Recent studies have shown that many cell-signaling networks contain interactions and feedback loops that give rise to complex dynamics. Synthetic biology has allowed researchers to construct and analyze well-defined signaling circuits exhibiting behavior that can be predicted and quantitatively understood. Combining these approaches--wiring natural network components together with engineered interactions--has the potential to precisely modulate the dynamics of endogenous signaling processes and control the cell decisions they influence. Here, we focus on the p53 signaling pathway as a template for constructing a tunable oscillator comprised of both natural and synthetic components in mammalian cells. We find that a reduced p53 circuit implementing a single feedback loop preserves some features of the full network's dynamics, exhibiting pulses of p53 with tightly controlled timing. However, in contrast to the full natural p53 network, these pulses are damped in individual cells, with amplitude that depends on the input strength. Guided by a computational model of the reduced circuit, we constructed and analyzed circuit variants supplemented with synthetic positive and negative feedback loops and subjected to chemical perturbation. Our work demonstrates that three important features of oscillator dynamics--amplitude, period, and the rate of damping--can be controlled by manipulating stimulus level, interaction strength, and feedback topology. The approaches taken here may be useful for the rational design of synthetic networks with defined dynamics, and for identifying perturbations that control dynamics in natural biological circuits for research or therapeutic purposes.