This article focuses on combustion instabilities (CI) driven by entropy fluctuations which is of great importance in practical devices. A simplified geometry is introduced. It keeps the essential features of an aeronautical combustion chamber (swirler, dilution holes, and outlet nozzle), while it is simplified sufficiently to ease the analysis (rectangular vane, one row of holes of the same diameter, no diffuser at the inlet of the chamber, and circular nozzle at the outlet). A large eddy simulation (LES) is carried out on this geometry and the limit cycle of a strong CI involving the convection of an entropy spot is obtained. The behavior of the instability is analyzed using phenomenological description and classical signal analysis. One shows that the system can be better described by considering two reacting zones: a rich mainly premixed flame is located downstream of the swirler and an overall lean diffusion flame is stabilized next to the dilution holes. In a second step, dynamic mode decomposition (DMD) is used to visualize, analyze, and model the complex phasing between different processes affecting the reacting zones. Using these data, a zero-dimensional (0D) modeling of the premixed flame and of the diffusion flame is proposed. These models provide an extended understanding of the combustion process in an aeronautical combustor and could be used or adapted to address mixed acoustic-entropy CI in an acoustic code.