The emergence of second-harmonic generation (SHG) is a pivotal issue to the development of nano-optical devices and interfaces. Here, we perform a classical analysis of the SHG dynamics with the aim of determining critical values of the electric field. The signals of the SHG process are nonlinear, so it seems reasonable that chaos theory can be a suitable tool to analyze their dynamics. For this purpose the bifurcation analysis and a new measure known as the scale index were employed. Obtained results show that the dynamics approaches from stable to self-pulsing and eventually to chaos through a period-doubling route. Moreover, the scale index can detect the critical strength of the electric field, where above it the SHG dynamics is chaotic.