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Rocket engine combustors are prone to transverse instabilities that are characterized by large amplitude high frequency oscillations in the acoustic pressure and the heat release rate. In this work, we study the coupled interaction between the acoustic pressure and the CH* intensity (representative of heat release rate) oscillations in a 2D multi-element self-excited model rocket combustor during the transition from a stable state to thermoacoustic instability through intermittency. We show the emergence of synchronization between these oscillations from desynchronization through intermittent phase synchronization during the onset of thermoacoustic instability. We find substantial evidence that the intensities of the jet flames close to the end wall is higher than that observed near the center of the combustor as a result of its strong coupling to the local acoustic field. Using concepts from recurrence theory, we distinguish the type of synchronization between the acoustic pressure and the CH* intensity oscillations at the end wall and the center of the combustor during thermoacoustic instability. Analyzing the local CH* intensity oscillations, we observe that the longitudinally propagating jet flames experience substantial transverse displacement with flame merging effects during thermoacoustic instability. Furthermore, we differentiate the interaction of the jet flames with the shock wave during intermittency and thermoacoustic instability at both the end wall and the center of the combustor. We also discerned the change from stochastic to deterministic nature in the local CH* intensity oscillations during the transition to thermoacoustic instability. Our results demonstrate that only the first few transverse modes contribute to the generation of acoustic power from the reacting flow, in spite of the pressure oscillations featuring several harmonics. The insights gained from such coupled analysis during the transition from stable state to thermoacoustic instability can be leveraged to aid modeling efforts of thermoacoustic instability in rocket engines with multiple injectors. [ABSTRACT FROM AUTHOR]