Biological systems exhibit numerous oscillatory behaviors from calcium oscillations to circadian rhythms that recur daily. These autonomous oscillators contain complex feedbacks with nonlinear dynamics that enable spontaneous oscillations. The detailed nonlinear dynamics of such systems remains largely unknown. In this paper, we investigate robustness and dynamical differences of five minimal systems that may underlie fundamental molecular processes in biological oscillatory systems. Bifurcation analyses of these five models demonstrate an increase of oscillatory domains with a positive feedback mechanism that incorporates a reversible reaction, and dramatic changes in dynamics with small modifications in the wiring. Furthermore, our parameter sensitivity analysis and stochastic simulations reveal different rankings of hierarchy of period robustness that are determined by the number of sensitive parameters or network topology. In addition, systems with autocatalytic positive feedback loop are shown to be more robust than those with positive feedback via inhibitory degradation regardless of noise type. We demonstrate that robustness has to be comprehensively assessed with both parameter sensitivity analysis and stochastic simulations.