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 main component interaction in such a reaction can be interpreted by a phenomenologically similar to it predator–prey model. Thereby, we consider a parabolic boundary value problem consisting of three Volterra-type equations, which is the mathematical model of this reaction. We carry out a local study of the neighborhood of the system’s non-trivial equilibrium state and 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 considered system and the form of its coefficients, that define the qualitative behavior of the model, and show the graphical representation of these coefficients depending on the main system parameters. The obtained normal form makes it possible to prove a theorem on the existence of an orbitally asymptotically stable limit cycle, which bifurcates from the equilibrium state, and find its asymptotics. To identify the applicability limits of the found asymptotics, 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 resulting behavior of solutions near the equilibrium state. In the second part of the paper, we consider the problem of diffusion loss of stability of the spatially homogeneous cycle obtained in the first part. We find a critical value of the diffusion parameter at which this cycle of the distributed system loses its stability.