Stability analysis of an autonomous system in loop quantum cosmology

We discuss the stability properties of an autonomous system in loop quantum cosmology. The system is described by a self-interacting scalar field $ϕ$ with positive potential $V$, coupled with a barotropic fluid in the Universe. With $Γ=VV"/V'^2$ considered as a function of $λ=V'/V$, the autonomous system is extended from three dimensions to four dimensions. We find that the dynamic behaviors of a subset, not all, of the fixed points are independent of the form of the potential. Considering the higher-order derivatives of the potential, we get an infinite-dimensional autonomous system which can describe the dynamical behavior of the scalar field with more general potential. We find that there is just one scalar-field-dominated scaling solution in the loop quantum cosmology scenario.

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