Brusselator model

European Foundation for Quality Management. The Excellence Model. Brussels European Foundation for Quality Management. Available at http //www.efqm.org.html. Accessed 21 July 2001. 

Estimating environmental concentrations of chemicals using fate and exposure models. Brussels, European Centre for Ecotoxicology and Toxicology of Chemicals (Technical Report No. 50). 

Cramer, H. E., Improved techniques for modeling the dispersion of tall stack plumes. "Proceedings of the Seventh International Technical Meeting on Air Pollution Modeling and Its Application." North Atlantic Treaty Organization Committee on Challenges of Modern Society. Pub. No. 51. Brussels, 1976. (National Technical Information Service PB-270 799.)... 

Turner, D. B, Zimmerman, J. R., and Busse, A. D., An evaluation of some climatological dispersion models, in "Proceedings of the Third Meeting of the Expert Panel on Air Pollution Modeling." North Atlantic Treaty Organization Committee on the Challenges of Modem Society Pub. No. 14. Brussels, 1972. (National Technical Information Service PB 240-574.)... 

E. Skiiret, Advanced Design of Ventilation System, Ventilation Models Lecture Series, Brussels Von Karman Institute for Fluid Dvnamics, 1993. 

ECETOC, The Value of Model Ecosystem Studies in Ecotoxicology. Technical Report, Brussel, 1997. 

In order to propose a satisfactory solution for ED structure analysis, we adapted a newly designed electron diffractometry device to a commercial TEM microscope (model EM 400 Philips working at 120 kv) in the Lab of Electrochemistry, ULB, Brussels. 

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. 

Classical dynamics and orthodox quantum mechanics are constructed along the model of integrable systems in the sense of Poincare. Our aim is to construct dynamics for nonintegrable systems. As far as we know, this is a new attempt, which has its roots in the early work of the Brussels School [1-9]. The main result is that we have to replace the unitary transformation f/ by a nonunitary... 

In this simplified version of the Brusselator model, the trimolecular autocatalytic step, which is a necessary condition for the existence of instabilities, is, of course, retained. However, the linear source-sink reaction steps A—>X—>E are suppressed. A continuous flow of X inside the system may still be ensured through the values maintained at the boundaries. The price of this simplification is that (36) can never lead to a homogeneous time-periodic solution. The homogeneous steady states are... 

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. 

This is namely the main reason why fnmolecular models (e.g., Brusselator [16]) being often physically non-transparent but having only two intermediate stages have attracted such great attention in synergetic studies. 

Finally, the basic model could be also constructed ad hoc just to reproduce the kinetic phenomena observed experimentally in time and in space the well-known examples are the Brusselator or Prigogine-Lefever model (see [2]) and the model by Smoes [7]). Practically any basic model is oriented for a simplest and transparent description of a particular kind of the autowave processes. 

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). 

Reprint F is an example of analyzing a reaction in formal kinetics. Gray and Scott introduced the autocatalytic A + 2B = 3B as a simple model reaction that proved to have a rich behavior, much richer than the Brusselator for example. However, A + 2B smacks of a three-body interaction, which is a sufficiently rare occurrence as to be avoided. I had done a pseudo-steady-state analysis before I visited Leeds at Gray s invitation, and the chance of working with the fons et origo of this reaction, so to speak, was an opportunity to make sure that the limiting behavior was not lost when certain parameters were small, but not actually zero. For another analysis of autocatalytic behavior, see [107]. 

Gray, B. F. and Morley-Buchanan, T., 1985, Some criticism concerning the Brusselator model of an oscillating chemical reaction. J. Chem. Soc. Faraday Trans. 2 81, 77. 

This work is centred around the study of the response to periodic forcing of systems that, when autonomous, had a stable limit cycle surrounding an unstable steady state in their phase plane. For the sake of simplicity—and since many of the fundamental phenomena are the same—we studied two-dimensional systems. We chose two examples of isothermal reactor models the first is an autocatalytic homogeneous Brusselator (Glansdorff and Prigog-ine, 1971) ... 

FIGURE 7 Typical shapes of subharmonic trajectories. A subharmonic period 4 within the 4/3 resonance horn is a three-peaked oscillation in time (a), has three loops in its phase plane projection (b), and four loops in its x-cos 0 projection (c) (Brusselator, a = 0.0072, o = 4/3). The subharmonic period 4 within the 4/1 resonance horn has one loop in its phase plane projection (e), four loops in the x-cos projection (f) and is a one-peaked oscillation in time (d). Stroboscopic points are denoted by O. Try to imagine them winding around the doughnut in three-dimensional space An interesting shape shows up at the period 2 resonance in the 2/3 resonance horn (surface model >/aio = 2/3, alao = 0.1, o0 = 0.001) (g, h). These shapes are comparatively simple because of the shape of the unperturbed limit cycle which for all cases was a simple closed curve. 

In figure 2b, there are clearly folds in the left-hand side of the 3/2 and 2/1 resonance horns. This phenomenon had not (when we observed it) been seen in other forced oscillators such as the Brusselator model (Kai Tomita 1979) and the non-isothermal cstr (Kevrekidis et al. 1986), although it may have been missed in previous numerical studies that did not use arc-length continuation. It is however also to be found in unpublished work of Marek s group. The cusp points at M and L are quite different from the apparent cusp ... 

The model samples were synthesized and characterized in the Analytical Chemistry Dept of the Universite Libre de Bruxelles under the direction of Prof. C. Buess F.R. is grateful to P. Kons and E. Silberberg for the preparation of the samples. AES sputter profiles and factor analysis was performed at the Vrije Universiteit Brussel, Dept, of Metallurgy, Electrochemistry and Materials Science. Many thanks to Prof. Vereecken, Hubin and Terryn for the discussions concerning the results and to N. Roose and O. Steenhaut for the Auger sputter profiles. The technical collaboration of L. Binst (ULB) is greatly appreciated. 

To confirm the above conjectures we have performed a numerical simulation of equation (29) on the Brusselator model chemical reaction.46 The results are shown in Fig. 7. We start with an initial condition corresponding to a clockwise wave. Under the effect of the counterclockwise field this wave is deformed and eventually its sense of rotation is reversed. In other words, the system shows a clear-cut preference for one chirality. As a matter of fact we are witnessing an entrainment phenomenon of a new kind, whereby not only the frequency but also the sense of rotation of the system are adjusted to those of the external field. More complex situations, including chaotic behavior, are likely to arise when the resonance condition w = fl, is not satisfied, but we do not address ourselves to this problem here. 

Fig. 7. Numerical simulation of equation (29) for the Brusselator model on a ring. At t = 0 a clockwise wave propagates along the ring at t = 4.49 the effect of the counterclockwise external field deforms this wave appreciably after a while the sense of rotation is reversed and for t > 22.49 one obtains a stable wave solution in the counterclockwise direction. The period of both the external field and of the linearized solution of the unperturbed system is 4.28 time units.

Recently, a whole "zoo of models has been investigated. Its most known inhabitants, the "brussellator and "oregonator Ref. [2], contain the steps... 

Bartus R, Dean RL (1985). Developing and utilizing animal models in the search for an effective treatment for age-related memory disturbances. In Gottfries CG (ed) Normal Alzheimer s disease and senior dementia. Aspects of etiology, pathogenesis diagnosis and treatment. EVB, Brussels, pp 231-267... 

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