The oscillatory behavior of the photoelectrochemical system p-CuInSe2/H2O2 is investigated under galvanostatic conditions. To simplify the experimental system (elimination of light as additional control parameter), we first demonstrate the qualitative equivalence between polished electrodes in the dark and etched electrodes under illumination with respect to the nonmonotonous current-voltage curve and then focus on the investigation of the oscillation characteristics at a polished surface. Typical features of the system CuInSe2/H2O2 are presented, concerning the dependence of the global oscillatory regime on H2O2 concentration, mass transport, and current density. As in other systems, a large variety of oscillation characteristics is observed, including a transition to chaos via period-doubling sequences (Feigenbaum scenario), mixed-mode oscillations (MMOs), and simple relaxation-type oscillations. Within the mixed-mode regime a kind of "self-similarity" in the oscillation characteristics is found, i.e., the global oscillation hierarchy seems to repeat within one single MMO sequence. Under conditions of constant current, slow time-dependent changes of the oscillation characteristics are typically observed. This time-transient behavior is attributed to time-dependent chemical surface modifications, acting as a "slowly varying control parameter". This assumption is supported by the observation of a striking phenomenological similarity between time-transient and current-transient (current sweep) oscillation behavior, which is interpreted in terms of a time-dependent inhomogeneous surface passivation, leading to an increase in the local current density under time-transient conditions.