Quintessence models with an oscillating equation of state and their potentials

In this paper, we investigate the quintessence models with an oscillating equation of state (EoS) and their potentials. From the constructed potentials, which have the EoS of $\omega_{\phi}=\omega_0+\omega_1\sin z$, we find they are all the oscillating functions of the field $\phi$, and the oscillating amplitudes are decreasing (or increasing) with $\phi$. From the evolutive equation of the field $\phi$, we find this is caused by the expansion of the universe. This also makes that it is very difficult to build a model whose EoS oscillates forever. However one can build a model with EoS oscillating for a period. Then we discuss three quintessence models, which are the combinations of the invert power law functions and the oscillating functions of the field $\phi$. We find they all follow the oscillating EoS.