Belousov Reaction Research Articles

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

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.

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

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.

Invariant characteristics of self-organization modes in Belousov reaction modeling

We consider the problem of mathematical modeling of oxidation-reduction oscillatory chemical reactions based on the mechanism of Belousov reaction. The process of the main components interaction in such 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 a mathematical model of this reaction. We carry out a local study of the neighborhood of the system’s non-trivial equilibrium state and construct the normal form of the considering system. Finally, we do a numerical analysis of the coexisting chaotic oscillatory modes of the boundary value problem in a flat area, which have different nature and occur as the diffusion coefficient decreases.

Relaxation oscillations and diffusion chaos in the Belousov reaction

Asymptotic and numerical analysis of relaxation self-oscillations in a three-dimensional system of Volterra ordinary differential equations that models the well-known Belousov reaction is carried out. A numerical study of the corresponding distributed model-the parabolic system obtained from the original system of ordinary differential equations with the diffusive terms taken into account subject to the zero Neumann boundary conditions at the endpoints of a finite interval is attempted. It is shown that, when the diffusion coefficients are proportionally decreased while the other parameters remain intact, the distributed model exhibits the diffusion chaos phenomenon; that is, chaotic attractors of arbitrarily high dimension emerge.

Non-classical relaxation cycle of a three-dimensional system of Lotka-Volterra equations

A mathematical model of the well-known Belousov's reaction is the object of study. It is reasonable to assume that one coefficient in the corresponding system of differential equations is large, while the other parameters are finite. Non-standard tools taking account of the peculiarities of the problem bring one to a theorem on the existence of a relaxation cycle, allowing at the same time to reveal its characteristic features.

Unsaturated dicarboxylic acids as substrates in oscillating reactions involving bromate, sulfuric acid and a catalyst

A new type of homogeneous oscillating reaction has been observed in the potassium bromate-acetylenedicarboxylic acid-ferroin-sulfuric acid system. This proves that the presence of an “active” methylene group is not a necessary condition for oscillating reactions involving carboxylic acids. Malonic acid can be partly replaced by maleic and fumaric acid in the Belousov reaction. Under the mild experimental conditions of the oscillating reaction, maleic acid undergoes isomerization to fumaric acid.

On the appearance of chaos in a model of the Belousov reaction

A modified version of the Oregonator is analysed as a model of oscillations, bistability and chaos in the bromate-malonic acid-cerium reaction.

Asymptotic Solutions of the Field‐Noyes Model for the Belousov Reaction—II. Plane Waves

The spatially homogeneous limit cycle solutions found by Stanshine and Howard [7] for the Field‐Noyes model of the Belousov reaction are used to obtain plane wave solutions for the related reaction‐diffusion equations. Perturbation techniques are implemented first to find long wavelength waves and then successively to find shorter wavelength waves. Plane wave solutions are then shown to exist for sufficiently short wavelengths for parametric values which are physically plausible but for which spatially homogeneous oscillations do not (apparently) exist.

Asymptotic Solutions of the Field‐Noyes Model for the Belousov Reaction—I. Homogeneous Oscillations

Multiple‐time‐scale techniques are used to solve the non‐linear autonomous system used by Field and Noyes to model the chemical oscillations of the Belousov reaction. An asymptotic representation, valid for a wide range of parameters, is found for a spatially homogeneous limit‐cycle solution. For certain values of the parameters, two limit‐cycle solutions are shown (asymptotically) to exist. For parameter values for which the limit cycle appears to be unique, it is shown to be linearly stable. The asymptotic solution is shown to correspond excellently to the numerical solution calculated by Field and Noyes for one set of parameters.

Synchronization in two interacting oscillatory systems

Nonlinear phenomena arising from the interaction of two oscillating systems of chemical reactions are studied experimentally. The system of two connected flow-through continuous stirred tank reactors (cells) with controlled exchange of reaction mixture is used. The Belousov reaction (oxidation of malonic acid by bromate in sulphuric acid with ceric/cerous ions as catalyst) served as model system. The frequency of oscillations was controlled by change of the reaction temperature. Phenomena such as synchronization of oscillations at a common frequency, synchronization at multiples of a common frequency, rhythm splitting and amplitude amplification were observed, depending on the degree of interaction and the differences in the original oscillation frequencies. Mathematical modelling of the above phenomena failed, probably due to insufficient knowledge of a kinetic model.

On the temporal characteristics of a model for the Zhabotinskii-Belousov reaction

The number and stability of stationary and periodic solutions in a semi- phenomenological reaction mechanism that qualitatively models the Zhabotinskii- Belousov reaction are studied as functions of parameters in the model. Both multiple stationary solutions and multiple periodic solutions can exist simultaneously. Periodic solutions bifurcate in one of three ways: at zero amplitude as predicted from the Hopf theorem; pairwise at finite amplitude, the so-called “hard” bifurcation; or by coalescense with separatrix loops. Detailed computations for several cases reveal the dependence of the period and amplitude of the periodic solutions on the parameters. The results show that the model exhibits all the qualitative features that might be expected in intracellular reactions and so can serve as a model system for theoretical studies of pattern formation in developing systems.

Investigation into the effect of temperature and added t-butyl alcohol on the dynamic properties of the belousov reaction

Both the induction period and frequency of oscillation, ν, for the Belousov reaction depend on temperature. A plot of log ν against reciprocal temperature is a straight line, yielding an activation energy of 70 kJ mol–1. However, neither the induction period nor the frequency are very sensitive to added t-butyl alcohol. These observations are examined in the context of the Noyes mechanism for this reaction and by reference to an analysis of the dependence of frequency of oscillation on rate constants in the reaction scheme called the Oregonator.

On the velocity of spatial wave propagation in the Belousov reaction

First Page

Oscillations in chemical systems. IV. Limit cycle behavior in a model of a real chemical reaction

The chemical mechanism of Field, Körös, and Noyes for the oscillatory Belousov reaction has been generalized by a model composed of five steps involving three independent chemical intermediates. The behavior of the resulting differential equations has been examined numerically, and it has been shown that the system traces a stable closed trajectory in three dimensional phase space. The same trajectory is attained from other phase points and even from the point corresponding to steady state solution of the differential equations. The model appears to exhibit limit cycle behavior. By stiffly coupling the concentrations of two of the intermediates, the limit cycle model can be simplified to a system described by two independent variables; this coupled system is amenable to analysis by theoretical techniques already developed for such systems.

Horizontal Bands in the Belousov Reaction

The formation and propagation of spatial patterns in the presence of an initial gradient in temperature or the concentration of one or more of the reactants is explained kinematically. Experimental verification of this mechanism is given, showing that diffusion plays a very minor role in the formation of the bands.

Explanation of Spatial Band Propagation in the Belousov Reaction

BELOUSOV1 reported that the ratio [Ce(IV)]/[Ce(III)] undergoes repeated temporal oscillations in a stirred sulphuric acid solution containing bromate ion and malonic acid. Busse2 and Zaikin and Zhabotinsky3 reported that spatial bands of alternating oxidizing and reducing character propagated through an unstirred solution if it were initially subject to a concentration gradient. Existence of these bands was demonstrated by the redox indicator ferroin (the 1,10-phenanthroline complex of ferrous ion), and bands could be generated in the absence of cerium ions if indicator were present, because the ferroin indicator itself serves as an oxidation-reduction couple in the absence of cerium ions.

A reaction periodic in time and space. A lecture demonstration

The Belousov reaction provides a fascinating and easily staged lecture demonstration that is suitable for use in conjunction with discussions of free energy and spontaneity or complex reaction kinetics.