Synthetic biology opens up the possibility of creating circuits that would not survive in the natural world and studying their behaviors in living cells, expanding our notion of biology. Based on this, we analyze on a synthetic biological system the effect of coupling between two instability-generating mechanisms. The systems considered are two topologically equivalent synthetic networks. In addition to simple periodic oscillations and stable steady state, the system can exhibit a variety of new modes of dynamic behavior: coexistence between two stable periodic regimes (birhythmicity) and coexistence of a stable periodic regime with a stable steady state (hard excitation). Birhythmicity and hard excitation have been proved to exist in biochemical networks. Through bifurcation analysis on these two synthetic cellular networks, we analyze the function of network structure for the collapse and revival of birhythmicity and hard excitation with the variation of parameters. The results have illustrated that the bifurcation space can be divided into four subspaces for which the dynamical behaviors of the system are generically distinct. Our analysis corroborates the results obtained by numerical simulation of the dynamics.