Modelling Coupled Thermoacoustic Systems: Synchronization and Amplitude Death

Control of thermoacoustic (TA) instabilities in multiple combustion systems, such as those in can-annular combustors of gas turbines, has been a challenging problem in propulsion and power generation industries. Recent experiments show that coupling two TA systems can lead to synchronization, switching between in-phase and anti-phase synchronized states via phase-flip bifurcation (PFB), or even complete suppression of TA oscillations in both systems through amplitude death (AD). However, the mechanism through which TA systems attain AD or exhibit PFB is not yet well understood. In this study, we examine the model of two time-delay coupled prototypical TA systems called Rijke tubes. Through numerical simulations and analytical approximations, we predict the critical parameter values for the occurrence of PFB and AD in the model. The results from the model qualitatively match with previous experiments on coupled Rijke tubes. Using bifurcation analysis, we show the system to transition from oscillatory state to AD through fold bifurcation, and vice-versa through subcritical Hopf bifurcation. These findings may provide insight into the coupled interaction and suppression of oscillatory instabilities in engineering systems, ecological networks, financial markets, and other systems.