A salient feature of nonlinear systems is that several distinct steady‐state solutions can coexist: Different initial conditions can lead asymptotically to different attractors. The way in which parameter variations affect the geometry of the basins of attraction is central to an understanding of stability transition phenomena, including the onset of chaotic vibrations. A computer program is described that allows steady‐state solutions and their basins of attraction to be rapidly obtained and visualized for given parameter values. The code was implemented on a vector‐parallel architecture supercomputer. The results of analyses carried out on several mechanical systems are presented, including movies of basin‐boundary evolution as the parameters are varied over a curve in the parameter space. Of particular interest is the way in which animation reveals phenomena not readily seen in still images, such as rapid transitions between smooth and fractal basin boundaries, or the rapid creation and annihilation of entire basins of attraction.