Low-frequency combustion instabilities arise in an aero-engine gas turbine combustor under idle/sub-idle conditions, particularly when pilot-mode non-premixed combustion is sustained. Despite the fundamental importance of such instabilities, little is known about how they are initiated and grow in the system. Here, we present nonlinear mode transition processes using several analysis methods, including Fourier/Hilbert transforms, phase portraits, spectrograms, low-order analytic modeling, and high-speed flame visualization methods. Our results demonstrate that a non-premixed Jet A-1 spray flame yields an intermediate-amplitude, quasi-periodic L1 mode oscillation at 50 Hz for a low pilot equivalence ratio, and the frequency gradually increases with increasing fuel flowrates. Subsequent to a critical point (103 Hz), the system undergoes a discontinuous mode transition, giving rise to the formation of an extremely large pressure oscillation with an L2 mode structure at 244 Hz. Several key triggers were found to induce the mode shift: (i) generation of a high intensity pulse in the flame's heat release rate due to the spontaneous ignition of unburned reactant mixtures, (ii) emergence of a large-scale vortical structure and its interaction with a partially premixed flame front, and (iii) development of self-sustained limit cycle oscillations driven by periodic convection of a hot spot – a mechanism known as entropy wave propagation. Our study identifies the occurrence of a large-amplitude peak followed by a local minimum intensity, analogous to the activation energy concept, as an essential step for entry into a new state with large-amplitude limit cycle oscillations.