Thermoacoustic instability problems are widely existed in many real-world applications such as gas turbines and rocket motors. A Rijke tube is a typical thermoacoustic system, and it is difficult to analyze such a system due to the nonlinearity and time delay. In this paper, a set of nonlinear ordinary differential equations with time delay which represent a Rijke tube system will be studied. The state space of such a tube system is consisted of velocity and pressure, and the periodic motion can be discretized based on an implicit midpoint scheme. Through Newton-Raphson method, the node points on the periodic motion will be solved, and the analytical solution of such a periodic motion for Rijke tube system can be recovered using a set of Fourier representations. According to the theory of discrete maps, the stability of the periodic motion will be obtained. Finally, specific system parameters will be adopted in order to carry out numerical studies to show different periodic motions for such a tube system. The analytical bifurcations which show how period-1 motion involves to period-m motion and then becomes chaos will be demonstrated. With such a technique, some interesting nonlinear phenomena will be explained analytically, which will be of great help to understand and control such a Rijke tube system.
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