Silicic volcanism seems chaotic: the styles, magnitudes, and/or timing of successive eruptions may be without apparent pattern, or patterns may merge, fade, or abruptly change. However, geophysical monitoring of recent eruptions shows that some silicic volcanoes can exhibit cyclic eruptive behavior wherein periods of explosive activity or rapid extrusion alternate with periods of repose. The cycles are commonly observed in time-averaged amplitudes of eruption-related seismicity and also have been observed in ground-surface tilt data. When tilt and seismicity are both observed during oscillatory behavior, as at Soufriere Hills volcano on Montserrat, British West Indies, they correlate in time. Cycle periods range from hours to days, and cycle amplitudes and waveforms vary widely. Complex oscillatory behavior is also sometimes observed during high-pressure (tens of MPa) extrusion of industrial polymer melts. With this phenomenon as a guide, we construct a simple dynamic model for the oscillatory behavior of erupting volcanoes. We propose that cyclic eruptions result from Newtonian flow of compressible magma through the volcanic conduit combined with a stick-slip condition along the conduit wall, in analogy to the behavior of industrial polymers. If magma is forced into the conduit at a constant rate, pressure and flow rate rise. If the flow rate through the conduit exceeds a threshold value, the flow resistance abruptly drops as the magma slips along a shallow portion of the conduit wall. This reduces resistance to flow and causes the flow rate to jump to a higher value. If this enhanced flow rate exceeds the supply rate, both pressure and flow rate decline as the compressed magma in the conduit expands. Eventually slip ceases as the magma reattaches to the conduit wall at a flow rate less than the supply rate. Consequently pressure begins to increase, and the cycle begins again. This simple model reconciles a variety of disparate phenomena associated with cyclic silicic volcanism and provides a paradigm to interpret cyclic eruptive behavior.