Model oregonator

The behaviour of the BZ system ean be modelled semi-quantitatively by the oregonator model [15] ... 

Oscillations are familiar phenomena in mechanical systems and in electric circuits. Noyes and Field discussed the possibilities for concentration oscillations in closed systems and illustrated the principles by means of Oregonator model consists of the following five steps ... 

In order to understand the BZ system Field, Koros and Noyes developed the so-called FKN mechanism. From this, Field and Noyes later derived the Oregonator model, an especially convenient kinetic model to match individual experimental observations and predict experimental conditions under which oscillations might arise. 

Figure 3-36. The BZ reaction as represented by the Oregonator model. The species Br, HBr02 and Mox display regular oscillations while the species BrO3, HOBr and MA change their concentrations slowly and more steadily.

The famous Oregonator model (ref. 9) is a highly simplified (but very successful) description of the Belousov - Zhabotinsky oscillating reaction ... 

Try to solve the Oregonator model using a non - stiff integrator as the module M70. Comment an the step size needed for a reasonable accuracy. 

Fig. 1. Partial (schematic) phase diagram for the reversible Oregonator model of the Belousov-Zha-botinski reaction.

FIGURE 7.18 The BZ reaction as represented by the Oregonator model. Calculated concentration profiles forHBr02 (—) and BrOf (...) toward the thermodynamic equilibrium. Note the different ordinates for [HBr02] and [BrOf],... 

K.Showalter, R.M.Noyes and K.Bar-Eli, A Modified Oregonator Model Exhibiting Complicated Limit Cycle Behavior In a Flow System, Journal of Chemical Physics, 69, 2514-2524(1978). 

IIIC) Sakanoue, S., Endo, M. The Existence of an Unstable Limit Cycle in the Oregonator Model... 

W. C. Troy, Mathematical analysis of the oregonator model of the Belousov-Zhabotinsky reaction, in Field and Burger (ref. Gl), Chapter 4. 

Oscillatory reactions provide one of the most active areas of research in contemporary chemical kinetics and two published studies on the photochemistry of Belousov-Zhabotinsky reaction are very significant in this respect. One deals with Ru(bpy)3 photocatalysed formation of spatial patterns and the other is an analysis of a modified complete Oregonator (model scheme) system which accounts for the O2 sensitivity and photosensitivity. ... 

The Oregonator model for the light-sensitive Belousov-Zhabotinsky reaction... 

The model that we take into account has been firstly proposed by H. J. Krug and coworkers in 1990 [30], to properly account for the photochemically-induced production of inhibitor bromide in the Belousov-Zhabotinsky reaction (BZ) catalyzed with the ruthenium complex Ru(bpy)3 [31, 32]. The Oregonator model was proposed in 1974 [16] on the basis Tyson-Fife reduction of the more complicated Field-Koros-Noyes mechanism [15] for the BZ reaction, the following modified model has been derived ... 

To validate numerically the above reported results, we perform calculations with the two-component Oregonator model introduced in subsection... 

Fig. 1.11. Normalized fluctuations of the inter-spike time versus the correlation time of the noise for the Oregonator model, Eqs. 1.3 with tpa = 0.007,5. Black curve A = 0.1, green A = 0.8, gray A = 0.6, blue A = 0.4. red A = 0,3. Bach point is an average over 5-10 luter-spike intervals. [7]...

Fig. 1.13. Coherence resonance in tlie Oregonator model (Eqs. 1.37) with respect to the correlation lengtlr for different values of the correlation time r gray curve r = 0.3. red curve T = 2, blue curve r = 5, yellow curve t = 20. Excitability j>arauieter — 0.01, system size L 45, noise intensity <7 = 0.25 [5].

We study in this subsection the effect of noise on pulses propagating in the two-component Oregonator model. Also here the excitability parameter (j) fluctuates, thus the noise enters multiplicatively in the system. The two-component Oregonator model supplied with local diffusive coupling reads... 

Fig. 1.14. Activator (black curve) and inhibitor (gray line) profile of typical pulse solution of the two-component Oregonator model (Eqs. 1.41) in the excitable kinetic regime (00 = 0.01) and with diffusion coefficient Du = 1. Calculations were performed in a one-dimensional domain of size L = 50 applying periodic boundary conditions. The propagation speed is c = 4.648. [6]...

As an example we state the two-component Oregonator model of the photosensitive BZ reaction with external forcing, see for example [84] ... 

W. Jahnke, W. Skaggs, and A. Winfree. Chemical vortex dynamics in the Belousov-Zhabotinskii reaction and in the two-variable Oregonator model. 

Basic features of the light-sensitive BZ reaction are reproduced by the Oregonator model given as a system of two coupled reaction-diffusion equations ... 

The term I = I t) describes the additional bromide production that is induced by external illumination [32]. Below all numerical simulations with the Oregonator model are performed using an explicit Euler method on a 380 X 380 array with a grid spacing Ax = 0.14 and time steps At = 0.002. 

It is important to stress that the above three descriptions of the spiral wave practically coincide far away from the rotation center. Moreover, already at a relatively small distance ta from the rotation center, the Archimedean spiral becomes very close to the curvature affected spiral obtained from Eq. (9.3), as can bee seen in Fig. 9.1(d). In this example ta can be estimated as ss 9.0 A. Recent computations performed with the Oregonator model [40] and experiments with the BZ reaction [43] also confirm that an Archimedean spiral provides a suitable approximation of the wave front except in a relatively small region of radius A near the rotation center. Even the shape of a slightly meandering spiral waves exhibits only small oscillations around an Archimedean shape, and the amplitude of these oscillations vanishes very quickly with r [44]. Therefore, the Archimedean spiral approximation will be used below to specify the shape of the wave front. 

This drift velocity field is shown in Fig. 9.2 corresponding to r = 0. The constant ip is taken as (/> == —1.8 in accordance with the chosen parameters for the Oregonator model (9.1). The field has rotational symmetry, but the drift angle 7 monotonously increases with the distance z from the detector point. Hence, there is a discrete set of sites along any radial direction, where the drift direction is orthogonal to the radial one i.e. 

Fig. 9.2. Drift velocity field under one-channel feedback for t = 0. The thick solid line shows the trajectory of the spiral center computed from the Oregonator model (9.1). ...

An example of feedback-induced drift computed for the Oregonator model... 

Fig. 9.5. Velocity field for spiral wave drift induced near a straight line detector (a) and a line segment (b). Solid lines show trajectories of the spiral center computed for the Oregonator model (9.1). r = 0.

To confirm these conclusions we have performed numerical simulations of the feedback mediated drift within the Oregonator model (9.1). The thick solid lines in Fig. 9.9 show the obtained trajectories of a spiral wave center for two different initial locations. The spiral center moves in very good agreement with the predicted drift velocity field and stops near the place where the velocity field vanishes. 

The constant (/ specifies the drift direction induced in the case r = 0 and 0 = 0. In the Oregonator model (9.1) one finds with the parameters indicated before ip = —0.5 [47]. Hence, the drift velocity field can be written as ... 

Fig. 9.10. Drift velocity field determined from Eqs.(9.43), (9.42), (9.48) for (a) dp/A = 0.45, (b) dp/X = 1.0. Thin solid lines represent lines of fixed points that satisfy Eq. (9.49) (compare text). Thick solid lines depict trajectories of the spiral center computed for the Oregonator model (9.1) with fc/ , = 0.02 and t = 0 [53].

For 0.5 < dp/X < 1.5, the drift velocity field changes dramatically, see Fig. 9.10(b). There are three spatially unbounded fixed lines that destroy the circular-shaped attractors existing as long as dp/ < 0.5. In numerical simulations with the Oregonator model (9.1), the spiral wave center follows an approximately circular trajectory until it stops practically at a fixed line in complete agreement with the predictions from the drift velocity field. 

It is very important to stress that changes in the geometrical shape of the integration domain can induce bifurcations in the drift velocity field [31, 47, 50, 52]. Let us consider, for example, the drift velocity field computed for an elliptical domain with major axis o = 3A and minor axis b = a/1.1. As shown in Fig. 9.13(a), instead of the stable limit cycle of the resonance attractor in the circular domain of radius Rg, = 1.5A we have two pairs of fixed points where the drift velocity vanishes. In each pair, one fixed point is a saddle and the other one is a stable node. Depending on the initial conditions, the spiral wave approaches one of the two stable nodes. Trajectories of the spiral center obtained by numerical integration of the Oregonator model (9.1) are in perfect agreement with the predicted drift... 

The experimental system employed in this paper is the light-sensitive Belousov-Zhabotinsky reaction. For numerical simulations only the underlying Oregonator model was used. However, our theoretical approach is based on a very general description of excitable media and does not rely on specific features of any experimental or model system. Therefore, our results are of general character, and can be applied to control spiral wave... 

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