Dynamical Stability and Attractor of the Variable Generalized Chaplygin Gas Model

Abstract

For the variable generalized Chaplygin gas (VGCG) as a dynamical system, its stability is analyzed and the related dynamical attractors are investigated. By analysis it is shown that there are two critical points corresponding to the matter-dominated phase and the VGCG dark energy-dominated phase, respectively. Moreover, when the parameters n, α and γ take some fixed values, the phase with ωVGCG = −0.92 is a dynamical attractor and the equation of state of VGCG reaches it from either ωVGCG > −1 or ωVGCG < −1, independent of the initial values of the dynamical system. This shows a satisfactory cosmological model: the early matter-dominated era, followed by the dark energy-dominated era. Meanwhile, the evolutions of density parameters Ωγ and ΩVGCG are quite different from each other. For different initial values of x and y, Ωγ decreases and ΩVGCG increases as the time grows, they will eventually approach Ωγ = 0 and ΩVGCG = 1. Furthermore, since different values of n or α may lead to different equation-of-state parameters ωVGCG, we also discuss the constraints on the parameters n and α by the observation data.