Stability analysis of periodic orbits in thermoacoustic oscillation using state space reconstruction with maximum predictability

Combustion instability arises from the coupling between unsteady pressure fluctuations and heat release rate. Stability analysis provides an understanding of the factors that cause this instability. One of transition of stability occurs between discrete periodic oscillations, and previous studies addressing this issue have primarily relied on stability analysis of fixed points. In this study, pressure fluctuations measured in a Helmholtz pulse combustor are analyzed from the perspective of stability for periodic orbits, with experiments conducted by varying the fuel supply pressure from 4 to 23 kPa. To capture the nonlinear characteristics of combustion instability, the dimensionality of the pressure fluctuation series is enhanced using Takens’ embedding, and singular value decomposition is applied to the delay series to identify periodic orbits, focusing on the dominant factors driving combustion oscillations, while the stability of these periodic orbits is determined using Floquet exponents. The main conclusions are as follows: as the fuel supply pressure increases, the oscillations transition from low-amplitude to high-amplitude states(average pulsation value from 7.4 to 14.5 kPa), with the transition process exhibiting mixed-mode oscillations. For low-amplitude oscillations, the system exhibits multi-periodic orbits characterized by a pie-shaped phase space and a unimodal Floquet exponent distribution, confirming that temporal amplitude variations do not significantly alter the stability of the periodic oscillations. In contrast, high-amplitude oscillations approximate a unified limit cycle with a similarly unimodal Floquet exponent distribution, indicating stable periodic oscillations. The transition is marked by a sharp focusing of the period distribution, rapid synchronization, and eventual phase-locking into a unified stable limit cycle. The Floquet multiplier distribution shifts from a negative to a positive and back to a negative peak(-2 to +2 back to 1.8), while the number of dominant modes changes from two to one, clearly signifying that the transition is a dynamical bifurcation process driven by strong nonlinear effects. This phenomenon is explained by potential function.