Stability, convergence and bifurcation in some models of chemical kinetics

TL;DR

This paper investigates the stability, bifurcation, and dynamic behavior of the Boissonade-De Kepper chemical model, providing analytical and numerical insights into its nonlinear dynamics and potential wider applications.

Contribution

It offers explicit conditions for stability, analyzes Hopf bifurcations using advanced mathematical tools, and extends findings beyond the specific model.

Findings

Identification of stability conditions for the BD model

Derivation of explicit formulas for Hopf bifurcation type

Numerical validation of analytical results

Abstract

In this paper, we analyze the stability, convergence, and bifurcation properties of the Boissonade-De Kepper (BD) model which played a key role in the development of nonlinear chemical dynamics. We first outline conditions for local stability, which may help guide design considerations. Then, we show that the BD model undergoes a Hopf bifurcation when the stability condition gets violated. Using Poincar\'{e} normal forms and center manifold theory, we derive explicit analytic expressions for determining the type of the Hopf bifurcation and the stability of the limit cycles. This provides insights on the system dynamics just beyond the stable regime. Some of the analytical insights are corroborated with numerical computations. We also show that the mathematical results obtained in this paper may have wider applicability beyond the BD model.