In the present study, we utilize spatially high-order convergence rate methods with complex reaction kinetics in resolving the nonlinear dynamics associated with one-dimensional unstable detonations. For a spark-induced detonation, as the detonation decays towards the self-sustaining Chapman-Jouguet mode from an over-driven mode, one obtains a sequence of physical oscillations between the flame and shock front, with different frequency ranges (categorized as high frequency-low amplitude and low frequency-high amplitude), dependent on the time after initiation of the detonation. The present studies indicate that one must use sufficient spatial resolution as well as realistic, complex kinetics to accurately simulate the preferred re-explosion and instability modes. Metrics that characterize the instabilities include, in addition to peak pressure, dominant high and low frequencies and time to re-explosion. A simple model for the transmission of acoustic and entropy waves is used to interpret physical phenomena creating the different instability modes, with reasonable quantitative correspondence to simulation results.