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 $(w_{\varphi },{\Omega }_{\varphi },\lambda )$ 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.

20 pages, 8 figures,some references added. Minor changes according to the suggestions from referees

References in corpus (17)

• Dynamics of dark energy
• Can dark energy evolve to the Phantom?
• Thawing quintessence with a nearly flat potential
• Scaling attractors in interacting teleparallel dark energy
• Dynamics of Quintessence Models of Dark Energy with Exponential Coupling to the Dark Matter
• Dynamical analysis of generalized Galileon cosmology
• Dynamical analysis of anisotropic scalar-field cosmologies for a wide range of potentials
• Cosmological dynamics of tachyonic teleparallel dark energy
• Exact Analysis of Scaling and Dominant Attractors Beyond the Exponential Potential
• A New Approach to Quintessence and a Solution of Multiple Attractors
• Phase-space analysis of a class of k-essence cosmology
• Stability analysis of an autonomous system in loop quantum cosmology
• Non-chaotic dynamics in general-relativistic and scalar-tensor cosmology
• Full Investigation on the Dynamics of Power-Law Kinetic Quintessence
• Dynamics of Tachyon and Phantom Field beyond the Inverse Square Potentials
• Dynamics and chaos in the unified scalar field Cosmology
• Chaotic dynamics of the Bianchi IX universe in Gauss-Bonnet gravity

Cited by in corpus (4)

• Dynamical systems applied to cosmology: dark energy and modified gravity
• Dynamical system analysis of LRS-BI Universe with f(Q) gravity theory
• Inflationary equilibrium configurations of scalar-tensor theories of gravity
• Phase space analysis of Bianchi III Universe with $f(R,T)$ gravity theory