In this paper, an innovative method of adopting a piping insert with an abrupt contraction or expansion structure has been proposed to eliminate undesirable thermally driven gas oscillations in cryogenic systems. In addition, the influences of the piping inserts’ geometrical parameters and position on the performance of thermoacoustic oscillation in cryogenic systems have been investigated. A series of simulations has been conducted, and simulation bifurcation diagrams show that thermoacoustic oscillation can be completely stopped, which, in other words, is called the amplitude death (AD) phenomenon. When the pipeline inserts are equipped in the cold part, AD can be achieved regardless of the pipeline inserts’ radius. When the pipeline inserts are located in the warm part, AD can be realized by the pipe inserts with an abrupt contraction. The identification of nonlinear characteristics of different state transitions with different insert geometries is further conducted by unfolding these original time series in an embedding space. The nonlinear change process of thermoacoustic oscillation is demonstrated in the phase space diagrams. As the insert radius increases, the system transforms from a periodic state to a quasi-periodic state and then from a quasi-periodic state to an AD state. This work will have practical impacts on the design of oscillation-free piping systems.