Numerical Methods for the Investigation of the Nonlinear Equations of Chemical Dynamics, and Their Application to a Model of the Belousov-Zhabotinskii Reaction.

Many dynamical systems of current interest are described by sets of coupled nonlinear ordinary differential equations which are largely insusceptible to analytical methods. Straightforward numerical integration realizes the dynamics latent in the equations, but by itself this technique is inadequate to illuminate the structures which support and determine the dynamical behavior. Method which wrestle information more purposely from the equations are required. In response to this need, algorithms are here developed for the computation of families of steady states, of local bifurcations, and of periodic orbits; algorithms which do not rely on dynamical relaxation to attracting invariant sets. Dynamically unstable fixed points and closed orbits which are immune to analysis by direct integration are computed with ease. Marginally stable sets are located and characterized efficiently. When applied to a model of the Belousov-Zhabotinskii reaction, these algorithms reveal and clarify aspects of the dynamical behavior, provide criteria for the assessment of the adequacy of the model, yield data which serve to guide investigation of the more exotic behavior exhibited, and suggest questions which may be answered analytically.