Dynamical Stability and Attractor of the Variable Generalized Chaplygin Gas Model

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 ω_VGCG= -0.92 is a dynamical attractor and the equation of state of VGCG reaches it from either ω_VGCG > -1 or ω_VGCG< -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 Ω_γ and Ω_VGCG are quite different from each other. For different initial values of x and y,Ω_γ decreases and Ω_VGCG increases as the time grows, they will eventually approach Ω_γ=0 and Ω_VGCG= 1. Furthermore, since different values of n or α may lead to different equation-of-state parameters Ω_VGCG, we also discuss the constraints on the parameters n and α by the observation data.