The present study focuses on a time-domain approach for investigating the response of laminar premixed flames to acoustic flow perturbations in a straight duct of a uniform cross-section. This alternative perspective has recently been found to facilitate the analysis and interpretation of thermo-acoustic instabilities by providing a more intuitive description of the interaction of the acoustics with the heat-release rate. A framework is first introduced to generate accurate time-series that provide a complete description of the thermo-acoustic system. Advantages include the flexibility to account for different models (delayed, local, and non-local terms), as well as the separation of numerical and modeling errors. Advanced dynamical system techniques are then employed to analyze the low-amplitude (linear) regime. Detailed representation of the acoustic signal inside the duct allows for accurate characterization of the transition from the linear to the non-linear regime, and give access to the coupled dynamics of complex limit-cycles. In particular, the effect of the heat release on the linear and non-linear response of the duct is analyzed in detail. This information forms the basis for an effective use of system reduction techniques. The presented capabilities are found to yield physical insight with relative ease, and are expected to produce interesting results by revisiting existing experimental and numerical strategies for the study of thermo-acoustic instabilities.