The Andronov–Hopf Bifurcation in a Biophysical Model of the Belousov Reaction

Abstract

We consider the problem of mathematical modeling of oxidation-reduction oscillatory chemical reactions based on the Belousov reaction mechanism. The process of the main components interaction in such a reaction can be interpreted by a “predator – prey” model phenomenologically similar to it. Thereby, we consider a parabolic boundary value problem consisting of three Volterratype equations, which is a mathematical model of this reaction. We carry out a local study of the neighborhood of the system non-trivial equilibrium state, define a critical parameter, at which the stability is lost in this neighborhood in an oscillatory manner. Using standard replacements, we construct the normal form of the considering system and the form of its coefficients defining the qualitative behaviour of the model and show the graphical representation of these coefficients depending on the main system parameters. On the basis of it, we prove a theorem on the existence of an orbitally asymptotically stable limit cycle, which bifurcates from the equilibrium state, and find its asymptotics. To identificate the limits of found asymptotics applicability, we compare the oscillation amplitudes of one periodic solution component obtained on the basis of asymptotic formulas and by numerical integration of the model system. Along with the main case of Andronov–Hopf bifurcation, we consider various combinations of normal form coefficients obtained by changing the parameters of the studied system, and the corresponding to them solutions behaviour near the equilibrium state. In the second part of the paper, we consider the problem of the diffusion loss of stability of a spatially homogeneous cycle obtained in the first part. We find a critical value of diffusion parameter, at which this cycle of distributed system loses the stability.