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Studying global dynamics and stability of biological network is of importance in order to understand its function and behavior. In this paper, we consider the p53-Mdm2 oscillator module with PDCD5 as a core part of p53 signaling pathway after the DNA damage, and explore the dynamics and stability of the tumor suppressor p53. The dynamics of p53 may decide the cell fate after the DNA damage, while the oscillation of p53 may induce cell cycle arrest and so promote the repair of DNA, and the high levels of p53 can trigger apoptosis. However, p53 activity may be inhibited by its negative regulator Mdm2 in some cancer cells, as Mdm2 is of overexpression due to the increase in Mdm2 production rate. So we first investigate the effect of Mdm2 production rate on the kinetics of p53 through bifurcation analysis. after the DNA damage. With the increase in Mdm2 production rate, p53 can display a steady state, a stable-limit cycle and the coexistence of a stable-limit cycle and a stable steady state. Furthermore, the potential landscapes for oscillation show that the lower concentration of p53 means a stronger stability, whereas those for bistability of the higher steady state and the oscillatory state illustrate that stability of the higher steady state increases with the increasing Mdm2 production rate. In addition, noise strength can greatly affect the stability of p53 oscillations, so we explore the effect of noise strength on potential landscapes, barrier heights and periods. A smaller noise strength leads to a higher barrier height associated with more stable-limit cycle, and the harmonic oscillation with more uniform period and smaller variance is helpful to have more stable maintainance. Our results may be useful for understanding regulation of p53 signaling pathway after DNA damage.