About Bifurcational Parametric Simplification

Abstract

A concept of "critical" simplification was proposed by Yablonsky and Lazman (1996) for the oxidation of carbon monoxide over a platinum catalyst using a Langmuir-Hinshelwood mechanism. The main observation was a simplification of the mechanism at ignition and extinction points. The critical simplification is an example of a much more general phenomenon that we call a bifurcational parametric simplification. Ignition and extinction points are points of steady state multiplicity bifurcations, i.e., they are points of a corresponding bifurcation set for parameters.Any bifurcation produces a dependence between system parameters. This is a mathematical explanation and/or justification of the "parametric simplification". It leads us to a conjecture that "maximal bifurcational parametric simplification" corresponds to the "maximal bifurcation complexity."This conjecture can have practical applications for experimental study, because at points of "maximal bifurcation complexity" the number of independent system parameters is minimal and all other parameters can be evaluated analytically or numerically.We illustrate this method by the case of the simplest possible bifurcation, that is a multiplicity bifurcation of steady state and we apply this analysis to the Langmuir mechanism. Our analytical study is based on a coordinate-free version of the method of invariant manifolds by Bykov, Goldfarb, Gol'dshtein (2006). As a result we obtain a more accurate description of the "critical (parametric) simplifications."With the help of the "bifurcational parametric simplification" kinetic mechanisms and reaction rate parameters may be readily identified from a monoparametric experiment (reaction rate vs. reaction parameter).

Model reduction in chemical dynamics: slow invariant manifolds, singular perturbations, thermodynamic estimates, and analysis of reaction graph

“…This is a very general phenomenon known for various bifurcations. For the equations of chemical kinetics, the theory of critical simplification was developed recently [48] using the constructive theory of invariant manifolds.…”