Introduction In recent years, nonlinear, oscillating chemical reactions have been widely studied, characterized, and modeled. Such chemical reactions share remarkable similarities with biological and neuronal processing that are capable of timing, switching, and signal propagation (Strizhak and Menzinger, 1996). Briggs-Rauscher reaction - colorless liquids are mixed to form an amber solution which turns a blue-black color, and cycles between the two colors. BZ reaction - the reaction mixture cycles through steps of green, blue, purple, and red colors. The oscillatory nature of these reactions was experimentally observed as shown below.
Introduction The Brusselator and the Belousov-Zhabotinskii(BZ) reaction both display similar Oscillating behavior and can be modeled with related differential equations. The Brusselator” is an oscillating chemical reaction that was first described by Ilya Prigogine, a Belgian-American physical chemist and Lefever in 1968. The brusselator is the minimal mathematical model that can incorporate the oscillating behaviour. This model is usually studied in the mean-field approximation. The mean-field approximation neglects the effect of fluctuations and correlations.
The Brusselator• Autocatalytic chemical reaction sequence: A X 2X Y 3X B X Y C X D Ilya Prigogine (1917 - 2003) •The Brusselator is an autocatalytic reaction, which means that species in the reaction also can act as a catalyst of the reaction. • The reactions takes place in an open system, where the reactant species A, B, C and D are in large excess. D and C are removed from the system as they are produced. •The two species of interest to us are X and Y, the autocatalytic species.
The rate of the above set of reactions with respect to X is simply the sum of the individual rates and is given by: rx = [A] + [X]^2[Y] –[B][X] –[X]• A positive value denotes when X is being produced, while a negative value is used for the rate when X is being used. Similarly, the rate of the total reaction with respect to Y is given by: ry = [B][X] – [X]^2[Y] The differential equations modeling this system are thus given by: dx/dt = A + (X^2)Y –BX - X dy/dt = BX – (X^2)Y
Stationary state and stability ODE- system Equilibria points for the system of differential equations given can be calculated by solving the system of equations below. x˙ = f(x, y) = 1− (1 + b)x + ax^2y y˙ = g(x, y) = bx − ax^2y The properties of the system do to a large extent depend on its stationary states, which we get by solving the ODE-system for left hand sides zero. Fixed Points : f(x., y.) = 0 , g(x., y.) = 0 . a – bx – x + x^2y = 0 bx – x^2y = 0 Such oscillating reactions, must satisfy two conditions-1. The reaction must be far from equilibrium by large excesses in both concentrations of reactants. This guarantees that the concentration of the reactants A and B do not change with time (Ault and Holmgreen, 2003).2. The second condition, at least one autocatalytic step must be present in the mechanism of the reaction. A product of the chemical system that is involved in its own synthesis is defined as autocatalytic, and is thus what makes the equation nonlinear.
• By adding the two equations, x is found to equal a, and by plugging that into fixed-point eq., the equilibrium point is determined to be (a, b/a). Therefore, this is the only equilibrium point of the system. To determine the stability of the system, the Jacobian matrix is calculated. The eigenvalues of the Jacobian matrix will tell us the behavior of small perturbations to the fixed point(i.e. will the disturbances grow or decay?).• The eigenvalues of the Jacobian matrix evaluated at the stationary point provide us with information about the stability of the equilibrium. •When evaluated at the equilibrium point, the Jacobian becomes:
We can expect several stationary point types for the dynamic system depending on a and b (in last formula), and we can also state that b – a <1 implies stability, manifested by the stationary point beeing attractive. On the other hand, if b – a>1 the chemical system will evolve from the equilibrium state. To track the oscillations in the chemical reaction system, we need a neutral centre in the linearized system, that is λ=0<->b=a+1 . The imaginary parts of the eigenvalues are the sources of oscillating solutions (cf. Eulers formula for imaginary numbers), and pure imaginary eigenvalues equips the linearization with a centre in the phase plane.
The trace and determinant of fixed point eq. calso be computed. Trace(Df(a, b/a)) = - a^2 + b - 1 Det (Df(a, b/a)) = a^2 Because the determinant is greater than zero (a is always positive), and a is greater than zero, the equilibrium point is locally never a saddle point (Ault and Holmgreen, 2003). In order for the trace to be less than zero (stable), b < a^2 + 1. In this situation, the equilibrium point is an attractor or stable. When b > a^2 +1, the equilibrium point is a repellor and thus the local phase portrait is an unstable spiral. Thus the Brusselator has the possibility of Hopf bifurcation.
The stability diagram showing the stable and unstable regions as a function of a and b is given in figure:•The solid curve, satisfying b = a^2 + 1 delimits the stability region.•The dotted curve corresponding to b = (a + 1)^2 and b =(a − 1)^2separate the node from the focus in the stable and unstable regionsrespectively.
Reference: Ault, S. and E. Holmgreen. (2003). Dynamics of the Brusselator. Benini, O., Cervellati, R., and P. Fetto. (1996). The BZ Reaction: Experimental and Model Studies in the Physical Chemistry Laboratory. Journal of Chemical Education. Vol. 73. 865-868. Michael Giver. Fixed Points, Linear Stability, Limit Cycles. P. Gray and S.K. Scott. Chemical Oscillations and Instabilities: Non-linear Chemical Kinetics. Oxford University Press, 1994.