About Bifurcational Parametric Simplification

¹ Department of Mathematics, Ben Gurion University of the Negev P.O.B. 653, 84105 Beer-Sheva, Israel
² Parks College of Engineering, Aviation and Technology Saint Louis University, Lindell Blvd, 3450, St. Louis MO 63103, USA

^⋆ Corresponding author. E-mail: vladimir@bgu.ac.il

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).