Qualitative Instruments

Abstract

Numerical calculations are primarily quantitative in nature. Appropriate interpretation transforms quantitative information into meaningful results.The diversity of nonlinear phenomena requires more than just the ability to understand numerical output. The preceding chapters have concentrated on one-parameter problems, with only brief excursions (in Sections 2.10 and 5.9) to two-parameter models, mainly to show how one-parameter studies can be applied to investigate certain aspects of two-parameter models. Reducing a parameter space to lower-dimensional subsets will remain a standard approachfor obtainingquantitativeinsight.However,afullqualitativeinterpretation of multiparameter models requires instruments that have not yet been introduced in the previous chapters. These instruments are provided by singularity theory and catastrophetheory. Both these fields are qualitative in nature; in no way do they replace numerical parameter studies. Knowledge of singularity theory and catastrophe theory helps organize a series of partial results into a global picture.