This paper explores the complex phenomenon of combustion instability in Rijke tubes, paying special attention to the energy content of the first two oscillation modes under varied operating conditions. Taking Galerkin method as basis, the work explores the complicated dynamics of nonlinear oscillatory behavior in such systems. The research focuses on how system parameter changes like heater position, time lag and damping influence mode 1 and mode 2 interaction. By undertaking a thorough analysis of parameter variation, the research identifies how the first two modes’ strength differ with respect to heater position, while variation in time lag influences both modes the same way. In addition to this, it has been noted that damping in the second mode is reduced when its strength of damping is minimized and effectively contributes little to the first mode. All these phenomena are not only significant from theoretical perspectives but are also of practical importance to improve combustion stability in engineering devices such as in gas turbines and other combustion plants.