Swirling jets undergoing vortex breakdown are widely used in combustion applications, due to their ability to provide aerodynamic flame stabilization. It is well known that vortex breakdown is accompanied by a dominant coherent structure, the so-called precessing vortex core (PVC). Reports on the impact of the PVC on the combustion process range from beneficial to detrimental. In any event, efficient methods for the analysis of the PVC help to increase the benefit or reduce the penalty resulting from it. This study uses particle image velocimetry (PIV) measurements of a generic nonisothermal swirling jet to demonstrate the use of advanced data analysis techniques. In particular, the finite time Lyapunov exponent (FTLE) and the local linear stability analysis (LSA) are shown to reveal deep insight into the physical mechanisms that drive the PVC. Particularly, it is demonstrated that the PVC amplitude is strongly reduced, if heating is applied at the wavemaker of the flow. These techniques are complemented by the traditionally used proper orthogonal decomposition (POD) and spatial correlation techniques. It is demonstrated how these methods complement each other and lead to a comprehensive understanding of the PVC that lays out the path to efficient control strategies.