Global sensitivity analysis of a nonlinear system using animated basins of attraction

Abstract

A computer program is described that allows the effect of parameter changes on the global behavior of a nonlinear system to be studied. Steady‐state motions and their basins of attraction can be rapidly obtained and visualized for given parameter values. The program is applied to the study of a differential equation modeling a nonlinear oscillator forced parametrically through its bifurcation parameter. The algorithm used in our program is based on the Interpolated Cell Mapping (ICM) method developed by Tongue [B. H. Tongue, Physica D 28(3), 401–408 (1987)]. It is shown how this method, when implemented on a vector/parallel architecture machine, can be combined with animation to perform global sensitivity analyses of nonlinear systems. A high‐resolution (1152×900 pixel) animation is created which shows the evolution of the basins of attraction for a specific pair of periodic solutions as the forcing frequency and forcing amplitude are varied over a closed path in the parameter space. The movie consists of 720 pictures, each of which requires 518 400 separate simulations. In order to accomplish this task, the basic algorithm is modified, using IBM Parallel Fortran, to exploit the architecture of the IBM 3090‐600S. Of particular interest is the way the animation reveals phenomena that are difficult or impossible to see in the still images. Detailed benchmarks are presented which show the performance of different configurations of the code in a typical university supercomputing environment.