When action is not least for systems with action-dependent Lagrangians

The dynamics of some non-conservative and dissipative systems can be derived by calculating the first variation an action-dependent action, according to variational principle Herglotz. This is directly analogous Hamilton commonly used derive conservative systems. In a similar fashion, just as second system's action infer whether that possible trajectories are dynamically stable, so too stable. this paper I show, generalizing earlier analyses for systems, how calculate apply it two physically important systems: time-independent harmonic oscillator time-dependent oscillator.