A criterion based on chemical explosive mode analysis (CEMA) is proposed to identify different local combustion modes, such as auto-ignition, diffusion-assisted ignition, and local extinction in laminar and turbulent premixed flames. The criterion is employed to distinguish between two different propagation modes of one-dimensional (1-D) freely propagating premixed flames, that is the canonical deflagration wave controlled by heat and species back-diffusion, which features a unique flame speed, and the auto-ignition controlled reaction front propagation, which may feature an arbitrary propagation speed. In the CEMA based diagnostic, the local chemistry and diffusion source terms are projected to the chemical explosive mode (CEM), such that the roles of diffusion and chemistry in the CEM can be quantified and compared for identification of the flame propagation mode. The new criterion is further applied to analyze two-dimensional (2-D) DNS datasets of homogeneous charge compression ignition (HCCI) combustion of n-heptane/air with different levels of thermal stratification. Both flame propagation modes are observed and contribute significantly to the overall flame burning in the case with a high level of thermal stratification, while the flame propagation is found dominated by the auto-ignition mode in the case with a low level of thermal stratification, which agrees with the finding in previous studies.