The exotic dynamical behaviors exhibited in chemical reaction systems, such as multiple steady states, undamped oscillations, chaos, and so on, often result from unstable steady states. A bacterial glycolysis model is studied, which involves the generation of adenosine triphosphate (ATP) in a flow system and consists of eight species and ten reactions. A minimum subnetwork of the bacterial glycolysis model is determined to exhibit an unstable steady state with a positive real eigenvalue, which gives rise to undamped oscillations for a small perturbation. A set of rate constants and the corresponding unstable steady state are computed by using a positive real eigenvalue condition. The phenomena of oscillations and bifurcation are discussed. These results are extended to the bacterial glycolysis model and several parent networks.