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 phi with positive potential V, coupled with a barotropic fluid in the Universe. With Gamma = VV ''/V'(2) considered as a function of lambda = 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.