We examine the non-stationary response of a one-degree-of-freedom non-linear system to a non-periodic parametric excitation with varying frequency. We use the method of multiple scales to obtain equations governing the stationary and non-stationary responses of the system, and we analyze the stability of the stationary responses. The response displays several phenomena, including penetration of the trivial response into the unstable trivial region, oscillation of the response about the non-trivial stationary solution, convergence of the non-stationary response to the stationary solution, lingering of the non-trivial response into the stable trivial region, and rebounding of the non-trivial response. These phenomena are affected by the sweep rate, the initial conditions, and the system parameters. Digital and analog computers are used to solve the original governing differential equation. The results of the simulations agree with each other and with those obtained by using the method of multiple scales.