Parametrized data-driven decomposition for bifurcation analysis, with application to thermo-acoustically unstable systems

Dynamic mode decomposition (DMD) belongs to a class of data-driven decomposition techniques, which extracts spatial modes of a constant frequency from a given set of numerical or experimental data. Although the modal shapes and frequencies are a direct product of the decomposition technique, the determination of the respective modal amplitudes is non-unique. In this study, we introduce a new algorithm for defining these amplitudes, which is capable of capturing physical growth/decay rates of the modes within a transient signal and is otherwise not straightforward using the standard DMD algorithm. In addition, a parametric DMD algorithm is introduced for studying dynamical systems going through a bifurcation. The parametric DMD alleviates multiple applications of the DMD decomposition to the system with fixed parametric values by including the bifurcation parameter in the decomposition process. The parametric DMD with amplitude correction is applied to a numerical and experimental data sequence taken from thermo-acoustically unstable systems. Using DMD with amplitude correction, we are able to identify the dominant modes of the transient regime and their respective growth/decay rates leading to the final limit-cycle. In addition, by applying parametrized DMD to images of an oscillating flame, we are able to identify the dominant modes of the bifurcation diagram.