Brusselator scheme

Example 12.7 Linear stability analysis Brusselator scheme This example is from Kondepudi and Prigogine (1999). Consider the chemical reaction scheme in Eq. (12.86). Assume that the concentration values of A, B, and E, F are maintained at uniform values, and X and Y are the only remaining variables. Then the kinetic equations are 

Here the concentrations A and B are specified. The stationary solutions for which f =f2 = 0 are 

The eigenvalues are obtained from the characteristic equation dct. / - AT] = 0, or for a 2 X 2 matrix we have the following quadratic form  

For a specified concentration of A, the system becomes unstable and oscillates if the condition in terms of the concentration value B becomes 

This kind of instability is space independent and leads to order in time. 

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Example 13.3 The Brusselator system and oscillations One theoretical model displaying order in time is the Brusselator scheme, which was introduced in Section 12.7.1. In the limit of irreversible reactions the Brusselator scheme is... 

Figure 13.5. The oscillations and the limit cycle obtained from the Brusselator scheme in Example 13.3 (a) Xand /versus time, where the bold line displays the concentration of Y, and (b) limit cycle with /c, = 1.3, k2 = 1.0, k3 = 1.0, k4 = 1.0, A= 1.1, B = 3.0 (c) X and Y versus time, where the bold line displays the concentration of Y, and (d) limit cycle with /c, = 1.0, k2 = 1.0, k3 = 1.0,...

The Schnakenberg model is a modification of the Brusselator scheme, and its mechanism consists of the following four steps ... 

Describe the linear stability of the Brusselator scheme (Kondepudi and Prigogine, 1999). 

Once the door was opened to these new perspectives, the works multiplied rapidly. In 1968 an important paper by Prigogine and Rene Lefever was published On symmetry-breaking instabilities in dissipative systems (TNC.19). Clearly, not any nolinear mechanism can produce the phenomena described above. In the case of chemical reactions, it can be shown that an autocatalytic step must be present in the reaction scheme in order to produce the necessary instability. Prigogine and Lefever invented a very simple model of reactions which contains all the necessary ingerdients for a detailed study of the bifurcations. This model, later called the Brusselator, provided the basis of many subsequent studies. 

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Three model kinetic schemes have been studied relatively intensively with periodic forcing the first-order non-isothermal CSTR of chapter 7 the Brusselator model, which is closely related to the cubic autocatalysis of chapters 2 and 3 and the surface reaction model discussed in 12.6. We will use the last of these to introduce some of the general features. 

As it was mentioned in Section 2.1.1, the concentration oscillations could be simulated quite well by a set of even two ordinary differential equations of the first order but paying the price of giving up the rigid condition imposed on interpretation of mechanisms of chemical reactions namely that they are based on mono- and bimolecular stages only (remember the Hanusse theorem [19]) An example of what Smoes [7] called the heuristic-topological model is the well-known Brusselator [2], Its scheme was discussed in Section 2.1.1 see equations (2.1.33) to (2.1.35). 

Oregonator and "brusselator studied in detail by the Prigogine school were nevertheless extremely speculative schemes. A study of the behaviour of classical chemical kinetics equations assumed a high priority in order to select the structure responsible for the appearance of critical effects. The results of such a study, described in Chap. 3, can be applied to interpret critical effect experiments. 

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Example 12.4 Stability of an autocatalytic reaction For a simple example, we may consider the following autocatalytic reaction, which appears in the reaction scheme of the Brusselator... 

Some autocatalytic chemical reactions such as the Brusselator and the Belousov-Zhabotinsky reaction schemes can produce temporal oscillations in a stirred homogeneous solution. In the presence of even a small initial concentration inhomogeneity, autocatalytic processes can couple with diffusion to produce organized systems in time and space. 

A well-known oscillating reaction scheme is the Brusselator system, representing a trimolecular model given by... 

The net reaction is A I B -> E I F. This reaction scheme has been developed by the Brussels School of Thermodynamics, and consists of a trimolecular collision and an autocatalytic step. This reaction may take place in a well-stirred medium leading to oscillations, or the diffusions of the components A and B may be considered. In the latter case, the system may produce Turing structures. 

Figure 13.6. Brusselator reaction scheme and order in time and space produced by Brussode demo of matlab.

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Much use has been, and continues to be, made of simplified model schemes representative of general classes of chemical or thermal feedback. The oregonator and Lengyel-Epstein models for the BZ and CDIMA systems have been given earlier. Pre-eminent among the more abstracted caricature models is the brusselator introduced by Prigogine and Lefever [47] which has the following form ... 

Building on similar ideas but starting from a more detailed reaction scheme, Tyson (1991) proposed a model for the mitotic oscillator based on the formation of a complex between cycUn and cdc2 kinase, followed by the activation of this complex. Essential to the oscillatory mechanism is the assumption that the active complex, i.e. MPF, promotes its own activation in a nonlinear memner. The kinetic equations, of a polynomial form, reduce under some simplifying assumptions to the equations of the two-variable Brusselator model. Inactivation of MPF is not... 

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The general treatment given above will now be illustrated by considering a simple two-variable chemical model. We shall examine the pattern formation that occurs in the Schnackenberg model, which is closely related to the Gray-Scott modeP and a member of the family of cubic autocatalysis models for chemical systems (a family that includes the Brusselator ). A detailed study of pattern formation in the Schnackenberg scheme has been carried out by Dufiet and Boissonade.3 ... 

Prigogine and Lefever introduced a simple two-variable scheme in 1968 [354, 243] that displays sustained oscillatory behavior. It was subsequently dubbed the Brusselator by Tyson [441] and consists of four steps ... 

The Brusselator provides a simple mass-action kinetic scheme that displays sustained, stable oscillatory behavior. 

The kinetic terms for DDRWs, see Sect. 5.6.2, depend on the details of the underlying mechanism of the reacting system, and it is impossible to write down a general form for them. We consider here the Brusselator model. In keeping with the spirit of the model, we consider two possible direction-dependent kinetic schemes, namely an activated U2 complex or an activated UV complex, to investigate the effect of activation energy on the Turing instability [206]. 

If the termolecular step in the Brusselator proceeds via an activated U2 complex, we have the following kinetic scheme for the Brusselator DDRW ... 

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