The addition of a small amount of noise in nonlinear systems can play an important role, inducing transitions between dynamical regime and producing more complex dynamics. In this work, we present the analysis of a second order circuit with a nonvolatile memristive device, subject to additive noise perturbations. Memristive devices are two terminal elements suitable for implementing circuits with complex dynamic behaviors, that can be used to perform bio-inspired computational tasks. We show that, depending on noise intensity, the dynamic behavior of the system undergoes multiple qualitative changes. We use numerical simulations and nonlinear dynamics concepts to analyze these transitions.